The document provides an overview of various techniques for describing and exploring data, including dot plots, stem-and-leaf displays, measures of position such as percentiles, box plots, skewness, scatterplots, and contingency tables. It defines each technique and provides examples of their construction and interpretation. Learning objectives cover how to construct and interpret each type of graph or statistical measure.
The document provides an overview of techniques for describing and exploring data, including dot plots, stem-and-leaf displays, measures of central tendency, box plots, coefficients of skewness, scatterplots, and contingency tables. It defines each technique and provides examples to illustrate how to construct and interpret the visualizations. The learning objectives cover how to use each technique to analyze and draw conclusions from sets of quantitative data.
The document discusses various methods for describing and exploring data, including dot plots, stem-and-leaf displays, percentiles, box plots, and skewness. It provides examples of each method using sample data sets and step-by-step calculations. Contingency tables are also introduced as a way to study relationships between nominal or ordinal variables.
This chapter discusses various methods for describing and exploring quantitative data, including dot plots, stem-and-leaf displays, percentiles, box plots, measures of skewness, scatter diagrams, and contingency tables. It provides examples and explanations of how to construct and interpret each method. Key goals are to develop an understanding of distributions and relationships within data sets.
This chapter discusses various methods for summarizing and exploring data, including dot plots, stem-and-leaf displays, percentiles, box plots, and scatter plots. Dot plots and stem-and-leaf displays organize data in a way that shows the distribution while maintaining each data point. Percentiles such as the median and quartiles divide data into equal portions. Box plots graphically show the center, spread, and outliers of data. Scatter plots reveal relationships between two variables, while contingency tables summarize categorical data relationships.
business statistics . In this chapter, you learn:
To construct and interpret confidence interval estimates for the population mean
To determine the sample size necessary to develop a confidence interval for the population mean
A point estimate is a single number,
a confidence interval provides additional information about the variability of the estimate
Suppose confidence level = 95%
Also written (1 - ) = 0.95, (so = 0.05)
A relative frequency interpretation:
95% of all the confidence intervals that can be constructed will contain the unknown true parameter
A specific interval either will contain or will not contain the true parameter
No probability involved in a specific interval. A sample of 11 circuits from a large normal population has a mean resistance of 2.20 ohms. We know from past testing that the population standard deviation is 0.35 ohms.
Determine a 95% confidence interval for the true mean resistance of the population.
A sample of 11 circuits from a large normal population has a mean resistance of 2.20 ohms. We know from past testing that the population standard deviation is 0.35 ohms.
We are 95% confident that the true mean resistance is between 1.9932 and 2.4068 ohms
Although the true mean may or may not be in this interval, 95% of intervals formed in this manner will contain the true mean
Interpreting this interval requires the assumption that the population you are sampling from is approximately a normal distribution (especially since n is only 25).
This condition can be checked by creating a:
Normal probability plot or
Boxplot
The required sample size can be found to reach a desired margin of error (e) with a specified level of confidence (1 - )
The margin of error is also called sampling error
the amount of imprecision in the estimate of the population parameter
the amount added and subtracted to the point estimate to form the confidence interval
Chap 04 - Describing Data_Displaying and Exploring Data.pdfSouravHasan8
This document provides an overview of various methods for displaying and exploring data, including frequency distributions, histograms, dot plots, stem-and-leaf displays, measures of position such as quartiles and percentiles, box plots, coefficients of skewness and kurtosis, scatterplots, and contingency tables. It discusses how to construct and interpret each method, provides examples, and outlines the key learning objectives for each topic. The document aims to help readers gain additional insight into the distribution and relationships within data through visual and statistical analysis techniques.
This document discusses various methods for summarizing and presenting data, including frequency tables, bar charts, pie charts, frequency distributions, histograms, and frequency polygons. It provides examples using vehicle sales profit data from Applewood Auto Group to demonstrate how to create these different data summaries and visualizations. The learning objectives cover how to make a frequency table, organize data into bar and pie charts, create a frequency distribution, understand relative frequency distributions, and present data from distributions in histograms and frequency polygons.
The document provides an overview of various techniques for describing and exploring data, including dot plots, stem-and-leaf displays, measures of position such as percentiles, box plots, skewness, scatterplots, and contingency tables. It defines each technique and provides examples of their construction and interpretation. Learning objectives cover how to construct and interpret each type of graph or statistical measure.
The document provides an overview of techniques for describing and exploring data, including dot plots, stem-and-leaf displays, measures of central tendency, box plots, coefficients of skewness, scatterplots, and contingency tables. It defines each technique and provides examples to illustrate how to construct and interpret the visualizations. The learning objectives cover how to use each technique to analyze and draw conclusions from sets of quantitative data.
The document discusses various methods for describing and exploring data, including dot plots, stem-and-leaf displays, percentiles, box plots, and skewness. It provides examples of each method using sample data sets and step-by-step calculations. Contingency tables are also introduced as a way to study relationships between nominal or ordinal variables.
This chapter discusses various methods for describing and exploring quantitative data, including dot plots, stem-and-leaf displays, percentiles, box plots, measures of skewness, scatter diagrams, and contingency tables. It provides examples and explanations of how to construct and interpret each method. Key goals are to develop an understanding of distributions and relationships within data sets.
This chapter discusses various methods for summarizing and exploring data, including dot plots, stem-and-leaf displays, percentiles, box plots, and scatter plots. Dot plots and stem-and-leaf displays organize data in a way that shows the distribution while maintaining each data point. Percentiles such as the median and quartiles divide data into equal portions. Box plots graphically show the center, spread, and outliers of data. Scatter plots reveal relationships between two variables, while contingency tables summarize categorical data relationships.
business statistics . In this chapter, you learn:
To construct and interpret confidence interval estimates for the population mean
To determine the sample size necessary to develop a confidence interval for the population mean
A point estimate is a single number,
a confidence interval provides additional information about the variability of the estimate
Suppose confidence level = 95%
Also written (1 - ) = 0.95, (so = 0.05)
A relative frequency interpretation:
95% of all the confidence intervals that can be constructed will contain the unknown true parameter
A specific interval either will contain or will not contain the true parameter
No probability involved in a specific interval. A sample of 11 circuits from a large normal population has a mean resistance of 2.20 ohms. We know from past testing that the population standard deviation is 0.35 ohms.
Determine a 95% confidence interval for the true mean resistance of the population.
A sample of 11 circuits from a large normal population has a mean resistance of 2.20 ohms. We know from past testing that the population standard deviation is 0.35 ohms.
We are 95% confident that the true mean resistance is between 1.9932 and 2.4068 ohms
Although the true mean may or may not be in this interval, 95% of intervals formed in this manner will contain the true mean
Interpreting this interval requires the assumption that the population you are sampling from is approximately a normal distribution (especially since n is only 25).
This condition can be checked by creating a:
Normal probability plot or
Boxplot
The required sample size can be found to reach a desired margin of error (e) with a specified level of confidence (1 - )
The margin of error is also called sampling error
the amount of imprecision in the estimate of the population parameter
the amount added and subtracted to the point estimate to form the confidence interval
Chap 04 - Describing Data_Displaying and Exploring Data.pdfSouravHasan8
This document provides an overview of various methods for displaying and exploring data, including frequency distributions, histograms, dot plots, stem-and-leaf displays, measures of position such as quartiles and percentiles, box plots, coefficients of skewness and kurtosis, scatterplots, and contingency tables. It discusses how to construct and interpret each method, provides examples, and outlines the key learning objectives for each topic. The document aims to help readers gain additional insight into the distribution and relationships within data through visual and statistical analysis techniques.
This document discusses various methods for summarizing and presenting data, including frequency tables, bar charts, pie charts, frequency distributions, histograms, and frequency polygons. It provides examples using vehicle sales profit data from Applewood Auto Group to demonstrate how to create these different data summaries and visualizations. The learning objectives cover how to make a frequency table, organize data into bar and pie charts, create a frequency distribution, understand relative frequency distributions, and present data from distributions in histograms and frequency polygons.
This document provides information on different types of graphs and displays that can be used to represent quantitative and qualitative data, including stem-and-leaf plots, dot plots, pie charts, Pareto charts, scatter plots, and time series charts. Examples are given for how to construct each type of graph or display using sample data sets. Key aspects like labeling axes, plotting data points, and interpreting trends are discussed.
This document discusses various methods for describing and exploring data, including dot plots, box plots, and measures of position such as percentiles and quartiles. Dot plots display each observation along a number line to show the distribution of values. Box plots use the minimum, quartiles, median and maximum to graphically depict a dataset. Percentiles and quartiles split a sorted dataset into equal groups to indicate location within the data. The interquartile range and coefficients of skewness are also introduced to analyze the shape, spread and outliers of distributions. Examples are provided to demonstrate computing and interpreting these descriptive statistics.
In our increasingly data-driven world, it's more important than ever to have accessible ways to view and understand data.
After all, employees' demand for data skills steadily increases each year.
Employees and Business owners at every level need to understand data and its impact.
That's where Data Visualization comes in handy.
To make Data more accessible and understandable, Data Visualization in Dashboards is the go-to tool for many businesses to Analyze and share Information.
This chapter discusses various methods for describing and exploring data, including dot plots, percentiles, box plots, and scatter diagrams. Dot plots display each data point along a number line and are useful for small data sets. Percentiles divide a data set into equal percentages and are used to calculate quartiles. Box plots graphically depict the center, spread, and outliers of a data set. Scatter diagrams show the relationship between two variables by plotting one on the x-axis and one on the y-axis. Contingency tables organize counts of observations into categories to study relationships between nominal or ordinal variables.
Business statistics takes the data analysis tools from elementary statistics and applies them to business. For example, estimating the probability of a defect coming off a factory line, or seeing where sales are headed in the future. Many of the tools used in business statistics are built on ones you’ve probably already come across in basic math: mean, mode and median, bar graphs and the bell curve, and basic probability. Hypothesis testing (where you test out an idea) and regression analysis (fitting data to an equation) builds on this foundation.
This document describes chapter 4 of a statistics textbook. It provides 7 learning objectives related to describing and exploring data through various statistical techniques. These include constructing and interpreting dot plots, stem-and-leaf displays, box plots, and scatterplots. It also covers measures of central tendency, percentiles, skewness, and contingency tables. Examples are provided for each statistical technique to help illustrate how to compute and interpret the various descriptive statistics.
This document discusses methods for describing and exploring data including dot plots, stem-and-leaf displays, measures of position like percentiles and quartiles, and box plots. It provides examples of each method and how to interpret the results. Key learning objectives covered are constructing and interpreting dot plots, stem-and-leaf displays, computing measures of position like percentiles and quartiles, and developing and analyzing box plots from given data.
This document provides information about data interpretation and different ways to present data. It discusses numerical data tables including time series tables, spatial tables, frequency distribution tables, and cumulative frequency tables. Examples are given to show how to calculate capacity utilization, sales growth percentages, and solve other problems using the data in tables. Cartesian graphs are also introduced as a way to show the variation of a quantity with respect to two parameters on the X and Y axes.
This document provides information about data interpretation and different ways of presenting data. It discusses numerical data tables like time series tables, spatial tables, frequency distribution tables, and cumulative frequency tables. It then provides examples of data interpretation questions involving calculating percentages from tables showing production, sales, and capacity for three automobile companies from 1988-1998. Another example involves interpreting a table showing the number of members in different age and gender categories across six societies. Finally, it briefly discusses Cartesian graphs and types like single dependent variable graphs and graphs with more than one dependent variable.
This document discusses methods for organizing and presenting qualitative and quantitative data using frequency tables, charts, and graphs. It covers:
1. Creating frequency tables to organize qualitative and quantitative data, and presenting qualitative data as bar charts or pie charts.
2. Constructing frequency distributions to organize quantitative data into class intervals and determining class frequencies, and presenting quantitative data using histograms, frequency polygons, and cumulative frequency polygons.
3. An example of creating a frequency table and histogram based on sales price data from 80 vehicles to compare typical selling prices on dealer lots.
TSTD 6251 Fall 2014SPSS Exercise and Assignment 120 PointsI.docxnanamonkton
TSTD 6251 Fall 2014
SPSS Exercise and Assignment 1
20 Points
In this class, we are going to study descriptive summary statistics and learn how to construct box plot. We are still working with univariate variable for this exercise.
Practice Example:
Admission receipts (in million of dollars) for a recent season are given below for the
n =
30 major league baseball teams:
19.4 26.6 22.9 44.5 24.4 19.0 27.5 19.9 22.8 19.0 16.9 15.2 25.7 19.0 15.5 17.1 15.6 10.6 16.2 15.6 15.4 18.2 15.5 14.2 9.5 9.9
10.7 11.9 26.7 17.5
Require:
a. Compute the mean, variance and standard deviation.
b. Find the sample median, first quartile, and third quartile.
c. Construct a boxplot and interpret the distribution of the data.
d. Discuss the distribution of this set of data by examining kurtosis and skewness
statistics, such as if the distribution is skewed to one side of the distribution, and if the
distribution shows a peaked/skinny curve or a spread out/flat curve.
SPSS Procedures for Computing Summary Statistics
:
Enter the 30 data values in the first column of SPSS
Data View
Tab
Variable View
and name this variable
receipts
Adjust
Decimals
to 3 decimal points
Type
Admission Receipts
($ mn)
in the
Label
column for output viewer
Return to
Data View
and click
A
nalyze
on the menu bar
Click the second menu
D
e
scriptive Statistics
Click
F
requencies …
Move
Admission Receipts
to the
Variable(s)
list by clicking the arrow button
Click
S
tatistics …
button at the top of the dialog box
Now, you can select the descriptive statistics according to what the question requires. For this practice question, it requires central tendency, dispersion, percentile and distribution statistics, so we click all the boxes
except for
P
ercentile(s): and Va
l
ues are group midpoints
.
Click
Continue
to return to the
Frequencies
dialog box
Click
OK
to generate descriptive statistic output which is pasted below:
The first table provides summary statistics and the second table lists frequencies, relative frequencies and cumulative frequencies. The statistics required for solving this problem are highlighted in red.
Statistics
Admission Receipts
N
Valid
30
Missing
0
Mean
18.76333
Std. Error of Mean
1.278590
Median
17.30000
Mode
19.000
Std. Deviation
7.003127
Variance
49.043782
Skewness
1.734
Std. Error of Skewness
.427
Kurtosis
5.160
Std. Error of Kurtosis
.833
Range
35.000
Minimum
9.500
Maximum
44.500
Sum
562.900
Percentiles
10
10.61000
20
14.40000
25
15.35000
30
15.50000
40
15.84000
50
17.30000
60
19.00000
70
19.75000
75
22.82500
80
24.10000
90
26.69000
Admission Receipts
Frequency
Percent
Valid Percent
Cumulative Percent
Valid
9.500
1
3.3
3.3
3.3
9.900
1
3.3
3.3
6.7
10.600
1
3.3
3.3
10.0
10.700
1
3.3
3.3
13.3
11.900
1
3.3
3.3
16.7
14.200
1
3.3
3.3
20.0
15.2.
ByPREFERENCES FOR CAR CHOICE IN UNITED STATES.docxclairbycraft
By
PREFERENCES FOR CAR CHOICE
IN UNITED STATES
Thank you
PREFERENCES FOR CAR CHOICE IN THE UNITED STATES 2
PREFERENCES FOR CAR CHOICE IN THE UNITED STATES 2
Table of Contents
Introduction………………………………………………………………………………………..3
Background3
Data Analysis4
Data Visualization9
Conclusion16
References17
Introduction
The most common applications of Statistics is describing a set of descriptive data statistics, regression, and hypothesis testing and inferential statistics. The two main branches are descriptive and inferential statistics. People who do not have any formal training in statistics are more familiar with inferential statistics than with descriptive statistics. In this paper, the data will analyze using descriptive statistics. So we will focus on the descriptive branch of the statistics.
Descriptive Statistics Definition
The descriptive statistics are the type of statistical analysis that helps to describe the data in some meaningful way. The statistics are helpful to describe quantitatively about the essential features of the data or information. The descriptive statistics give the summaries of the given sample as well as the observations done. These summaries or descriptions can either be graphical or quantitative.Background
This study will focus on and analyzing & Visualizing the data set about Preferences For Car Choice In The United States. The data set contained 4654 observations and 71 columns. There are several different types of graphs that help describe the statistical data. These graphs are histogram, bar graph, box and whisker plot, line graph, scatter plot, ogive, pie chart, and many more. Generally, the kinds of measurements that can use with descriptive statistics are:
The measure of central tendency describes the data which lies in the center of a given frequency distribution. The main steps of central tendency are mean and median and mode (Nick, 2020).
The spread measure describes how the scores are spread across the entire distribution. In the spread, measurements that are included standard deviation, variance, quartiles, range, absolute difference.Data Analysis
One of the essential concepts of statistics is data analysis. It is the process that is observing the data, analyzing, and modeling the data. The purpose of data analysis is to obtain useful data information and state conclusions which support decision-making. The data analysis can be performed under several techniques using different approaches. The method of data assessment and analysis can be achieved by using analytical and logical approaches to examine each component of the data provided. Data from various sources are collected, reviewed, and then explained for decision making or conclusions. There are several methods for analyzing the results. Data mining, text analytics, and business intelligence are some of the most commonly used techniques and data visualizations.
The data an.
By
PREFERENCES FOR CAR CHOICE
IN UNITED STATES
Thank you
PREFERENCES FOR CAR CHOICE IN THE UNITED STATES 2
PREFERENCES FOR CAR CHOICE IN THE UNITED STATES 2
Table of Contents
Introduction………………………………………………………………………………………..3
Background3
Data Analysis4
Data Visualization9
Conclusion16
References17
Introduction
The most common applications of Statistics is describing a set of descriptive data statistics, regression, and hypothesis testing and inferential statistics. The two main branches are descriptive and inferential statistics. People who do not have any formal training in statistics are more familiar with inferential statistics than with descriptive statistics. In this paper, the data will analyze using descriptive statistics. So we will focus on the descriptive branch of the statistics.
Descriptive Statistics Definition
The descriptive statistics are the type of statistical analysis that helps to describe the data in some meaningful way. The statistics are helpful to describe quantitatively about the essential features of the data or information. The descriptive statistics give the summaries of the given sample as well as the observations done. These summaries or descriptions can either be graphical or quantitative.Background
This study will focus on and analyzing & Visualizing the data set about Preferences For Car Choice In The United States. The data set contained 4654 observations and 71 columns. There are several different types of graphs that help describe the statistical data. These graphs are histogram, bar graph, box and whisker plot, line graph, scatter plot, ogive, pie chart, and many more. Generally, the kinds of measurements that can use with descriptive statistics are:
The measure of central tendency describes the data which lies in the center of a given frequency distribution. The main steps of central tendency are mean and median and mode (Nick, 2020).
The spread measure describes how the scores are spread across the entire distribution. In the spread, measurements that are included standard deviation, variance, quartiles, range, absolute difference.Data Analysis
One of the essential concepts of statistics is data analysis. It is the process that is observing the data, analyzing, and modeling the data. The purpose of data analysis is to obtain useful data information and state conclusions which support decision-making. The data analysis can be performed under several techniques using different approaches. The method of data assessment and analysis can be achieved by using analytical and logical approaches to examine each component of the data provided. Data from various sources are collected, reviewed, and then explained for decision making or conclusions. There are several methods for analyzing the results. Data mining, text analytics, and business intelligence are some of the most commonly used techniques and data visualizations.
The data an.
B409 W11 Sas Collaborative Stats Guide V4.2marshalkalra
This document provides an overview of numerical summaries and variation within data. It defines key terms like mean, median, mode, range, standard deviation, and variance. It also discusses sources of variation within data like process inputs and conditions versus random temporary events. The document demonstrates how to use SAS software to analyze a cars dataset and create reports and bar charts to describe the data and identify trends and variation.
This document provides an overview of key concepts in statistics including:
- Descriptive statistics such as frequency distributions which organize and summarize data
- Inferential statistics which make estimates or predictions about populations based on samples
- Types of variables including quantitative, qualitative, discrete and continuous
- Levels of measurement including nominal, ordinal, interval and ratio
- Common measures of central tendency (mean, median, mode) and dispersion (range, standard deviation)
This document provides an introduction to descriptive statistics. It discusses organizing and presenting both qualitative and quantitative data. For qualitative data, it describes frequency distribution tables, relative frequencies, percentages, and graphs like bar charts and pie charts. For quantitative data, it covers stem-and-leaf displays, frequency distributions, class widths and midpoints, relative frequencies and percentages. It also discusses histograms for presenting grouped quantitative data. Examples are provided to illustrate these concepts and techniques.
It is the type of data defined in Statistics & it can also used in the process of knowledge discovery or pattern searching such as data mining, web data mining which is important for the purpose of decision making. The presentation focus on the type of data known as four level of measurement in Statistics.
Outline
1.What is Statistics ?
2.Type of Statistics
3.Type of Sampling
4.Four Level of Measurement
5.Describing Data: Frequency Distributions and Graphic Presentation
Simple Random Sample
all members of the population has the same chance of being selected for a sample.
Systematic Sample
A random starting point is selected, then every k item is selected for the sample.
Stratified Sample
Population is divided into several groups or strata and then a sample is selected from each stratum.
Cluster Sample
Primary units and then samples are drawn from the primary unit.
The document provides an introduction to QQ plots and explains how they can be used to check if collected data is normally distributed. It defines what a quantile is, explaining that a QQ plot, or quantile-quantile plot, plots the quantiles of the collected data against the quantiles of a normal distribution to see if the data follows a normal pattern. The document also includes a brief example QQ plot and explains how to interpret it to assess normality.
The document discusses different types of statistical data and analysis methods. It covers descriptive and inferential statistics, different levels of measurement for variables, sampling methods, and approaches for organizing and presenting data through frequency distributions and graphs. Specific topics include nominal, ordinal, interval and ratio levels of measurement, constructing frequency distributions, calculating class intervals, and using histograms and frequency polygons to portray the distribution of vehicle selling prices at a car dealership.
Major Benefits and Drivers of IoT.Background According to T.docxjesssueann
Major Benefits and Drivers of IoT.
Background: According to Turban (2015),The major objective of IoT systems is to improve productivity, quality, speed, and the quality of life. There are potentially several major benefits from IoT, especially when combined with Artificial Intelligence (AI).
Reference: Sharda, R., Delen, Dursun, and Turban, E. (2020). Analytics, Data Science, & Artificial Intelligence: Systems for Decision Support. 11th Edition. By PEARSON Education. Inc.
ISBN-13: 978-0-13-519201-6
Assignment/Research: Go to pages 694 to 695 of your recommended textbook and familiarize yourself with the contents therein. Go ahead and make a list of the major benefits and drivers of IoT, thereafter pick two from each list and discuss them briefly.
Your research paper should be at least three pages (800 words), double-spaced, have at least 4 APA references, and typed in an easy-to-read font in MS Word
.
Major Assessment 2 The Educated Person” For educators to be ef.docxjesssueann
Major Assessment 2: The “Educated Person” For educators to be effective in supporting diverse learners, they need to develop, possess, and continually refine their vision of the “educated person.” In other words, they need to have a vision of their goals and outcomes for educating students. Prepare a statement of your image of and beliefs and values about the educated person. Explain your beliefs about the role of the teacher in valuing and encouraging others to value the image of an educated person. Be certain to address the roles of cultural diversity in achieving a viable vision of the educated person. Begin by reading the key documents discussed in the chapters in this section. Reference at least five additional current professional references to illustrate your position. Organize your presentation by sections and use American Psychological Association (APA) style for citing references in the body of the text and for developing your reference list. Include the following sections in your paper:
1. Introduction
2. Vision of learning and the educated person (critical knowledge, skills, dispositions)
3. Role of the teacher in providing an effective instructional program and applying best practices to student learning
4. Critical issues in promoting the success of all students and responding to diverse community needs
5. Capacity to translate the image of the educated person into educational aims and organizational goals and processes
6. Conclusion
7. References
.
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This document provides information on different types of graphs and displays that can be used to represent quantitative and qualitative data, including stem-and-leaf plots, dot plots, pie charts, Pareto charts, scatter plots, and time series charts. Examples are given for how to construct each type of graph or display using sample data sets. Key aspects like labeling axes, plotting data points, and interpreting trends are discussed.
This document discusses various methods for describing and exploring data, including dot plots, box plots, and measures of position such as percentiles and quartiles. Dot plots display each observation along a number line to show the distribution of values. Box plots use the minimum, quartiles, median and maximum to graphically depict a dataset. Percentiles and quartiles split a sorted dataset into equal groups to indicate location within the data. The interquartile range and coefficients of skewness are also introduced to analyze the shape, spread and outliers of distributions. Examples are provided to demonstrate computing and interpreting these descriptive statistics.
In our increasingly data-driven world, it's more important than ever to have accessible ways to view and understand data.
After all, employees' demand for data skills steadily increases each year.
Employees and Business owners at every level need to understand data and its impact.
That's where Data Visualization comes in handy.
To make Data more accessible and understandable, Data Visualization in Dashboards is the go-to tool for many businesses to Analyze and share Information.
This chapter discusses various methods for describing and exploring data, including dot plots, percentiles, box plots, and scatter diagrams. Dot plots display each data point along a number line and are useful for small data sets. Percentiles divide a data set into equal percentages and are used to calculate quartiles. Box plots graphically depict the center, spread, and outliers of a data set. Scatter diagrams show the relationship between two variables by plotting one on the x-axis and one on the y-axis. Contingency tables organize counts of observations into categories to study relationships between nominal or ordinal variables.
Business statistics takes the data analysis tools from elementary statistics and applies them to business. For example, estimating the probability of a defect coming off a factory line, or seeing where sales are headed in the future. Many of the tools used in business statistics are built on ones you’ve probably already come across in basic math: mean, mode and median, bar graphs and the bell curve, and basic probability. Hypothesis testing (where you test out an idea) and regression analysis (fitting data to an equation) builds on this foundation.
This document describes chapter 4 of a statistics textbook. It provides 7 learning objectives related to describing and exploring data through various statistical techniques. These include constructing and interpreting dot plots, stem-and-leaf displays, box plots, and scatterplots. It also covers measures of central tendency, percentiles, skewness, and contingency tables. Examples are provided for each statistical technique to help illustrate how to compute and interpret the various descriptive statistics.
This document discusses methods for describing and exploring data including dot plots, stem-and-leaf displays, measures of position like percentiles and quartiles, and box plots. It provides examples of each method and how to interpret the results. Key learning objectives covered are constructing and interpreting dot plots, stem-and-leaf displays, computing measures of position like percentiles and quartiles, and developing and analyzing box plots from given data.
This document provides information about data interpretation and different ways to present data. It discusses numerical data tables including time series tables, spatial tables, frequency distribution tables, and cumulative frequency tables. Examples are given to show how to calculate capacity utilization, sales growth percentages, and solve other problems using the data in tables. Cartesian graphs are also introduced as a way to show the variation of a quantity with respect to two parameters on the X and Y axes.
This document provides information about data interpretation and different ways of presenting data. It discusses numerical data tables like time series tables, spatial tables, frequency distribution tables, and cumulative frequency tables. It then provides examples of data interpretation questions involving calculating percentages from tables showing production, sales, and capacity for three automobile companies from 1988-1998. Another example involves interpreting a table showing the number of members in different age and gender categories across six societies. Finally, it briefly discusses Cartesian graphs and types like single dependent variable graphs and graphs with more than one dependent variable.
This document discusses methods for organizing and presenting qualitative and quantitative data using frequency tables, charts, and graphs. It covers:
1. Creating frequency tables to organize qualitative and quantitative data, and presenting qualitative data as bar charts or pie charts.
2. Constructing frequency distributions to organize quantitative data into class intervals and determining class frequencies, and presenting quantitative data using histograms, frequency polygons, and cumulative frequency polygons.
3. An example of creating a frequency table and histogram based on sales price data from 80 vehicles to compare typical selling prices on dealer lots.
TSTD 6251 Fall 2014SPSS Exercise and Assignment 120 PointsI.docxnanamonkton
TSTD 6251 Fall 2014
SPSS Exercise and Assignment 1
20 Points
In this class, we are going to study descriptive summary statistics and learn how to construct box plot. We are still working with univariate variable for this exercise.
Practice Example:
Admission receipts (in million of dollars) for a recent season are given below for the
n =
30 major league baseball teams:
19.4 26.6 22.9 44.5 24.4 19.0 27.5 19.9 22.8 19.0 16.9 15.2 25.7 19.0 15.5 17.1 15.6 10.6 16.2 15.6 15.4 18.2 15.5 14.2 9.5 9.9
10.7 11.9 26.7 17.5
Require:
a. Compute the mean, variance and standard deviation.
b. Find the sample median, first quartile, and third quartile.
c. Construct a boxplot and interpret the distribution of the data.
d. Discuss the distribution of this set of data by examining kurtosis and skewness
statistics, such as if the distribution is skewed to one side of the distribution, and if the
distribution shows a peaked/skinny curve or a spread out/flat curve.
SPSS Procedures for Computing Summary Statistics
:
Enter the 30 data values in the first column of SPSS
Data View
Tab
Variable View
and name this variable
receipts
Adjust
Decimals
to 3 decimal points
Type
Admission Receipts
($ mn)
in the
Label
column for output viewer
Return to
Data View
and click
A
nalyze
on the menu bar
Click the second menu
D
e
scriptive Statistics
Click
F
requencies …
Move
Admission Receipts
to the
Variable(s)
list by clicking the arrow button
Click
S
tatistics …
button at the top of the dialog box
Now, you can select the descriptive statistics according to what the question requires. For this practice question, it requires central tendency, dispersion, percentile and distribution statistics, so we click all the boxes
except for
P
ercentile(s): and Va
l
ues are group midpoints
.
Click
Continue
to return to the
Frequencies
dialog box
Click
OK
to generate descriptive statistic output which is pasted below:
The first table provides summary statistics and the second table lists frequencies, relative frequencies and cumulative frequencies. The statistics required for solving this problem are highlighted in red.
Statistics
Admission Receipts
N
Valid
30
Missing
0
Mean
18.76333
Std. Error of Mean
1.278590
Median
17.30000
Mode
19.000
Std. Deviation
7.003127
Variance
49.043782
Skewness
1.734
Std. Error of Skewness
.427
Kurtosis
5.160
Std. Error of Kurtosis
.833
Range
35.000
Minimum
9.500
Maximum
44.500
Sum
562.900
Percentiles
10
10.61000
20
14.40000
25
15.35000
30
15.50000
40
15.84000
50
17.30000
60
19.00000
70
19.75000
75
22.82500
80
24.10000
90
26.69000
Admission Receipts
Frequency
Percent
Valid Percent
Cumulative Percent
Valid
9.500
1
3.3
3.3
3.3
9.900
1
3.3
3.3
6.7
10.600
1
3.3
3.3
10.0
10.700
1
3.3
3.3
13.3
11.900
1
3.3
3.3
16.7
14.200
1
3.3
3.3
20.0
15.2.
ByPREFERENCES FOR CAR CHOICE IN UNITED STATES.docxclairbycraft
By
PREFERENCES FOR CAR CHOICE
IN UNITED STATES
Thank you
PREFERENCES FOR CAR CHOICE IN THE UNITED STATES 2
PREFERENCES FOR CAR CHOICE IN THE UNITED STATES 2
Table of Contents
Introduction………………………………………………………………………………………..3
Background3
Data Analysis4
Data Visualization9
Conclusion16
References17
Introduction
The most common applications of Statistics is describing a set of descriptive data statistics, regression, and hypothesis testing and inferential statistics. The two main branches are descriptive and inferential statistics. People who do not have any formal training in statistics are more familiar with inferential statistics than with descriptive statistics. In this paper, the data will analyze using descriptive statistics. So we will focus on the descriptive branch of the statistics.
Descriptive Statistics Definition
The descriptive statistics are the type of statistical analysis that helps to describe the data in some meaningful way. The statistics are helpful to describe quantitatively about the essential features of the data or information. The descriptive statistics give the summaries of the given sample as well as the observations done. These summaries or descriptions can either be graphical or quantitative.Background
This study will focus on and analyzing & Visualizing the data set about Preferences For Car Choice In The United States. The data set contained 4654 observations and 71 columns. There are several different types of graphs that help describe the statistical data. These graphs are histogram, bar graph, box and whisker plot, line graph, scatter plot, ogive, pie chart, and many more. Generally, the kinds of measurements that can use with descriptive statistics are:
The measure of central tendency describes the data which lies in the center of a given frequency distribution. The main steps of central tendency are mean and median and mode (Nick, 2020).
The spread measure describes how the scores are spread across the entire distribution. In the spread, measurements that are included standard deviation, variance, quartiles, range, absolute difference.Data Analysis
One of the essential concepts of statistics is data analysis. It is the process that is observing the data, analyzing, and modeling the data. The purpose of data analysis is to obtain useful data information and state conclusions which support decision-making. The data analysis can be performed under several techniques using different approaches. The method of data assessment and analysis can be achieved by using analytical and logical approaches to examine each component of the data provided. Data from various sources are collected, reviewed, and then explained for decision making or conclusions. There are several methods for analyzing the results. Data mining, text analytics, and business intelligence are some of the most commonly used techniques and data visualizations.
The data an.
By
PREFERENCES FOR CAR CHOICE
IN UNITED STATES
Thank you
PREFERENCES FOR CAR CHOICE IN THE UNITED STATES 2
PREFERENCES FOR CAR CHOICE IN THE UNITED STATES 2
Table of Contents
Introduction………………………………………………………………………………………..3
Background3
Data Analysis4
Data Visualization9
Conclusion16
References17
Introduction
The most common applications of Statistics is describing a set of descriptive data statistics, regression, and hypothesis testing and inferential statistics. The two main branches are descriptive and inferential statistics. People who do not have any formal training in statistics are more familiar with inferential statistics than with descriptive statistics. In this paper, the data will analyze using descriptive statistics. So we will focus on the descriptive branch of the statistics.
Descriptive Statistics Definition
The descriptive statistics are the type of statistical analysis that helps to describe the data in some meaningful way. The statistics are helpful to describe quantitatively about the essential features of the data or information. The descriptive statistics give the summaries of the given sample as well as the observations done. These summaries or descriptions can either be graphical or quantitative.Background
This study will focus on and analyzing & Visualizing the data set about Preferences For Car Choice In The United States. The data set contained 4654 observations and 71 columns. There are several different types of graphs that help describe the statistical data. These graphs are histogram, bar graph, box and whisker plot, line graph, scatter plot, ogive, pie chart, and many more. Generally, the kinds of measurements that can use with descriptive statistics are:
The measure of central tendency describes the data which lies in the center of a given frequency distribution. The main steps of central tendency are mean and median and mode (Nick, 2020).
The spread measure describes how the scores are spread across the entire distribution. In the spread, measurements that are included standard deviation, variance, quartiles, range, absolute difference.Data Analysis
One of the essential concepts of statistics is data analysis. It is the process that is observing the data, analyzing, and modeling the data. The purpose of data analysis is to obtain useful data information and state conclusions which support decision-making. The data analysis can be performed under several techniques using different approaches. The method of data assessment and analysis can be achieved by using analytical and logical approaches to examine each component of the data provided. Data from various sources are collected, reviewed, and then explained for decision making or conclusions. There are several methods for analyzing the results. Data mining, text analytics, and business intelligence are some of the most commonly used techniques and data visualizations.
The data an.
B409 W11 Sas Collaborative Stats Guide V4.2marshalkalra
This document provides an overview of numerical summaries and variation within data. It defines key terms like mean, median, mode, range, standard deviation, and variance. It also discusses sources of variation within data like process inputs and conditions versus random temporary events. The document demonstrates how to use SAS software to analyze a cars dataset and create reports and bar charts to describe the data and identify trends and variation.
This document provides an overview of key concepts in statistics including:
- Descriptive statistics such as frequency distributions which organize and summarize data
- Inferential statistics which make estimates or predictions about populations based on samples
- Types of variables including quantitative, qualitative, discrete and continuous
- Levels of measurement including nominal, ordinal, interval and ratio
- Common measures of central tendency (mean, median, mode) and dispersion (range, standard deviation)
This document provides an introduction to descriptive statistics. It discusses organizing and presenting both qualitative and quantitative data. For qualitative data, it describes frequency distribution tables, relative frequencies, percentages, and graphs like bar charts and pie charts. For quantitative data, it covers stem-and-leaf displays, frequency distributions, class widths and midpoints, relative frequencies and percentages. It also discusses histograms for presenting grouped quantitative data. Examples are provided to illustrate these concepts and techniques.
It is the type of data defined in Statistics & it can also used in the process of knowledge discovery or pattern searching such as data mining, web data mining which is important for the purpose of decision making. The presentation focus on the type of data known as four level of measurement in Statistics.
Outline
1.What is Statistics ?
2.Type of Statistics
3.Type of Sampling
4.Four Level of Measurement
5.Describing Data: Frequency Distributions and Graphic Presentation
Simple Random Sample
all members of the population has the same chance of being selected for a sample.
Systematic Sample
A random starting point is selected, then every k item is selected for the sample.
Stratified Sample
Population is divided into several groups or strata and then a sample is selected from each stratum.
Cluster Sample
Primary units and then samples are drawn from the primary unit.
The document provides an introduction to QQ plots and explains how they can be used to check if collected data is normally distributed. It defines what a quantile is, explaining that a QQ plot, or quantile-quantile plot, plots the quantiles of the collected data against the quantiles of a normal distribution to see if the data follows a normal pattern. The document also includes a brief example QQ plot and explains how to interpret it to assess normality.
The document discusses different types of statistical data and analysis methods. It covers descriptive and inferential statistics, different levels of measurement for variables, sampling methods, and approaches for organizing and presenting data through frequency distributions and graphs. Specific topics include nominal, ordinal, interval and ratio levels of measurement, constructing frequency distributions, calculating class intervals, and using histograms and frequency polygons to portray the distribution of vehicle selling prices at a car dealership.
Similar to LEARNING OBJECTIVESWhen you have completed this chapter, you.docx (20)
Major Benefits and Drivers of IoT.Background According to T.docxjesssueann
Major Benefits and Drivers of IoT.
Background: According to Turban (2015),The major objective of IoT systems is to improve productivity, quality, speed, and the quality of life. There are potentially several major benefits from IoT, especially when combined with Artificial Intelligence (AI).
Reference: Sharda, R., Delen, Dursun, and Turban, E. (2020). Analytics, Data Science, & Artificial Intelligence: Systems for Decision Support. 11th Edition. By PEARSON Education. Inc.
ISBN-13: 978-0-13-519201-6
Assignment/Research: Go to pages 694 to 695 of your recommended textbook and familiarize yourself with the contents therein. Go ahead and make a list of the major benefits and drivers of IoT, thereafter pick two from each list and discuss them briefly.
Your research paper should be at least three pages (800 words), double-spaced, have at least 4 APA references, and typed in an easy-to-read font in MS Word
.
Major Assessment 2 The Educated Person” For educators to be ef.docxjesssueann
Major Assessment 2: The “Educated Person” For educators to be effective in supporting diverse learners, they need to develop, possess, and continually refine their vision of the “educated person.” In other words, they need to have a vision of their goals and outcomes for educating students. Prepare a statement of your image of and beliefs and values about the educated person. Explain your beliefs about the role of the teacher in valuing and encouraging others to value the image of an educated person. Be certain to address the roles of cultural diversity in achieving a viable vision of the educated person. Begin by reading the key documents discussed in the chapters in this section. Reference at least five additional current professional references to illustrate your position. Organize your presentation by sections and use American Psychological Association (APA) style for citing references in the body of the text and for developing your reference list. Include the following sections in your paper:
1. Introduction
2. Vision of learning and the educated person (critical knowledge, skills, dispositions)
3. Role of the teacher in providing an effective instructional program and applying best practices to student learning
4. Critical issues in promoting the success of all students and responding to diverse community needs
5. Capacity to translate the image of the educated person into educational aims and organizational goals and processes
6. Conclusion
7. References
.
Major Assessment 4 Cultural Bias Investigation Most educators agree.docxjesssueann
Major Assessment 4: Cultural Bias Investigation Most educators agree that major influences on the achievement of students are the activities and support materials; environment; and types of expectations, interactions, and behaviors to which they are exposed. Therefore, an understanding of bias and skill in discerning subtle and/or overt bias in curriculum, instruction, and assessment are extremely important. Conduct a cultural bias investigation to examine a particular textbook with which you are familiar. Your investigation will focus on identifying instructional and assessment practices that reflect cultural bias and inhibit learning. The investigation will include reflection on the impact of these practices on student learning. Procedure 1. Make sure you are familiar with the key authors and experts described in the chapters in this section. Review at least five research-based sources that clarify the research to expand your understanding of the influence of culture on teaching and learning and the presence of bias in curriculum, instruction, and assessment. 2. Select and analyze a textbook with which you are familiar. Use the Sadkers’ (Sadker & Zittleman, 2012) list of the seven prevalent forms of bias in the curriculum to conduct a critical analysis of the textbook. Look at such aspects as pictures, names of people, the relative marginalization or integration of groups of people throughout the text, examples used, and so on. Summarize and present your data in displays (charts, tables, etc.). 3. Include in a written report the following: Introduction (text selected; rationale for selection; description of the text and context in which it is used) Review of the research on the influence of culture in teaching and learning and bias in the curriculum Summary of your findings (data tables and appropriate narratives) Discussion of the findings, including: { resonance with the research on bias { your understanding of bias and the challenges it poses to teaching and learning { the implications of your findings for teaching and learning Relate your discussion of the findings to class discussions and readings of the philosophy of education and purposes of curriculum. Be sure to adhere to APA guidelines in writing the final paper. Use the following tables to display your data: SECTION IV ASSESSMENT SKILLS Table 2: Analysis of Four Chapters for Frequency of Mention of Each Search Category Whites/Caucasians (male/female) African Americans (male/female) Hispanics/Latinos/Latinas (male/female) Native Americans (male/female) Asian Americans (male/female) Disability and deaf culture Gay, lesbian, bisexual, and transgendered persons (male/female) Religious groups Language groups Other Example Table 2 Format: Textbook Chapter Analysis Search category 1 # mentions/ # pages 2 # mentions/ # pages 3 # mentions/ # pages 4 # mentions/ # pages Total # mentions/ # pages White males White females African Americans Hispanics/Latinos/Latinas Table 3.
Maintaining privacy and confidentiality always is also vital. Nurses.docxjesssueann
Maintaining privacy and confidentiality always is also vital. Nurses handle information that if misplaced can expose patient’s unnecessarily and thus cause a breach in confidentiality. Such information can include drug use, sexual activity and history of mental illness (Masters, 2020). Conversations regarding patient care and condition must be private and involve only those in direct care. A violation of patient’s privacy can result in fines and employment termination
.
Main content15-2aHow Identity Theft OccursPerpetrators of iden.docxjesssueann
Main content
15-2aHow Identity Theft Occurs
Perpetrators of identity theft follow a common pattern after they have stolen a victim’s identity. To help you understand this process, we have created the “identity theft cycle.” Although some fraudsters perpetrate their frauds in slightly different ways, most generally follow the stages in the cycle shown in Figure 15.1.
Stage 1. Discovery
1. Perpetrators gain information.
2. Perpetrators verify information.
Stage 2. Action
1. Perpetrators accumulate documentation.
2. Perpetrators conceive cover-up or concealment actions.
Stage 3. Trial
1. First dimensional actions—Small thefts to test the stolen information.
2. Second dimensional actions—Larger thefts, often involving personal interaction, without much chance of getting caught.
3. Third dimensional actions—Largest thefts committed after perpetrators have confidence that their schemes are working.
Figure 15.1The Identity Theft Cycle
Stage 1: Discovery
The discovery stage involves two phases: information gathering and information verification. This is the first step in the identity theft cycle because all other actions the perpetrator takes depend upon the accuracy and effectiveness of the discovery stage. A powerful discovery stage constitutes a solid foundation for the perpetrator to commit identity theft. The smarter the perpetrator, the better the discovery foundation will be.
During the gaining information phase, fraudsters do all they can to gather a victim’s information. Examples of discovery techniques include such information-gathering techniques as searching trash, searching someone’s home or computer, stealing mail, phishing, breaking into cars or homes, scanning credit card information, or using other means whereby a perpetrator gathers information about a victim.
During the information verification phase, a fraudster uses various means to verify the information already gathered. Examples include telephone scams, where perpetrators call the victim and act as a representative of a business to verify the information gathered (this is known as pretexting), and trash searches (when another means was used to gather the original information). Although some fraudsters may not initially go through the information verification process, they will eventually use information verification procedures at some point during the scam. The scams of perpetrators who don’t verify stolen information are usually shorter and easier to catch than scams of perpetrators who verify stolen information.
Step 2: Action
The action stage is the second phase of the identity theft cycle. It involves two activities: accumulating documentation and devising cover-up or concealment actions.
Accumulating documentation refers to the process perpetrators use to obtain needed tools to defraud the victim. For example, using the information already obtained, perpetrators may apply for a bogus credit card, fake check, or driver’s license in the victim’s name. Although the perpetra.
Macro Presentation – Australia Table of ContentOver.docxjesssueann
Macro Presentation – Australia
Table of Content:
Overview
Nominal GDP & Real GDP
GDP/Capita
Inflation rate
Exports & Imports
Unemployment Rate & Labor force
labor force participation & composition of labor force
Money Supply
pie-chart (composition of the economy)
strengths and weaknesses of this economy
Overview:
sixth-largest country in the world.
Australia is a continent & an island
located in Oceania
Population: 25.2 million
Australia is one of the wealthiest Asia
the world’s 14th largest (economically)
Overview:
GDP :
$1.3 trillion
2.8% growth
2.6% 5-year compound annual growth
$52,373 per capita
Unemployment: 5.4%
Inflation (CPI): 2.0%
Characterized by: diverse services, technology sectors & low government debt
five key reasons for investing in Australia: Robust Economy, Dynamic Industries, Innovation and Skills, Global Ties and Strong Foundations & compares Australia’s credentials with other countries.
GDP:
Nominal GDP & Real GDP:
Nominal GDP:
1.434 trillion
Real GDP:
45439.30 $
GDP/Capita:
57,373.687
Inflation Rate:
Inflation Rate 2018 = 1.9%
Inflation Rate 2017 = 1.9%
Inflation Rate 2016 = 1.3%
Inflation Rate 2015 = 1.5%
Inflation Rate 2014 = 2.5%
Inflation Rate 2013 = 2.5%
Inflation Rates over 5 years
عمود12013201420152016201720182.52.51.51.31.91.9عمود2201320142015201620172018
Exports & Imports:
Exports:
Bituminous coal
iron ores and concentrates
Gold
Petroleum oils and oils obtained from bituminous
Copper ores and concentrates
The total value of exports: is US$ 252,776 million.
Imports:
Petroleum oïl
Automobiles with reciprocating piston engine di
Transmission apparatus
Diesel powered trucks
The total value of imports: is US$ 235,519 million
Exports & Imports (partners) :
Exports:
China
Japan
Korea
India
United sates
Imports:
China
United states
Japan
Germany
Thailand
Unemployment Rate & Labor force:
Unemployment Rate:
5.4%
Labor force:
79%
labor force participation & composition of labor force:
labor force participation:
77.558
composition of labor force:
Employed = 12658.6
Unemployed = 671.0
Labour force =
12658.6 + 671.0 = 13329.6
Nationals = 29.7 %
foreigners+ = 70.3 %
Money Supply:
M1 = 1189.19
M3 = 2231.55
pie-chart (composition of the economy):
70% of coal, 54% of iron, service industry 70%, Agriculture 12%
المبيعاتcoalironindustryagriculture70547012
strengths and weaknesses of this economy:
Weaknesses:
The quality of life in Australia is high & not permanent
The size of their investment
Most concentrated investments: coal, gas, iron mining
Solution
s & Suggestion:
To sustain a high quality of life long-term:
Many investments with added value ‘not from their priorities’ : (workforce for education, high teach sector in nanotechnology + solar energy & agricultural innovation) > should focus on
strengths and weaknesses of this economy:
Strength:
Mining is a strong investment in Australia
References:
https://www.h.
M.S Aviation Pty Ltd TA Australian School of Commerce RTO N.docxjesssueann
This document is an assessment booklet for the unit BSBINN601 Lead and manage organisational change. It contains information on the assessment process, requirements, tasks and evidence to be collected to determine competency. The assessments will take place at the Australian School of Commerce campus and involve knowledge tests, project work and roleplays to demonstrate skills in leading and managing organizational change.
M4.3 Case StudyCase Study ExampleJennifer S. is an Army veter.docxjesssueann
M4.3 Case Study
Case Study Example:
Jennifer S. is an Army veteran of Operation Freedom. Since returning home, Jennifer has suffered from recurrent headaches, ringing in her ears, difficulty focusing, and dizziness. In addition, soon after returning home, she began to experience moments of panic when in open spaces; flashbacks reliving the blast and the death of fellow soldiers; feelings of emotional numbness and depression; and being easily startled. She was placed on medical leave and diagnosed with Post Traumatic Stress Disorder (PTSD) and is currently being seen by a psychiatrist at the VA hospital. Her husband understands the concept of PTSD but is unprepared to handle his wife’s deteriorating condition.
Recently, Jennifer was seen at the local urgent care center for recurrent headaches, complaints of shortness of breath, and chest pain. Her husband informed the urgent care nurse that for the past four weeks his wife has been unable to care for the children, remains in bed, complaining of headaches, and is very ‘jumpy’.
The nurse assesses Jennifer knowing that returning veterans with PTSD and their families face an array of challenges, with implications for the veterans, their partners, and their children. The nurse considers referring them to: a social worker specializing in crisis intervention for veterans, a family counselor, the school nurse, a family health care practitioner.
Key elements of the nurse’s assessment are as follows:
Jennifer is 33 year-old woman who enlisted in Reserve Officers’ Training Corps (ROTC) in college, where she majored in Journalism. Upon graduation, she obtained a position in the Army as public affairs broadcast specialist. Her first assignment was at a base in upstate New York. Three years ago, she was relocated to the St. Louis, Missouri area. Jennifer has been married to her husband, Zane, for 14 years and they have two children ages six and ten. Cameron is ten years-old and entering middle school and Zeta is six years-old and in kindergarten. Zane works as a civil engineer in the St. Louis area. Both Jennifer and Zane come from large families who reside in the Boston area. Jennifer’s family is Portuguese and Zane's is Irish, they were both raised Catholic. While Jennifer was deployed, her mother moved in with Zane and the children to provide additional support and child care.
One year ago, Jennifer was deployed to Afghanistan on a six month assignment to report on the events of the war: she thought she had a ‘safe’ assignment. While working on a story in the field an Improvised Explosive Device (IED) exploded near her: two soldiers and four citizens were killed including one child. Although she was unhurt, she was unable to sleep after this event. Upon returning stateside, she began experiencing vivid nightmares, sleeplessness, survivor guilt, and depression. She was recently diagnosed with PTSD and is attempting to find a support group and counseling. Unfortunately, she has found that treatment for fe.
make a histogram out of this information Earthquake Frequency .docxjesssueann
This document provides earthquake frequency data categorized by magnitude. Great earthquakes occur annually, major earthquakes occur 18 times annually, strong earthquakes occur 120 times annually, and moderate earthquakes occur 800 times annually.
Love Language Project FINAL PAPERLove Language Project Part .docxjesssueann
Love Language Project FINAL PAPER
Love Language Project Part I
Objective:
To demonstrate the principles of love languages and effective use of interpersonal communication skills through “gifting” a close interpersonal relationship.
Assignment:
Please research the 5 Love Languages. Set a time when you can interview your selected person, at least ½ hour. Choose a quiet, comfortable environment where you will be able to listen effectively. The goal of your interview is to learn how your selected person most likes to receive expressions of affection.
You might begin by sharing the five love languages with them and asking some versions of the following questions:
1. Based on the descriptions in this section and this piece, which of the five love languages is most appealing to you to receive?
2. Can you share a story/example of a time when you received affection this way?
3. Which is the most challenging/uncomfortable love language for you to receive?
4. Can you share a story/example of a time when you received affection this way?
5. What changes do you think you could make in the way you receive affectionate messages in your close relationships?
Please describe the person that you chose to interview and your relationship with them. Then, post their responses to the questions
Love Language Project Part II
Write a personal reflection paper, at least 1.5 pages long, double spaced, typed, include the following:
1. What did you learn about your selected person and their preferred love languages from your interview? What was challenging about the interview? What surprised you?
2. How does their preferred love languages differ from yours? Did this make it difficult to plan your special event?
3. Comment on planning your Love Language Event. How did you come up with your ideas? What was easy and what was challenging?
4. Comment on implementing your Love Language Event. What was enjoyable? What was challenging? Did it go as you’d planned?
5. Comment on the Love Language Project in general. What did you learn? About the other person? About yourself?
6. How might what you learned during this Love Language Project affect your expressions of affection in other relationships?
.
Major Computer Science What are the core skills and knowledge y.docxjesssueann
Major: Computer Science
What are the core skills and knowledge you hope to acquire by completing a degree in this major and how do you plan to apply these when you graduate?
Please provide any other information about yourself that you feel will help this college make an admission decision. This may include work, research, volunteer activities or other experiences pertaining to the degree program.
.
Major Crime in Your CommunityUse the Internet to search for .docxjesssueann
Major Crime in Your Community
Use the Internet to search for a recent major crime in your community.
Write a report (narrative only) based on the account of the incident, using the outline process mentioned in chapter three of the course text.
You may simulate interviews and "fill in" any unknown information required to complete the report.
Be sure to include the characteristics of an effective police report covered in chapter three.
Instructions
This report must be at
least 2 pages
of written text.
· The entire paper must be your original work
· This report will use 1-inch margins, Times New Roman 12-point font, and double spacing.
· Cite your source – where do you get the information for your report?
.
Major Assignment - Learning NarrativeWrite a learning narr.docxjesssueann
Major Assignment - Learning Narrative
Write a
learning narrative
that narrates a specific event from your life that helped you learn something new about yourself or others. Your narrative should focus on a specific event in a narrow timeframe, using vivid description, narration, detail, and dialogue to organize your memories and make the significance of what happened clear to an audience.
Assignment
A
narrative
is a specific type of essay that uses stories of particular moments to help audiences perceive, understand, and "appreciate the value of an idea" (
The Composition of Everyday Life
, Ch. 1, p. 19).
For this essay, you will write a
learning narrative
, a specific type of narrative that focuses on showing how a particular moment from your memory changed how you thought about yourself or others. The learning narrative requires you to organize your memories and decide which details best show an audience how the events from your past affected you. A learning narrative is broader than a "literacy narrative": while you can write about how language or education changed your life, you also can write about other things you learned through music, sports, business, or in any other relevant setting.
In order to write a strong, focused narrative, you will need to be attentive to the following expectations for the essay:
Find the significance:
Think of how your narrative connects your memories to feelings / concepts others have experienced
Tell a particular story:
Like Keller and Zimmer, choose a single moment or event that can reflect your process of learning
Choose relevant details:
Include only those details that contribute to the significance
Narrate and describe:
Add emotional weight and interest to your story by narrating events with dialogue, action, description, and sensory experiences
Caution
: Please keep in mind that writing in this class is public, and anything you write about yourself may be shared with other students and instructors. Please only write about details that you are comfortable making public within our classroom community. You should know that your teacher is required by the State of Texas
(Links to an external site.)
to report any suspected incidents of discrimination, harassment, Title IX sexual harassment, and sexual misconduct to the UNT Title IX coordinators. If you have any questions about anything personal that you might want to disclose, email your teacher first or consult with one of the resources listed on this page:
Information on Sexual Violence and Mandatory Reporting.
Format and Length
Format
: Typed, double-spaced, submitted as a word-processing document.
12 point,
serif font (Links to an external site.)
(i.e. Times New Roman; Garamond; Book Antiqua), 1-inch margins.
Length
: 750 - 1000 words (approx. 3-4 pages)
Objectives and Questions
These questions help to guide discussion and set up the objectives for this unit.
What is an experience? What are significant experience.
Looking to have this work done AGAIN. It was submitted several times.docxjesssueann
This student had submitted an assignment multiple times but it did not meet the professor's requirements and they are looking for help correcting it based on the feedback provided. They are including their previous submission, the assignment instructions, and the professor's feedback on what is still missing.
Major Assessment 1 Develop a Platform of Beliefs The following .docxjesssueann
Major Assessment 1: Develop a Platform of Beliefs
The following major assessment involves integrating your knowledge and skills around defining multicultural education and being a multicultural educator. You will write a platform of beliefs about teaching and learning. Your platform should be grounded in your growing understanding of teaching and learning, as well as the knowledge base about teaching and learning. You will also describe personal strengths and challenges as an educator in building an educational environment that reflects your beliefs. In assessing your own strengths and challenge areas, include an analysis of the findings from the assessment instruments and exercises that are included in the previous chapter. You may also access additional assessment instruments. Include in your platform the following sections: 1. Introduction 2. Your platform of beliefs about teaching and learning. Some essential questions that might be addressed in your platform are these: What do you believe is the purpose of education? What is the role of the teacher? What should be taught (the curriculum)? How do people learn? How do you view students as learners? Who controls the curriculum in schools? Whose knowledge is important to include? Are state standards and tests desirable? What is the impact of standardized testing on learning? How do issues of race, class, and gender influence what you do? What is your definition of effective teaching? Who and what have influenced your beliefs (e.g., people, experiences, readings)? What is the impact of your beliefs on teaching and learning for diverse students? Make specific and clear connections between your platform and course readings and discussions. 3. Personal strengths and challenges in advancing a school vision of learning; promoting the success of all students; responding to diverse student interests and needs; understanding and responding to social, economic, legal, and cultural contexts 4. Personal goals (knowledge, skills, dispositions) that you will be working on in the future 5. Conclusions
.
Macroeconomics PaperThere are currently three major political ap.docxjesssueann
Macroeconomics Paper
There are currently three major political approaches to fixing the problem with the national debt .
1) One group of advocates is asking that we cut down government expenditures and give more tax breaks and incentives to small and big business.
2) Another group of advocates is saying that we must emphasize our exports by lowering our dollar value or forcing our trade partners – China – to regulate more accurately it’s currency.
3) A third group of approaches by saying we should have a balance budget amendment.
i) Identify the notable political advocates of all three positions.
ii) Give the pro’s and con’s of each approach.
Length: 2-3 pages.
Please email the paper in either
Microsoft word *.doc (97-2003) format or
Rich text format *.rtf OR GOOGLE DOCS
font 12 double-space
1-inch margins
Bibliography need not be inclusive in writing size.
SOURCES
Agresti, James D. "National Debt." National Debt - Just Facts. N.p., 26 Apr. 2011. Web. 24 Apr. 2015.
"Americans for a Balanced Budget Amendment." Balanced Budget Amendment. N.p., n.d. Web. 01 May 2015.
"Bailout Timeline: Another Day, Another Bailout." ProPublica. N.p., n.d. Web. 26 Apr. 2015.
Bandow, Doug. "Federal Spending: Killing the Economy With Government Stimulus." Forbes. Forbes Magazine, 6 Aug. 2012. Web. 01 May 2015.
FROM UNIT 2 FOLDER
Macroeconomics Paper
There are currently three major political approaches to fixing the problem with the national debt .
1) One group of advocates is asking that we cut down government expenditures and give more tax breaks and incentives to small and big business.
2) Another group of advocates is saying that we must emphasize our exports by lowering our dollar value or forcing our trade partners – China – to regulate more accurately it’s currency.
3) A third group of approaches by saying we should have a balance budget amendment.
i) Identify the notable political advocates of all three positions.
ii) Give the pro’s and con’s of each approach.
Length: 2-3 pages.
Please email the paper in either
Microsoft word *.doc (97-2003) format or
Rich text format *.rtf
font 12
double-spaced
1-inch margins
Bibliography need not be inclusive in writing size.
"Federal Spending, Budget, and Debt."
Solution
s.heritage.org. N.p., n.d. Web. 01 May 2015.
Lee, Bonnie. "Tax Breaks Every Small Business Needs to Know About." Smallbusiness.foxbusiness.com. N.p., 24 June 2013. Web. 01 May 2015.
Rifkin, Jesse. "Advocates See 2015 As Year Of The Balanced Budget Amendment." The Huffington Post. TheHuffingtonPost.com, 3 Feb. 2015. Web. 01 May 2015.
Macroeconomics Paper
There are currently three major political approaches to fixing the problem with the national debt .
1) One group of advocates is asking that we cut down government expenditures and give more tax breaks and incentives to small and big business.
2) Another group of advocates is saying that we must emphasize our exports by lowering our dollar value or forcing our trade p.
M A T T D O N O V A NThings in the Form o f a Prayer in.docxjesssueann
M A T T D O N O V A N
Things in the Form o f
a Prayer in the Form
o f a Wail
H e r e ’s t h e j o u r n e y i n m i n i a t u r e .Oscar Hammerstein, not long before stomach cancer kills him,
writes the song as a duet between Marie and the Mother Abbess, for a
scene in which the plucky nun is told she’s being booted from the con
vent since she privileges melody over God. Marie doesn’t want to serve
as governess for the Von Trapp clan, but she’s already shown her hand
by giving rapturous voice to a song that summons the bliss and solace
o f secular joys. She needs to go. Although the film version of The Sound
of Music will shift “My Favorite Things” to the thunderstorm scene in
which Marie offers up raindrops on roses and warm woolen mittens as
balm to the terrified kids, John Coltrane’s classic jazz cover much more
radically revamps the Broadway hit, transfiguring mere catchiness into
complex modalities. Yet if this were simply a one-off recording, there
wouldn’t be much to say: turning cornball consolation into jazz isn’t
news. Instead, Coltrane can’t relinquish it. Instead, even throughout all
his late music-as-prayer work, he never lets go of the show tune.
“We played it every night for five years,” drummer Elvin Jones re
membered. “We played it every night like there would be no tomorrow.
Like it would be the last time we played it.” His son, Ravi Coltrane,
calculates that his father’s band played “My Favorite Things” thousands
o f times as a regular fixture in the set: “They worked a lo t— forty-five
weeks a year, six nights a week, three sets, sometimes even four sets on
the weekend. You’re talking about getting the blade as sharp as can be.”
But of all the blades to w het— especially one bedecked with ponies
and kittens— why that song in particular?
M y f i r s t e n c o u n t e r with Coltrane’s late free jazz work came from
an unlikely source: the writings o f cult rock critic Lester Bangs. At the age
o f fourteen, I stumbled upon a copy of his collected writings— Psychotic
632
Reactions and Carburetor Dung— and proceeded to treat it as less an assem
blage o f essays and music reviews than a checklist of writers and albums I
was obliged to track down if I might ever break free from my Ohio sub
urbs. The Velvet Underground, William Burroughs, Iggy and the Stooges’
Metallic K.O. (a live album in which you can hear beer bottles shattering
against guitar strings), and even Baudelaire all first came tumbling my
way through the same careening chute of Bangs’s writing. His claim that
Van Morrison’s Astral Weeks was fueled by many lifetimes o f wisdom
lured me into transcribing the entirety o f the album’s lyrics in my algebra
notebook, and the visible bottom edge of an Undertones poster in his
author photograph led me, without having heard a note o f the band’s
music, to bike six miles to Spin More records in Kent on a quest to
cobble together their discography.
Sandwiched between articl.
M A R C H 2 0 1 5F O R W A R D ❚ E N G A G E D ❚ .docxjesssueann
M A R C H 2 0 1 5
F O R W A R D ❚ E N G A G E D ❚ R E A D Y
A Cooperative Strategy for
21st Century Seapower
DRAFT/PRE-DECISIONAL - NOT FOR DISSEMINATION - 02 FEB
A COOPERATIVE STRATEGY FOR 21ST CENTURY SEAPOWER, MARCH 2015 [i]
America’s Sea Services—the U.S. Navy, Marine Corps, and Coast Guard—uniquely provide presence around the globe. During peacetime and times of conflict, across the full spectrum—from
supporting an ally with humanitarian assistance or disaster relief to
deterring or defeating an adversary in kinetic action—Sailors, Marines,
and Coast Guardsmen are deployed at sea and in far-flung posts to be
wherever we are needed, when we are needed. Coming from the sea, we
get there sooner, stay there longer, bring everything we need with us,
and we don’t have to ask anyone’s permission.
Our founders recognized the United States as a maritime nation and
the importance of maritime forces, including in our Constitution the re-
quirement that Congress “maintain a Navy.” In today’s dynamic security
environment, with multiple challenges from state and non-state actors
that are often fed by social disorder, political upheaval, and technological
advancements, that requirement is even more prescient.
The United States Navy, Marine Corps, and Coast Guard are our
Nation’s first line of defense, often far from our shores. As such, main-
taining America’s leadership role in the world requires our Nation’s Sea
Services to return to our maritime strategy on occasion and reassess
our approach to shifting relationships and global responsibilities. This
necessary review has affirmed our focus on providing presence around
the world in order to ensure stability, build on our relationships with allies
and partners, prevent wars, and provide our Nation’s leaders with options
in times of crisis. It has confirmed our continued commitment to main-
tain the combat power necessary to deter potential adversaries and to
fight and win when required.
Our responsibility to the American people dictates an efficient use of
our fiscal resources and an approach that adapts to the evolving security
environment. The adjustments made in this document do just that. Look-
ing at how we support our people, build the right platforms, power them
to achieve efficient global capability, and develop critical partnerships
will be central to its successful execution and to providing that unique
capability: presence.
PREFACE
[ii] Forward ✦ Engaged ✦ Ready
Seapower has been and will continue to be the critical foundation of
national power and prosperity and international prestige for the United
States of America. Our Sea Services will integrate with the rest of our
national efforts, and those of our friends and allies. This revision to A
Cooperative Strategy for 21st Century Seapower builds on the heritage
and complementary capabilities of the Navy-Marine Corps-Coast Guard
team to advan.
Lymphedema following breast cancer The importance of surgic.docxjesssueann
Lymphedema following breast cancer: The importance of
surgical methods and obesity
Rebecca J. Tsai, PhDa,*, Leslie K. Dennis, PhDa,b, Charles F. Lynch, MD, PhDa, Linda G.
Snetselaar, RD, PhD, LDa, Gideon K.D. Zamba, PhDc, and Carol Scott-Conner, MD, PhD,
MBAd
aDepartment of Epidemiology, College of Public Health, University of Iowa, Iowa City, IA, USA.
bDivision of Epidemiology and Biostatistics, College of Public Health, University of Arizona,
Tucson, AZ, USA.
cDepartment of Biostatistics, College of Public Health, University of Iowa, Iowa City, IA, USA.
dDepartment of Surgery, College of Medicine, University of Iowa, Iowa City, IA, USA.
Abstract
Background: Breast cancer-related arm lymphedema is a serious complication that can
adversely affect quality of life. Identifying risk factors that contribute to the development of
lymphedema is vital for identifying avenues for prevention. The aim of this study was to examine
the association between the development of arm lymphedema and both treatment and personal
(e.g., obesity) risk factors.
Methods: Women diagnosed with breast cancer in Iowa during 2004 and followed through 2010,
who met eligibility criteria, were asked to complete a short computer assisted telephone interview
about chronic conditions, arm activities, demographics, and lymphedema status. Lymphedema was
characterized by a reported physician-diagnosis, a difference between arms in the circumference
(> 2cm), or the presence of multiple self-reported arm symptoms (at least two of five major arm
symptoms, and at least four total arm symptoms). Relative risks (RR) were estimated using
logistic regression.
Results: Arm lymphedema was identified in 102 of 522 participants (19.5%). Participants treated
by both axillary dissection and radiation therapy were more likely to have arm lymphedema than
treated by either alone. Women with advanced cancer stage, positive nodes, and larger tumors
along with a body mass index > 40 were also more likely to develop lymphedema. Arm activity
level was not associated with lymphedema.
*Correspondence and Reprints to: Rebecca Tsai, National Institute for Occupational Safety and Health, 4676 Columbia Parkway,
R-17, Cincinnati, OH 45226. [email protected] Phone: (513)841-4398. Fax: (513) 841-4489.
Authorship contribution
All authors contributed to the conception, design, drafting, revision, and the final review of this manuscript.
Competing interest
Conflicts of Interest and Source of Funding: This study was funded by the National Cancer Institute Grant Number: 5R03CA130031.
All authors do not declare any conflict of interest.
All authors do not declare any conflict of interest.
HHS Public Access
Author manuscript
Front Womens Health. Author manuscript; available in PMC 2018 December 14.
Published in final edited form as:
Front Womens Health. 2018 June ; 3(2): .
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Lukas Nelson and his wife Anne and their three daughters had been li.docxjesssueann
Lukas Nelson and his wife Anne and their three daughters had been living in their house for over five years when they decided it was time to make some modest improvements. One area they both agreed needed an upgrade was the bath tub. Their current house had one standard shower bathtub combination. Lukas was 6 feet four, and could barely squeeze into it. In fact, he had taken only one bath since they moved in. He and Anne both missed soaking in the older, deep bath tubs they enjoyed when they lived back East.
(Rest of case not shown due to length.)
What factors and forces contributed to scope creep in this case?
Is this an example of good or bad scope creep? Explain.
How could scope creep been better managed by the Nelson
.
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বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...
Leveraging Generative AI to Drive Nonprofit InnovationTechSoup
In this webinar, participants learned how to utilize Generative AI to streamline operations and elevate member engagement. Amazon Web Service experts provided a customer specific use cases and dived into low/no-code tools that are quick and easy to deploy through Amazon Web Service (AWS.)
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
How to Make a Field Mandatory in Odoo 17Celine George
In Odoo, making a field required can be done through both Python code and XML views. When you set the required attribute to True in Python code, it makes the field required across all views where it's used. Conversely, when you set the required attribute in XML views, it makes the field required only in the context of that particular view.
Chapter wise All Notes of First year Basic Civil Engineering.pptxDenish Jangid
Chapter wise All Notes of First year Basic Civil Engineering
Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
Chapter 4
Buildings: Selection of site for Buildings, Layout of Building Plan, Types of buildings, Plinth area, carpet area, floor space index, Introduction to building byelaws, concept of sun light & ventilation. Components of Buildings & their functions, Basic concept of R.C.C., Introduction to types of foundation
Chapter 5
Transportation: Introduction to Transportation Engineering; Traffic and Road Safety: Types and Characteristics of Various Modes of Transportation; Various Road Traffic Signs, Causes of Accidents and Road Safety Measures.
Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
Text Books:
1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
4. BCP, Surveying volume 1
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Spark Good (walmart.com/sparkgood) is a charitable platform that enables nonprofits to receive donations directly from customers and associates.
Answers about how you can do more with Walmart!"
2. 95
INTRODUCTION
Chapter 2 began our study of descriptive statistics. In order to
transform raw or un-
grouped data into a meaningful form, we organize the data into
a frequency distribution.
We present the frequency distribution in graphic form as a
histogram or a frequency
polygon. This allows us to visualize where the data tend to
cluster, the largest and the
smallest values, and the general shape of the data.
In Chapter 3, we first computed several measures of location,
such as the mean,
median, and mode. These measures of location allow us to
report a typical value in the
set of observations. We also computed several measures of
dispersion, such as the
range, variance, and standard deviation. These measures of
dispersion allow us to de-
scribe the variation or the spread in a set of observations.
We continue our study of descriptive statistics in this chapter.
We study (1) dot plots,
(2) stem-and-leaf displays, (3) percentiles, and (4) box plots.
These charts and statistics
give us additional insight into where the values are concentrated
as well as the general
shape of the data. Then we consider bivariate data. In bivariate
data, we observe two
variables for each individual or observation. Examples include
the number of hours a
student studied and the points earned on an examination; if a
sampled product meets
3. quality specifications and the shift on which it is manufactured;
or the amount of electric-
ity used in a month by a homeowner and the mean daily high
temperature in the region
for the month. These charts and graphs provide useful insights
as we use business
analytics to enhance our understanding of data.
DOT PLOTS
Recall for the Applewood Auto Group data, we summarized the
profit earned on the
180 vehicles sold with a frequency distribution using eight
classes. When we orga-
nized the data into the eight classes, we lost the exact value of
the observations. A
dot plot, on the other hand, groups the data as little as possible,
and we do not lose
the identity of an individual observation. To develop a dot plot,
we display a dot for
each observation along a horizontal number line indicating the
possible values of the
data. If there are identical observations or the observations are
too close to be shown
individually, the dots are “piled” on top of each other. This
allows us to see the shape
of the distribution, the value about which the data tend to
cluster, and the largest and
smallest observations. Dot plots are most useful for smaller data
sets, whereas histo-
grams tend to be most useful for large data sets. An example
will show how to con-
struct and interpret dot plots.
LO4-1
Construct and interpret a
dot plot.
4. E X A M P L E
The service departments at Tionesta Ford Lincoln and Sheffield
Motors Inc., two
of the four Applewood Auto Group dealerships, were both open
24 days last
month. Listed below is the number of vehicles serviced last
month at the two
dealerships. Construct dot plots and report summary statistics to
compare the
two dealerships.
Tionesta Ford Lincoln
Monday Tuesday Wednesday Thursday Friday Saturday
23 33 27 28 39 26
30 32 28 33 35 32
29 25 36 31 32 27
35 32 35 37 36 30
96 CHAPTER 4
Sheffield Motors Inc.
Monday Tuesday Wednesday Thursday Friday Saturday
31 35 44 36 34 37
30 37 43 31 40 31
32 44 36 34 43 36
26 38 37 30 42 33
5. S O L U T I O N
The Minitab system provides a dot plot and outputs the mean,
median, maximum,
and minimum values, and the standard deviation for the number
of cars serviced
at each dealership over the last 24 working days.
The dot plots, shown in the center of the output, graphically
illustrate the distribu-
tions for each dealership. The plots show the difference in the
location and dis-
persion of the observations. By looking at the dot plots, we can
see that the
number of vehicles serviced at the Sheffield dealership is more
widely dispersed
and has a larger mean than at the Tionesta dealership. Several
other features of
the number of vehicles serviced are:
• Tionesta serviced the fewest cars in any day, 23.
• Sheffield serviced 26 cars during their slowest day, which is
4 cars less than
the next lowest day.
• Tionesta serviced exactly 32 cars on four different days.
• The numbers of cars serviced cluster around 36 for Sheffield
and 32 for Tionesta.
From the descriptive statistics, we see Sheffield serviced a
mean of 35.83 vehicles
per day. Tionesta serviced a mean of 31.292 vehicles per day
during the same
period. So Sheffield typically services 4.54 more vehicles per
day. There is also
more dispersion, or variation, in the daily number of vehicles
6. serviced at Sheffield
than at Tionesta. How do we know this? The standard deviation
is larger at Shef-
field (4.96 vehicles per day) than at Tionesta (4.112 cars per
day).
STEM-AND-LEAF DISPLAYS
In Chapter 2, we showed how to organize data into a frequency
distribution so we could
summarize the raw data into a meaningful form. The major
advantage to organizing the
data into a frequency distribution is we get a quick visual
picture of the shape of the
LO4-2
Construct and describe a
stem-and-leaf display.
DESCRIBING DATA: DISPLAYING AND EXPLORING DATA
97
distribution without doing any further calculation. To put it
another way, we can see
where the data are concentrated and also determine whether
there are any extremely
large or small values. There are two disadvantages, however, to
organizing the data into
a frequency distribution: (1) we lose the exact identity of each
value and (2) we are not
sure how the values within each class are distributed. To
explain, the Theater of the
Republic in Erie, Pennsylvania, books live theater and musical
performances. The the-
7. ater’s capacity is 160 seats. Last year, among the forty-five
performances, there were
eight different plays and twelve different bands. The following
frequency distribution
shows that between eighty up to ninety people attended two of
the forty-five perfor-
mances; there were seven performances where ninety up to one
hundred people at-
tended. However, is the attendance within this class clustered
about 90, spread evenly
throughout the class, or clustered near 99? We cannot tell.
Attendance Frequency
80 up to 90 2
90 up to 100 7
100 up to 110 6
110 up to 120 9
120 up to 130 8
130 up to 140 7
140 up to 150 3
150 up to 160 3
Total 45
One technique used to display quantitative information in a
condensed form
and provide more information than the frequency distribution is
the stem-and-leaf
display. An advantage of the stem-and-leaf display over a
frequency distribution is
we do not lose the identity of each observation. In the above
example, we would not
know the identity of the values in the 90 up to 100 class. To
illustrate the construc-
tion of a stem-and-leaf display using the number people
8. attending each perfor-
mance, suppose the seven observations in the 90 up to 100 class
are 96, 94, 93,
94, 95, 96, and 97. The stem value is the leading digit or digits,
in this case 9. The
leaves are the trailing digits. The stem is placed to the left of a
vertical line and the
leaf values to the right.
The values in the 90 up to 100 class would appear as follows:
9 ∣ 6 4 3 4 5 6 7
It is also customary to sort the values within each stem from
smallest to largest. Thus,
the second row of the stem-and-leaf display would appear as
follows:
9 ∣ 3 4 4 5 6 6 7
With the stem-and-leaf display, we can quickly observe that 94
people attended two
performances and the number attending ranged from 93 to 97. A
stem-and-leaf display
is similar to a frequency distribution with more information,
that is, the identity of the
observations is preserved.
STEM-AND-LEAF DISPLAY A statistical technique to present
a set of data. Each
numerical value is divided into two parts. The leading digit(s)
becomes the stem
and the trailing digit the leaf. The stems are located along the
vertical axis, and the
leaf values are stacked against each other along the horizontal
axis.
9. 98 CHAPTER 4
The following example explains the details of developing a
stem-and-leaf display.
E X A M P L E
Listed in Table 4–1 is the number of people attending each of
the 45 performances
at the Theater of the Republic last year. Organize the data into a
stem-and-leaf
display. Around what values does attendance tend to cluster?
What is the smallest
attendance? The largest attendance?
S O L U T I O N
From the data in Table 4–1, we note that the smallest attendance
is 88. So we will
make the first stem value 8. The largest attendance is 156, so
we will have the
stem values begin at 8 and continue to 15. The first number in
Table 4–1 is 96,
which has a stem value of 9 and a leaf value of 6. Moving
across the top row, the
second value is 93 and the third is 88. After the first 3 data
values are considered,
the chart is as follows.
Stem Leaf
8 8
10. 9 6 3
10
11
12
13
14
15
Organizing all the data, the stem-and-leaf chart looks as
follows.
Stem Leaf
8 8 9
9 6 3 5 6 4 4 7
10 8 7 3 4 6 3
11 7 3 2 7 2 1 9 8 3
12 7 5 7 0 5 5 0 4
13 9 5 2 9 4 6 8
14 8 2 3
15 6 5 5
The usual procedure is to sort the leaf values from the smallest
to largest. The last
line, the row referring to the values in the 150s, would appear
as:
15 ∣ 5 5 6
TABLE 4–1 Number of People Attending Each of the 45
Performances at the Theater of
the Republic
96 93 88 117 127 95 113 96 108 94 148 156
139 142 94 107 125 155 155 103 112 127 117 120
112 135 132 111 125 104 106 139 134 119 97 89
11. 118 136 125 143 120 103 113 124 138
DESCRIBING DATA: DISPLAYING AND EXPLORING DATA
99
The final table would appear as follows, where we have sorted
all of the leaf values.
Stem Leaf
8 8 9
9 3 4 4 5 6 6 7
10 3 3 4 6 7 8
11 1 2 2 3 3 7 7 8 9
12 0 0 4 5 5 5 7 7
13 2 4 5 6 8 9 9
14 2 3 8
15 5 5 6
You can draw several conclusions from the stem-and-leaf
display. First, the mini-
mum number of people attending is 88 and the maximum is 156.
There were two per-
formances with less than 90 people attending, and three
performances with 150 or
more. You can observe, for example, that for the three
performances with more than
150 people attending, the actual attendances were 155, 155, and
156. The concentra-
tion of attendance is between 110 and 130. There were fifteen
performances with at-
tendance between 110 and 119 and eight performances between
120 and 129. We
12. can also tell that within the 120 to 129 group the actual
attendances were spread
evenly throughout the class. That is, 120 people attended two
performances, 124 peo-
ple attended one performance, 125 people attended three
performances, and 127 peo-
ple attended two performances.
We also can generate this information on the Minitab software
system. We have
named the variable Attendance. The Minitab output is below.
You can find the Minitab
commands that will produce this output in Appendix C.
The Minitab solution provides some additional information
regarding cumulative totals.
In the column to the left of the stem values are numbers such as
2, 9, 15, and so on. The
number 9 indicates there are 9 observations that have occurred
before the value of 100.
The number 15 indicates that 15 observations have occurred
prior to 110. About halfway
down the column the number 9 appears in parentheses. The
parentheses indicate that the
middle value or median appears in that row and there are nine
values in this group. In this
case, we describe the middle value as the value below which
half of the observations oc-
cur. There are a total of 45 observations, so the middle value, if
the data were arranged
from smallest to largest, would be the 23rd observation; its
value is 118. After the median,
the values begin to decline. These values represent the “more
than” cumulative totals.
There are 21 observations of 120 or more, 13 of 130 or more,
and so on.
14. 8.3 9.6 9.5 9.1 8.8 11.2 7.7 10.1 9.9 10.8
10.2 8.0 8.4 8.1 11.6 9.6 8.8 8.0 10.4 9.8 9.2
S E L F - R E V I E W 4–1
DESCRIBING DATA: DISPLAYING AND EXPLORING DATA
101
Organize this information into a stem-and-leaf display.
(a) How many rates are less than 9.0?
(b) List the rates in the 10.0 up to 11.0 category.
(c) What is the median?
(d) What are the maximum and the minimum rates of return?
1. Describe the differences between a histogram and a dot plot.
When might a dot
plot be better than a histogram?
2. Describe the differences between a histogram and a stem-
and-leaf display.
3. Consider the following chart.
6 72 3 4 51
a. What is this chart called?
b. How many observations are in the study?
c. What are the maximum and the minimum values?
d. Around what values do the observations tend to cluster?
4. The following chart reports the number of cell phones sold at
a big-box retail store
for the last 26 days.
15. 199 144
a. What are the maximum and the minimum numbers of cell
phones sold in a day?
b. What is a typical number of cell phones sold?
5. The first row of a stem-and-leaf chart appears as follows: 62
| 1 3 3 7 9. Assume
whole number values.
a. What is the “possible range” of the values in this row?
b. How many data values are in this row?
c. List the actual values in this row of data.
6. The third row of a stem-and-leaf chart appears as follows: 21
| 0 1 3 5 7 9. Assume
whole number values.
a. What is the “possible range” of the values in this row?
b. How many data values are in this row?
c. List the actual values in this row of data.
7. The following stem-and-leaf chart shows the number of units
produced per day in a
factory.
Stem Leaf
3 8
4
5 6
6 0133559
7 0236778
8 59
9 00156
10 36
16. a. How many days were studied?
b. How many observations are in the first class?
E X E R C I S E S
102 CHAPTER 4
c. What are the minimum value and the maximum value?
d. List the actual values in the fourth row.
e. List the actual values in the second row.
f. How many values are less than 70?
g. How many values are 80 or more?
h. What is the median?
i. How many values are between 60 and 89, inclusive?
8. The following stem-and-leaf chart reports the number of
prescriptions filled per day
at the pharmacy on the corner of Fourth and Main Streets.
Stem Leaf
12 689
13 123
14 6889
15 589
16 35
17 24568
18 268
19 13456
20 034679
21 2239
22 789
23 00179
24 8
17. 25 13
26
27 0
a. How many days were studied?
b. How many observations are in the last class?
c. What are the maximum and the minimum values in the entire
set of data?
d. List the actual values in the fourth row.
e. List the actual values in the next to the last row.
f. On how many days were less than 160 prescriptions filled?
g. On how many days were 220 or more prescriptions filled?
h. What is the middle value?
i. How many days did the number of filled prescriptions range
between 170 and 210?
9. A survey of the number of phone calls made by a sample of
16 Verizon sub-
scribers last week revealed the following information. Develop
a stem-and-leaf
chart. How many calls did a typical subscriber make? What
were the maximum and
the minimum number of calls made?
52 43 30 38 30 42 12 46 39
37 34 46 32 18 41 5
10. Aloha Banking Co. is studying ATM use in suburban
Honolulu. Yesterday, for a
sample of 30 ATM's, the bank counted the number of times each
machine was
used. The data is presented in the table. Develop a stem-and-
leaf chart to summa-
rize the data. What were the typical, minimum, and maximum
number of times each
ATM was used?
18. 83 64 84 76 84 54 75 59 70 61
63 80 84 73 68 52 65 90 52 77
95 36 78 61 59 84 95 47 87 60
DESCRIBING DATA: DISPLAYING AND EXPLORING DATA
103
MEASURES OF POSITION
The standard deviation is the most widely used measure of
dispersion. However, there
are other ways of describing the variation or spread in a set of
data. One method is to
determine the location of values that divide a set of
observations into equal parts. These
measures include quartiles, deciles, and percentiles.
Quartiles divide a set of observations into four equal parts. To
explain further, think of
any set of values arranged from the minimum to the maximum.
In Chapter 3, we called the
middle value of a set of data arranged from the minimum to the
maximum the median.
That is, 50% of the observations are larger than the median and
50% are smaller. The
median is a measure of location because it pinpoints the center
of the data. In a similar
fashion, quartiles divide a set of observations into four equal
parts. The first quartile, usu-
ally labeled Q1, is the value below which 25% of the
observations occur, and the third
quartile, usually labeled Q3, is the value below which 75% of
the observations occur.
19. Similarly, deciles divide a set of observations into 10 equal
parts and percentiles
into 100 equal parts. So if you found that your GPA was in the
8th decile at your univer-
sity, you could conclude that 80% of the students had a GPA
lower than yours and 20%
had a higher GPA. If your GPA was in the 92nd percentile, then
92% of students had a
GPA less than your GPA and only 8% of students had a GPA
greater than your GPA. Per-
centile scores are frequently used to report results on such
national standardized tests
as the SAT, ACT, GMAT (used to judge entry into many master
of business administration
programs), and LSAT (used to judge entry into law school).
Quartiles, Deciles, and Percentiles
To formalize the computational procedure, let Lp refer to the
location of a desired percen-
tile. So if we want to find the 92nd percentile we would use
L92, and if we wanted the
median, the 50th percentile, then L50. For a number of
observations, n, the location of
the Pth percentile, can be found using the formula:
LO4-3
Identify and compute
measures of position.
LOCATION OF A PERCENTILE Lp = (n + 1)
P
100
[4–1]
20. An example will help to explain further.
E X A M P L E
Morgan Stanley is an investment company with offices located
throughout the
United States. Listed below are the commissions earned last
month by a sample of
15 brokers at the Morgan Stanley office in Oakland, California.
$2,038 $1,758 $1,721 $1,637 $2,097 $2,047 $2,205 $1,787
$2,287
1,940 2,311 2,054 2,406 1,471 1,460
Locate the median, the first quartile, and the third quartile for
the commissions
earned.
S O L U T I O N
The first step is to sort the data from the smallest commission to
the largest.
$1,460 $1,471 $1,637 $1,721 $1,758 $1,787 $1,940 $2,038
2,047 2,054 2,097 2,205 2,287 2,311 2,406
104 CHAPTER 4
In the above example, the location formula yielded a whole
number. That is, we
wanted to find the first quartile and there were 15 observations,
so the location formula
21. indicated we should find the fourth ordered value. What if there
were 20 observations
in the sample, that is n = 20, and we wanted to locate the first
quartile? From the loca-
tion formula (4–1):
L25 = (n + 1)
P
100
= (20 + 1)
25
100
= 5.25
We would locate the fifth value in the ordered array and then
move .25 of the distance
between the fifth and sixth values and report that as the first
quartile. Like the median,
the quartile does not need to be one of the actual values in the
data set.
To explain further, suppose a data set contained the six values
91, 75, 61, 101, 43,
and 104. We want to locate the first quartile. We order the
values from the minimum to
the maximum: 43, 61, 75, 91, 101, and 104. The first quartile is
located at
L25 = (n + 1)
P
100
= (6 + 1)
22. 25
100
= 1.75
The position formula tells us that the first quartile is located
between the first and the
second values and it is .75 of the distance between the first and
the second values. The
first value is 43 and the second is 61. So the distance between
these two values is 18.
To locate the first quartile, we need to move .75 of the distance
between the first and
second values, so .75(18) = 13.5. To complete the procedure, we
add 13.5 to the first
value, 43, and report that the first quartile is 56.5.
We can extend the idea to include both deciles and percentiles.
To locate the 23rd
percentile in a sample of 80 observations, we would look for the
18.63 position.
L23 = (n + 1)
P
100
= (80 + 1)
23
100
= 18.63
The median value is the observation in the
center and is the same as the 50th percen-
23. tile, so P equals 50. So the median or L50 is
located at (n + 1)(50/100), where n is the
number of observations. In this case, that is
position number 8, found by (15 + 1)
(50/100). The eighth-largest commission is
$2,038. So we conclude this is the median
and that half the brokers earned com-
missions more than $2,038 and half
earned less than $2,038. The result using
formula (4–1) to find the median is the same as the method
presented in
Chapter 3.
Recall the definition of a quartile. Quartiles divide a set of
observations into
four equal parts. Hence 25% of the observations will be less
than the first quartile.
Seventy-five percent of the observations will be less than the
third quartile. To
locate the first quartile, we use formula (4–1), where n = 15 and
P = 25:
L25 = (n + 1)
P
100
= (15 + 1)
25
100
= 4
and to locate the third quartile, n = 15 and P = 75:
25. provide summary statistics
that include quartiles. For example, the Minitab summary of the
Morgan Stanley com-
mission data, shown below, includes the first and third
quartiles, and other statistics.
Based on the reported quartiles, 25% of the commissions earned
were less than
$1,721 and 75% were less than $2,205. These are the same
values we calculated
using formula (4–1).
There are ways other than formula (4–1) to lo-
cate quartile values. For example, another method
uses 0.25n + 0.75 to locate the position of the first
quartile and 0.75n + 0.25 to locate the position of
the third quartile. We will call this the Excel Method.
In the Morgan Stanley data, this method would
place the first quartile at position 4.5 (.25 × 15 +
.75) and the third quartile at position 11.5 (.75 ×
15 + .25). The first quartile would be interpolated
as 0.5, or one-half the difference between the
fourth- and the fifth-ranked values. Based on this
method, the first quartile is $1739.5, found by
($1,721 + 0.5[$1,758 − $1,721]). The third quar-
tile, at position 11.5, would be $2,151, or one-half
the distance between the eleventh- and the
twelfth-ranked values, found by ($2,097 + 0.5[$2,205 −
$2,097]). Excel, as shown in
the Morgan Stanley and Applewood examples, can compute
quartiles using either of
the two methods. Please note the text uses formula (4–1) to
calculate quartiles.
Is the difference between the two methods important? No.
Usually it is just a nui-
26. sance. In general, both methods calculate values that will
support the statement that ap-
proximately 25% of the values are less than the value of the
first quartile, and approximately
75% of the data values are less than the value of the third
quartile. When the sample is
large, the difference in the results
from the two methods is small. For
example, in the Applewood Auto
Group data there are 180 vehicles.
The quartiles computed using both
methods are shown to the left. Based
on the variable profit, 45 of the
180 values (25%) are less than both
values of the first quartile, and 135 of
the 180 values (75%) are less than
both values of the third quartile.
When using Excel, be careful to
understand the method used to
STATISTICS IN ACTION
John W. Tukey (1915–2000)
received a PhD in mathe-
matics from Princeton in
1939. However, when he
joined the Fire Control Re-
search Office during World
War II, his interest in ab-
stract mathematics shifted
to applied statistics. He de-
veloped effective numerical
and graphical methods for
studying patterns in data.
27. Among the graphics he
developed are the stem-
and-leaf diagram and the
box-and-whisker plot or box
plot. From 1960 to 1980,
Tukey headed the statistical
division of NBC’s election
night vote projection team.
He became renowned in
1960 for preventing an
early call of victory for
Richard Nixon in the presi-
dential election won by
John F. Kennedy.
Morgan Stanley
Commissions
1460 Equation 4-1
2047
1471
Quartile 1
Quartile 3
1721
2205
Alternate Method
Quartile 1
Quartile 3
1739.5
2151
2054
29. $1,951
$2,692
$1,342
21
23
24
25
26
27
27
28
28
29
106 CHAPTER 4
calculate quartiles. Excel 2013 and Excel 2016 offer both
methods. The Excel function,
Quartile.exc, will result in the same answer as Equation 4–1.
The Excel function, Quar-
tile.inc, will result in the Excel Method answers.
The Quality Control department of Plainsville Peanut Company
is responsible for checking
the weight of the 8-ounce jar of peanut butter. The weights of a
sample of nine jars pro-
duced last hour are:
7.69 7.72 7.8 7.86 7.90 7.94 7.97 8.06 8.09
(a) What is the median weight?
(b) Determine the weights corresponding to the first and third
30. quartiles.
S E L F - R E V I E W 4–2
11. Determine the median and the first and third quartiles in
the following data.
46 47 49 49 51 53 54 54 55 55 59
12. Determine the median and the first and third quartiles in
the following data.
5.24 6.02 6.67 7.30 7.59 7.99 8.03 8.35 8.81 9.45
9.61 10.37 10.39 11.86 12.22 12.71 13.07 13.59 13.89 15.42
13. The Thomas Supply Company Inc. is a distributor of gas-
powered generators.
As with any business, the length of time customers take to pay
their invoices is im-
portant. Listed below, arranged from smallest to largest, is the
time, in days, for a
sample of The Thomas Supply Company Inc. invoices.
13 13 13 20 26 27 31 34 34 34 35 35 36 37 38
41 41 41 45 47 47 47 50 51 53 54 56 62 67 82
a. Determine the first and third quartiles.
b. Determine the second decile and the eighth decile.
c. Determine the 67th percentile.
14. Kevin Horn is the national sales manager for National
Textbooks Inc. He
has a sales staff of 40 who visit college professors all over the
United States.
Each Saturday morning he requires his sales staff to send him a
report. This re-
31. port includes, among other things, the number of professors
visited during the
previous week. Listed below, ordered from smallest to largest,
are the number
of visits last week.
38 40 41 45 48 48 50 50 51 51 52 52 53 54 55 55 55
56 56 57
59 59 59 62 62 62 63 64 65 66 66 67 67 69 69 71 77
78 79 79
a. Determine the median number of calls.
b. Determine the first and third quartiles.
c. Determine the first decile and the ninth decile.
d. Determine the 33rd percentile.
E X E R C I S E S
DESCRIBING DATA: DISPLAYING AND EXPLORING DATA
107
BOX PLOTS
A box plot is a graphical display, based on quartiles, that helps
us picture a set of data.
To construct a box plot, we need only five statistics: the
minimum value, Q1 (the first
quartile), the median, Q3 (the third quartile), and the maximum
value. An example will
help to explain.
LO4-4
Construct and analyze a
box plot.
32. E X A M P L E
Alexander’s Pizza offers free delivery of its pizza within 15
miles. Alex, the owner,
wants some information on the time it takes for delivery. How
long does a typical
delivery take? Within what range of times will most deliveries
be completed? For a
sample of 20 deliveries, he determined the following
information:
Minimum value = 13 minutes
Q1 = 15 minutes
Median = 18 minutes
Q3 = 22 minutes
Maximum value = 30 minutes
Develop a box plot for the delivery times. What conclusions can
you make about
the delivery times?
S O L U T I O N
The first step in drawing a box plot is to create an appropriate
scale along the
horizontal axis. Next, we draw a box that starts at Q1 (15
minutes) and ends at Q3
(22 minutes). Inside the box we place a vertical line to represent
the median (18
minutes). Finally, we extend horizontal lines from the box out
to the minimum
33. value (13 minutes) and the maximum value (30 minutes). These
horizontal lines
outside of the box are sometimes called “whiskers” because
they look a bit like a
cat’s whiskers.
12 14 16 18 20 22 24 26 28 30 32
Q1
Median
Q3
Minimum
value
Maximum
value
Minutes
The box plot also shows the interquartile range of delivery
times between
Q1 and Q3. The interquartile range is 7 minutes and indicates
that 50% of the
deliveries are between 15 and 22 minutes.
The box plot also reveals that the distribution of delivery times
is positively skewed.
In Chapter 3, we defined skewness as the lack of symmetry in a
set of data. How do we
know this distribution is positively skewed? In this case, there
are actually two pieces
of information that suggest this. First, the dashed line to the
right of the box from 22
minutes (Q3) to the maximum time of 30 minutes is longer than
34. the dashed line from
the left of 15 minutes (Q1) to the minimum value of 13 minutes.
To put it another way,
108 CHAPTER 4
the 25% of the data larger than the third quartile is more spread
out than the 25% less
than the first quartile. A second indication of positive skewness
is that the median is
not in the center of the box. The distance from the first quartile
to the median is smaller
than the distance from the median to the third quartile. We
know that the number of
delivery times between 15 minutes and 18 minutes is the same
as the number of de-
livery times between 18 minutes and 22 minutes.
E X A M P L E
Refer to the Applewood Auto Group data. Develop a box plot
for the variable age of
the buyer. What can we conclude about the distribution of the
age of the buyer?
S O L U T I O N
Minitab was used to develop the following chart and summary
statistics.
The median age of the purchaser is 46 years, 25% of the
purchasers are less than
40 years of age, and 25% are more than 52.75 years of age.
35. Based on the sum-
mary information and the box plot, we conclude:
• Fifty percent of the purchasers are between the ages of 40 and
52.75 years.
• The distribution of ages is fairly symmetric. There are two
reasons for this con-
clusion. The length of the whisker above 52.75 years (Q3) is
about the same
length as the whisker below 40 years (Q1). Also, the area in the
box between
40 years and the median of 46 years is about the same as the
area between
the median and 52.75.
There are three asterisks (*) above 70 years. What do they
indicate? In a box
plot, an asterisk identifies an outlier. An outlier is a value that
is inconsistent with
the rest of the data. It is defined as a value that is more than 1.5
times the inter-
quartile range smaller than Q1 or larger than Q3. In this
example, an outlier would
be a value larger than 71.875 years, found by:
Outlier > Q3 + 1.5(Q3 − Q1) = 52.75 + 1.5(52.75 − 40) =
71.875
An outlier would also be a value less than 20.875 years.
Outlier < Q1 − 1.5(Q3 − Q1) = 40 − 1.5(52.75 − 40) = 20.875
DESCRIBING DATA: DISPLAYING AND EXPLORING DATA
36. 109
The following box plot shows the assets in millions of dollars
for credit unions in Seattle,
Washington.
0 10 20 30 40 50 60 70 80 90 100
What are the smallest and largest values, the first and third
quartiles, and the median?
Would you agree that the distribution is symmetrical? Are there
any outliers?
S E L F - R E V I E W 4–3
From the box plot, we conclude there are three purchasers 72
years of age or
older and none less than 21 years of age. Technical note: In
some cases, a single
asterisk may represent more than one observation because of the
limitations of the
software and space available. It is a good idea to check the
actual data. In this in-
stance, there are three purchasers 72 years old or older; two are
72 and one is 73.
15. The box plot below shows the amount spent for books and
supplies per year by
students at four-year public colleges.
0 350 700 1,050 1,400 $1,750
a. Estimate the median amount spent.
b. Estimate the first and third quartiles for the amount spent.
c. Estimate the interquartile range for the amount spent.
37. d. Beyond what point is a value considered an outlier?
e. Identify any outliers and estimate their value.
f. Is the distribution symmetrical or positively or negatively
skewed?
16. The box plot shows the undergraduate in-state tuition per
credit hour at four-year
public colleges.
*
0 300 600 900 1,200 $1,500
a. Estimate the median.
b. Estimate the first and third quartiles.
c. Determine the interquartile range.
d. Beyond what point is a value considered an outlier?
e. Identify any outliers and estimate their value.
f. Is the distribution symmetrical or positively or negatively
skewed?
17. In a study of the gasoline mileage of model year 2016
automobiles, the mean miles
per gallon was 27.5 and the median was 26.8. The smallest
value in the study was
12.70 miles per gallon, and the largest was 50.20. The first and
third quartiles were
17.95 and 35.45 miles per gallon, respectively. Develop a box
plot and comment
on the distribution. Is it a symmetric distribution?
E X E R C I S E S
110 CHAPTER 4
38. SKEWNESS
In Chapter 3, we described measures of central location for a
distribution of data by re-
porting the mean, median, and mode. We also described
measures that show the amount
of spread or variation in a distribution, such as the range and
the standard deviation.
Another characteristic of a distribution is the shape. There are
four shapes com-
monly observed: symmetric, positively skewed, negatively
skewed, and bimodal. In a
symmetric distribution the mean and median are equal and the
data values are evenly
spread around these values. The shape of the distribution below
the mean and median
is a mirror image of distribution above the mean and median. A
distribution of values is
skewed to the right or positively skewed if there is a single
peak, but the values extend
much farther to the right of the peak than to the left of the peak.
In this case, the mean
is larger than the median. In a negatively skewed distribution
there is a single peak, but
the observations extend farther to the left, in the negative
direction, than to the right. In
a negatively skewed distribution, the mean is smaller than the
median. Positively
skewed distributions are more common. Salaries often follow
this pattern. Think of the
salaries of those employed in a small company of about 100
people. The president and
a few top executives would have very large salaries relative to
the other workers and
39. hence the distribution of salaries would exhibit positive
skewness. A bimodal distribu-
tion will have two or more peaks. This is often the case when
the values are from two or
more populations. This information is summarized in Chart 4–1.
LO4-5
Compute and interpret
the coefficient of
skewness.
M
ed
ia
n
M
ea
n
45
Fr
eq
ue
nc
y
Fr
eq
ue
41. n
M
ea
n
Median
Mean
Test Scores
Negatively Skewed
75 80 Score
Mean
Outside Diameter
Bimodal
.98 1.04 Inches$
CHART 4–1 Shapes of Frequency Polygons
There are several formulas in the statistical literature used to
calculate skewness.
The simplest, developed by Professor Karl Pearson (1857–
1936), is based on the differ-
ence between the mean and the median.
18. A sample of 28 time shares in the Orlando, Florida, area
revealed the follow-
ing daily charges for a one-bedroom suite. For convenience, the
data are ordered
42. from smallest to largest. Construct a box plot to represent the
data. Comment on
the distribution. Be sure to identify the first and third quartiles
and the median.
$116 $121 $157 $192 $207 $209 $209
229 232 236 236 239 243 246
260 264 276 281 283 289 296
307 309 312 317 324 341 353
DESCRIBING DATA: DISPLAYING AND EXPLORING DATA
111
Using this relationship, the coefficient of skewness can range
from −3 up to 3. A value
near −3, such as −2.57, indicates considerable negative
skewness. A value such as
1.63 indicates moderate positive skewness. A value of 0, which
will occur when the
mean and median are equal, indicates the distribution is
symmetrical and there is no
skewness present.
In this text, we present output from Minitab and Excel. Both of
these software pack-
ages compute a value for the coefficient of skewness based on
the cubed deviations
from the mean. The formula is:
SOFTWARE COEFFICIENT OF SKEWNESS
sk =
n
43. (n − 1) (n − 2)[
∑(
x − x
s )
3
] [4–3]
Formula (4–3) offers an insight into skewness. The right-hand
side of the formula is
the difference between each value and the mean, divided by the
standard deviation.
That is the portion (x − x )/s of the formula. This idea is called
standardizing. We will
discuss the idea of standardizing a value in more detail in
Chapter 7 when we describe
the normal probability distribution. At this point, observe that
the result is to report the
difference between each value and the mean in units of the
standard deviation. If this
difference is positive, the particular value is larger than the
mean; if the value is nega-
tive, the standardized quantity is smaller than the mean. When
we cube these values,
we retain the information on the direction of the difference.
Recall that in the formula for
the standard deviation [see formula (3–10)] we squared the
difference between each
value and the mean, so that the result was all nonnegative
values.
If the set of data values under consideration is symmetric, when
we cube the stan-
44. dardized values and sum over all the values, the result would be
near zero. If there are
several large values, clearly separate from the others, the sum
of the cubed differences
would be a large positive value. If there are several small values
clearly separate from
the others, the sum of the cubed differences will be negative.
An example will illustrate the idea of skewness.
PEARSON’S COEFFICIENT OF SKEWNESS sk =
3(x − Median)
s
[4–2]
STATISTICS IN ACTION
The late Stephen Jay Gould
(1941–2002) was a profes-
sor of zoology and professor
of geology at Harvard
University. In 1982, he was
diagnosed with cancer and
had an expected survival
time of 8 months. However,
never to be discouraged,
his research showed that
the distribution of survival
time is dramatically skewed
to the right and showed that
not only do 50% of similar
cancer patients survive
more than 8 months, but
that the survival time could
be years rather than months!
45. In fact, Dr. Gould lived an-
other 20 years. Based on
his experience, he wrote a
widely published essay
titled “The Median Is Not
the Message.”
E X A M P L E
Following are the earnings per share for a sample of 15 software
companies for the
year 2016. The earnings per share are arranged from smallest to
largest.
Compute the mean, median, and standard deviation. Find the
coefficient of
skewness using Pearson’s estimate and the software methods.
What is your
conclusion regarding the shape of the distribution?
S O L U T I O N
These are sample data, so we use formula (3–2) to determine the
mean
x =
Σx
n
=
$74.26
15
= $4.95
$0.09 $0.13 $0.41 $0.51 $ 1.12 $ 1.20 $ 1.49 $3.18
46. 3.50 6.36 7.83 8.92 10.13 12.99 16.40
112 CHAPTER 4
The median is the middle value in a set of data, arranged from
smallest to largest.
In this case, there is an odd-number of observations, so the
middle value is the
median. It is $3.18.
We use formula (3–10) on page 78 to determine the sample
standard deviation.
s = √
Σ(x − x )2
n − 1
= √
($0.09 − $4.95)2 + … + ($16.40 − $4.95)2
15 − 1
= $5.22
Pearson’s coefficient of skewness is 1.017, found by
sk =
3(x − Median)
s
=
3($4.95 − $3.18)
47. $5.22
= 1.017
This indicates there is moderate positive skewness in the
earnings per share data.
We obtain a similar, but not exactly the same, value from the
software method.
The details of the calculations are shown in Table 4–2. To
begin, we find the differ-
ence between each earnings per share value and the mean and
divide this result
by the standard deviation. We have referred to this as
standardizing. Next, we cube,
that is, raise to the third power, the result of the first step.
Finally, we sum the cubed
values. The details for the first company, that is, the company
with an earnings per
share of $0.09, are:
(
x − x
s )
3
= (
0.09 − 4.95
5.22 )
3
= (−0.9310)3 = −0.8070
When we sum the 15 cubed values, the result is 11.8274. That
48. is, the term
Σ[(x − x )/s]3 = 11.8274. To find the coefficient of skewness,
we use formula (4–3),
with n = 15.
sk =
n
(n − 1) (n − 2)
∑(
x − x
s )
3
=
15
(15 − 1) (15 − 2)
(11.8274) = 0.975
We conclude that the earnings per share values are somewhat
positively
skewed. The following Minitab summary reports the descriptive
measures, such as
TABLE 4–2 Calculation of the Coefficient of Skewness
Earnings per Share
(x − x )
s
(
x − x
49. s )
3
0.09 −0.9310 −0.8070
0.13 −0.9234 −0.7873
0.41 −0.8697 −0.6579
0.51 −0.8506 −0.6154
1.12 −0.7337 −0.3950
1.20 −0.7184 −0.3708
1.49 −0.6628 −0.2912
3.18 −0.3391 −0.0390
3.50 −0.2778 −0.0214
6.36 0.2701 0.0197
7.83 0.5517 0.1679
8.92 0.7605 0.4399
10.13 0.9923 0.9772
12.99 1.5402 3.6539
16.40 2.1935 10.5537
11.8274
DESCRIBING DATA: DISPLAYING AND EXPLORING DATA
113
A sample of five data entry clerks employed in the Horry
County Tax Office revised the fol-
lowing number of tax records last hour: 73, 98, 60, 92, and 84.
(a) Find the mean, median, and the standard deviation.
(b) Compute the coefficient of skewness using Pearson’s
method.
(c) Calculate the coefficient of skewness using the software
50. method.
(d) What is your conclusion regarding the skewness of the
data?
S E L F - R E V I E W 4–4
For Exercises 19–22:
a. Determine the mean, median, and the standard deviation.
b. Determine the coefficient of skewness using Pearson’s
method.
c. Determine the coefficient of skewness using the software
method.
19. The following values are the starting salaries, in $000, for a
sample of five
accounting graduates who accepted positions in public
accounting last year.
36.0 26.0 33.0 28.0 31.0
20. Listed below are the salaries, in $000, for a sample of 15
chief financial offi-
cers in the electronics industry.
$516.0 $548.0 $566.0 $534.0 $586.0 $529.0
546.0 523.0 538.0 523.0 551.0 552.0
486.0 558.0 574.0
E X E R C I S E S
the mean, median, and standard deviation of the earnings per
share data. Also in-
cluded are the coefficient of skewness and a histogram with a
bell-shaped curve
superimposed.
51. 114 CHAPTER 4
DESCRIBING THE RELATIONSHIP BETWEEN
TWO VARIABLES
In Chapter 2 and the first section of this chapter, we presented
graphical techniques
to summarize the distribution of a single variable. We used a
histogram in Chapter 2
to summarize the profit on vehicles sold by the Applewood Auto
Group. Earlier in
this chapter, we used dot plots and stem-and-leaf
displays to visually summarize a set of data. Because
we are studying a single variable, we refer to this as
univariate data.
There are situations where we wish to study and
visually portray the relationship between two vari-
ables. When we study the relationship between two
variables, we refer to the data as bivariate. Data ana-
lysts frequently wish to understand the relationship
between two variables. Here are some examples:
• Tybo and Associates is a law firm that advertises ex-
tensively on local TV. The partners are considering
increasing their advertising budget. Before doing
so, they would like to know the relationship be-
tween the amount spent per month on advertising
and the total amount of billings for that month. To
put it another way, will increasing the amount spent
on advertising result in an increase in billings?
53. DESCRIBING DATA: DISPLAYING AND EXPLORING DATA
115
• Coastal Realty is studying the selling prices of homes. What
variables seem to be
related to the selling price of homes? For example, do larger
homes sell for more
than smaller ones? Probably. So Coastal might study the
relationship between the
area in square feet and the selling price.
• Dr. Stephen Givens is an expert in human development. He is
studying the relation-
ship between the height of fathers and the height of their sons.
That is, do tall fathers
tend to have tall children? Would you expect LeBron James, the
6′8″, 250 pound
professional basketball player, to have relatively tall sons?
One graphical technique we use to show the relationship
between variables is called a
scatter diagram.
To draw a scatter diagram, we need two variables. We scale one
variable
along the horizontal axis (X-axis) of a graph and the other
variable along the vertical
axis (Y-axis). Usually one variable depends to some degree on
the other. In the
third example above, the height of the son depends on the
height of the father. So
we scale the height of the father on the horizontal axis and that
of the son on the
vertical axis.
54. We can use statistical software, such as Excel, to perform the
plotting function for
us. Caution: You should always be careful of the scale. By
changing the scale of either
the vertical or the horizontal axis, you can affect the apparent
visual strength of the
relationship.
Following are three scatter diagrams (Chart 4–2). The one on
the left shows a
rather strong positive relationship between the age in years and
the maintenance
cost last year for a sample of 10 buses owned by the city of
Cleveland, Ohio. Note
that as the age of the bus increases, the yearly maintenance cost
also increases. The
example in the center, for a sample of 20 vehicles, shows a
rather strong indirect rela-
tionship between the odometer reading and the auction price.
That is, as the number
of miles driven increases, the auction price decreases. The
example on the right de-
picts the relationship between the height and yearly salary for a
sample of 15 shift
supervisors. This graph indicates there is little relationship
between their height and
yearly salary.
$24,000
21,000
18,000
15,000
12,000A
uc
55. tio
n
pr
ic
e
10,000 30,000 50,000
Odometer
Auction Price versus Odometer
$10,000
8,000
6,000
4,000
2,000
0
Co
st
(a
nn
ua
l)
0 1 2 3 4 5 6
Age (years)
Age of Buses and
Maintenance Cost Height versus Salary
56. 125
120
115
110
105
100
95
90S
al
ar
y
($
00
0)
54 55 56 57 58 59 60 61 62 63
Height (inches)
CHART 4–2 Three Examples of Scatter Diagrams.
E X A M P L E
In the introduction to Chapter 2, we presented data from the
Applewood Auto
Group. We gathered information concerning several variables,
including the profit
earned from the sale of 180 vehicles sold last month. In addition
to the amount of
profit on each sale, one of the other variables is the age of the
purchaser. Is there a
relationship between the profit earned on a vehicle sale and the
age of the pur-
57. chaser? Would it be reasonable to conclude that more profit is
made on vehicles
purchased by older buyers?
116 CHAPTER 4
In the preceding example, there is a weak positive, or direct,
relationship between the
variables. There are, however, many instances where there is a
relationship between
the variables, but that relationship is inverse or negative. For
example:
• The value of a vehicle and the number of miles driven. As the
number of miles in-
creases, the value of the vehicle decreases.
• The premium for auto insurance and the age of the driver.
Auto rates tend to be the
highest for younger drivers and less for older drivers.
• For many law enforcement personnel, as the number of years
on the job increases,
the number of traffic citations decreases. This may be because
personnel become
more liberal in their interpretations or they may be in supervisor
positions and not
in a position to issue as many citations. But in any event, as age
increases, the num-
ber of citations decreases.
CONTINGENCY TABLES
A scatter diagram requires that both of the variables be at least
58. interval scale. In the
Applewood Auto Group example, both age and vehicle profit
are ratio scale variables.
Height is also ratio scale as used in the discussion of the
relationship between the
height of fathers and the height of their sons. What if we wish
to study the relationship
between two variables when one or both are nominal or ordinal
scale? In this case, we
tally the results in a contingency table.
LO4-7
Develop and explain a
contingency table.
S O L U T I O N
We can investigate the relationship between vehicle profit and
the age of the buyer
with a scatter diagram. We scale age on the horizontal, or X-
axis, and the profit on
the vertical, or Y-axis. We assume profit depends on the age of
the purchaser. As
people age, they earn more income and purchase more
expensive cars which, in
turn, produce higher profits. We use Excel to develop the
scatter diagram. The
Excel commands are in Appendix C.
The scatter diagram shows a rather weak positive relationship
between the two
variables. It does not appear there is much relationship between
the vehicle profit
and the age of the buyer. In Chapter 13, we will study the
relationship between
variables more extensively, even calculating several numerical
59. measures to ex-
press the relationship between variables.
0 10 20 30 40
Age (Years)
Profit and Age of Buyer at Applewood Auto Group
Pr
ofi
t p
er
V
eh
ic
le
($
)
50 60 70 80
$0
$500
$1,000
$1,500
$2,000
$2,500
60. $3,000
$3,500
DESCRIBING DATA: DISPLAYING AND EXPLORING DATA
117
A contingency table is a cross-tabulation that simultaneously
summarizes two variables
of interest. For example:
• Students at a university are classified by gender and class
(freshman, sophomore,
junior, or senior).
• A product is classified as acceptable or unacceptable and by
the shift (day, after-
noon, or night) on which it is manufactured.
• A voter in a school bond referendum is classified as to party
affiliation (Democrat,
Republican, other) and the number of children that voter has
attending school in the
district (0, 1, 2, etc.).
CONTINGENCY TABLE A table used to classify observations
according to two
identifiable characteristics.
E X A M P L E
There are four dealerships in the Applewood Auto Group.
Suppose we want to com-
61. pare the profit earned on each vehicle sold by the particular
dealership. To put it
another way, is there a relationship between the amount of
profit earned and the
dealership?
S O L U T I O N
In a contingency table, both variables only need to be nominal
or ordinal. In this
example, the variable dealership is a nominal variable and the
variable profit is a
ratio variable. To convert profit to an ordinal variable, we
classify the variable profit
into two categories, those cases where the profit earned is more
than the median
and those cases where it is less. On page 64, we calculated the
median profit for all
sales last month at Applewood Auto Group to be $1,882.50.
Contingency Table Showing the Relationship between Profit
and Dealership
Above/Below
Median Profit Kane Olean Sheffield Tionesta Total
Above 25 20 19 26 90
Below 27 20 26 17 90
Total 52 40 45 43 180
By organizing the information into a contingency table, we can
compare the profit
at the four dealerships. We observe the following:
• From the Total column on the right, 90 of the 180 cars sold
62. had a profit above
the median and half below. From the definition of the median,
this is
expected.
• For the Kane dealership, 25 out of the 52, or 48%, of the cars
sold were sold
for a profit more than the median.
• The percentage of profits above the median for the other
dealerships are 50%
for Olean, 42% for Sheffield, and 60% for Tionesta.
We will return to the study of contingency tables in Chapter 5
during the study of
probability and in Chapter 15 during the study of nonparametric
methods of analysis.
118 CHAPTER 4
The rock group Blue String Beans is touring the United States.
The following chart shows
the relationship between concert seating capacity and revenue in
$000 for a sample of
concerts.
5800 6300 6800
Seating Capacity
8
7
63. 6
5
4
3
2
Am
ou
nt
($
00
0)
7300
(a) What is the diagram called?
(b) How many concerts were studied?
(c) Estimate the revenue for the concert with the largest seating
capacity.
(d) How would you characterize the relationship between
revenue and seating capacity?
Is it strong or weak, direct or inverse?
S E L F - R E V I E W 4–5
23. Develop a scatter diagram for the following sample data.
How would you
describe the relationship between the values?
64. x-Value y-Value x-Value y-Value
10 6 11 6
8 2 10 5
9 6 7 2
11 5 7 3
13 7 11 7
24. Silver Springs Moving and Storage Inc. is studying the
relationship between the
number of rooms in a move and the number of labor hours
required for the move.
As part of the analysis, the CFO of Silver Springs developed the
following scatter
diagram.
1 2 3
Rooms
40
30
20
10
0
Ho
ur
s
54
65. E X E R C I S E S
DESCRIBING DATA: DISPLAYING AND EXPLORING DATA
119
a. How many moves are in the sample?
b. Does it appear that more labor hours are required as the
number of rooms
increases, or do labor hours decrease as the number of rooms
increases?
25. The Director of Planning for Devine Dining Inc. wishes to
study the relationship be-
tween the gender of a guest and whether the guest orders
dessert. To investigate the
relationship, the manager collected the following information
on 200 recent customers.
Gender
Dessert Ordered Male Female Total
Yes 32 15 47
No 68 85 153
Total 100 100 200
a. What is the level of measurement of the two variables?
b. What is the above table called?
c. Does the evidence in the table suggest men are more likely
to order dessert
66. than women? Explain why.
26. Ski Resorts of Vermont Inc. is considering a merger with
Gulf Shores Beach Resorts
Inc. of Alabama. The board of directors surveyed 50
stockholders concerning their
position on the merger. The results are reported below.
Opinion
Number of Shares Held Favor Oppose Undecided Total
Under 200 8 6 2 16
200 up to 1,000 6 8 1 15
Over 1,000 6 12 1 19
Total 20 26 4 50
a. What level of measurement is used in this table?
b. What is this table called?
c. What group seems most strongly opposed to the merger?
C H A P T E R S U M M A R Y
I. A dot plot shows the range of values on the horizontal axis
and the number of observa-
tions for each value on the vertical axis.
A. Dot plots report the details of each observation.
B. They are useful for comparing two or more data sets.
II. A stem-and-leaf display is an alternative to a histogram.
A. The leading digit is the stem and the trailing digit the leaf.
B. The advantages of a stem-and-leaf display over a histogram
include:
67. 1. The identity of each observation is not lost.
2. The digits themselves give a picture of the distribution.
3. The cumulative frequencies are also shown.
III. Measures of location also describe the shape of a set of
observations.
A. Quartiles divide a set of observations into four equal parts.
1. Twenty-five percent of the observations are less than the first
quartile, 50% are
less than the second quartile, and 75% are less than the third
quartile.
2. The interquartile range is the difference between the third
quartile and the first
quartile.
B. Deciles divide a set of observations into 10 equal parts and
percentiles into 100
equal parts.
120 CHAPTER 4
IV. A box plot is a graphic display of a set of data.
A. A box is drawn enclosing the regions between the first
quartile and the third quartile.
1. A line is drawn inside the box at the median value.
2. Dotted line segments are drawn from the third quartile to the
largest value to
show the highest 25% of the values and from the first quartile to
the smallest
68. value to show the lowest 25% of the values.
B. A box plot is based on five statistics: the maximum and
minimum values, the first and
third quartiles, and the median.
V. The coefficient of skewness is a measure of the symmetry of
a distribution.
A. There are two formulas for the coefficient of skewness.
1. The formula developed by Pearson is:
sk =
3(x − Median)
s
[4–2]
2. The coefficient of skewness computed by statistical software
is:
sk =
n
(n − 1) (n − 2)[
∑(
x − x
s )
3
] [4–3]
VI. A scatter diagram is a graphic tool to portray the
relationship between two variables.
69. A. Both variables are measured with interval or ratio scales.
B. If the scatter of points moves from the lower left to the upper
right, the variables un-
der consideration are directly or positively related.
C. If the scatter of points moves from the upper left to the lower
right, the variables are
inversely or negatively related.
VII. A contingency table is used to classify nominal-scale
observations according to two
characteristics.
P R O N U N C I A T I O N K E Y
SYMBOL MEANING PRONUNCIATION
Lp Location of percentile L sub p
Q1 First quartile Q sub 1
Q3 Third quartile Q sub 3
C H A P T E R E X E R C I S E S
27. A sample of students attending Southeast Florida
University is asked the number of so-
cial activities in which they participated last week. The chart
below was prepared from
the sample data.
41 2
Activities
30
70. a. What is the name given to this chart?
b. How many students were in the study?
c. How many students reported attending no social activities?
28. Doctor’s Care is a walk-in clinic, with locations in
Georgetown, Moncks Corner, and
Aynor, at which patients may receive treatment for minor
injuries, colds, and flu, as well
as physical examinations. The following charts report the
number of patients treated in
each of the three locations last month.
DESCRIBING DATA: DISPLAYING AND EXPLORING DATA
121
5020 30
Patients
4010
Location
Georgetown
Moncks Corner
Aynor
Describe the number of patients served at the three locations
each day. What are the
maximum and minimum numbers of patients served at each of
the locations?
71. 29. Below is the number of customers who visited Smith’s
True-Value hardware store
in Bellville, Ohio, over the last twenty-three days. Make a stem-
and-leaf display of this
variable.
46 52 46 40 42 46 40 37 46 40 52 32 37 32 52
40 32 52 40 52 46 46 52
30. The top 25 companies (by market capitalization) operating
in the Washington, DC,
area along with the year they were founded and the number of
employees are given
below. Make a stem-and-leaf display of each of these variables
and write a short de-
scription of your findings.
Company Name Year Founded Employees
AES Corp. 1981 30,000
American Capital Ltd. 1986 484
AvalonBay Communities Inc. 1978 1,767
Capital One Financial Corp. 1995 31,800
Constellation Energy Group Inc. 1816 9,736
Coventry Health Care Inc. 1986 10,250
Danaher Corp. 1984 45,000
Dominion Resources Inc. 1909 17,500
Fannie Mae 1938 6,450
Freddie Mac 1970 5,533
Gannett Co. 1906 49,675
General Dynamics Corp. 1952 81,000
Genworth Financial Inc. 2004 7,200
Harman International Industries Inc. 1980 11,246
Host Hotels & Resorts Inc. 1927 229
Legg Mason 1899 3,800
72. Lockheed Martin Corp. 1995 140,000
Marriott International Inc. 1927 151,000
MedImmune LLC 1988 2,516
NII Holdings Inc. 1996 7,748
Norfolk Southern Corp. 1982 30,594
Pepco Holdings Inc. 1896 5,057
Sallie Mae 1972 11,456
T. Rowe Price Group Inc. 1937 4,605
The Washington Post Co. 1877 17,100
31. In recent years, due to low interest rates, many
homeowners refinanced their
home mortgages. Linda Lahey is a mortgage officer at Down
River Federal Savings
122 CHAPTER 4
and Loan. Below is the amount refinanced for 20 loans she
processed last week.
The data are reported in thousands of dollars and arranged from
smallest to
largest.
59.2 59.5 61.6 65.5 66.6 72.9 74.8 77.3 79.2
83.7 85.6 85.8 86.6 87.0 87.1 90.2 93.3 98.6
100.2 100.7
a. Find the median, first quartile, and third quartile.
b. Find the 26th and 83rd percentiles.
c. Draw a box plot of the data.
32. A study is made by the recording industry in the United
States of the number
73. of music CDs owned by 25 senior citizens and 30 young adults.
The information is
reported below.
Seniors
28 35 41 48 52 81 97 98 98 99
118 132 133 140 145 147 153 158 162 174
177 180 180 187 188
Young Adults
81 107 113 147 147 175 183 192 202 209
233 251 254 266 283 284 284 316 372 401
417 423 490 500 507 518 550 557 590 594
a. Find the median and the first and third quartiles for the
number of CDs owned by
senior citizens. Develop a box plot for the information.
b. Find the median and the first and third quartiles for the
number of CDs owned by
young adults. Develop a box plot for the information.
c. Compare the number of CDs owned by the two groups.
33. The corporate headquarters of Bank.com, an on-line
banking company, is located
in downtown Philadelphia. The director of human resources is
making a study of the
time it takes employees to get to work. The city is planning to
offer incentives to each
downtown employer if they will encourage their employees to
use public transportation.
Below is a listing of the time to get to work this morning
according to whether the em-
74. ployee used public transportation or drove a car.
Public Transportation
23 25 25 30 31 31 32 33 35 36
37 42
Private
32 32 33 34 37 37 38 38 38 39
40 44
a. Find the median and the first and third quartiles for the time
it took employees using
public transportation. Develop a box plot for the information.
b. Find the median and the first and third quartiles for the time
it took employees who
drove their own vehicle. Develop a box plot for the
information.
c. Compare the times of the two groups.
34. The following box plot shows the number of daily
newspapers published in each
state and the District of Columbia. Write a brief report
summarizing the number pub-
lished. Be sure to include information on the values of the first
and third quartiles,
DESCRIBING DATA: DISPLAYING AND EXPLORING DATA
123
75. the median, and whether there is any skewness. If there are any
outliers, estimate
their value.
Number of Newspapers
****
0 20 40 60 80 100
35. Walter Gogel Company is an industrial supplier of
fasteners, tools, and springs. The
amounts of its invoices vary widely, from less than $20.00 to
more than $400.00. During
the month of January the company sent out 80 invoices. Here is
a box plot of these in-
voices. Write a brief report summarizing the invoice amounts.
Be sure to include infor-
mation on the values of the first and third quartiles, the median,
and whether there is any
skewness. If there are any outliers, approximate the value of
these invoices.
Invoice Amount
*
0 50 100 150 200 250
36. The American Society of PeriAnesthesia Nurses (ASPAN;
www.aspan.org) is a
national organization serving nurses practicing in ambulatory
surgery, preanesthesia, and
postanesthesia care. The organization consists of the 40
components listed below.
76. State/Region Membership
Alabama 95
Arizona 399
Maryland, Delaware, DC 531
Connecticut 239
Florida 631
Georgia 384
Hawaii 73
Illinois 562
Indiana 270
Iowa 117
Kentucky 197
Louisiana 258
Michigan 411
Massachusetts 480
Maine 97
Minnesota, Dakotas 289
Missouri, Kansas 282
Mississippi 90
Nebraska 115
North Carolina 542
Nevada 106
State/Region Membership
New Jersey, Bermuda 517
Alaska, Idaho, Montana,
Oregon, Washington 708
New York 891
Ohio 708
Oklahoma 171
Arkansas 68
California 1,165
New Mexico 79
Pennsylvania 575
77. Rhode Island 53
Colorado 409
South Carolina 237
Texas 1,026
Tennessee 167
Utah 67
Virginia 414
Vermont,
New Hampshire 144
Wisconsin 311
West Virginia 62
Use statistical software to answer the following questions.
a. Find the mean, median, and standard deviation of the number
of members per
component.
124 CHAPTER 4
b. Find the coefficient of skewness, using the software. What do
you conclude about
the shape of the distribution of component size?
c. Compute the first and third quartiles using formula (4–1).
d. Develop a box plot. Are there any outliers? Which
components are outliers? What are
the limits for outliers?
37. McGivern Jewelers is located in the Levis Square Mall just
south of Toledo, Ohio.
Recently it posted an advertisement on a social media site
78. reporting the shape, size,
price, and cut grade for 33 of its diamonds currently in stock.
The information is re-
ported below.
Shape Size (carats) Price Cut Grade Shape Size (carats) Price
Cut Grade
Princess 5.03 $44,312 Ideal cut Round 0.77 $2,828 Ultra ideal
cut
Round 2.35 20,413 Premium cut Oval 0.76 3,808 Premium cut
Round 2.03 13,080 Ideal cut Princess 0.71 2,327 Premium cut
Round 1.56 13,925 Ideal cut Marquise 0.71 2,732 Good cut
Round 1.21 7,382 Ultra ideal cut Round 0.70 1,915 Premium cut
Round 1.21 5,154 Average cut Round 0.66 1,885 Premium cut
Round 1.19 5,339 Premium cut Round 0.62 1,397 Good cut
Emerald 1.16 5,161 Ideal cut Round 0.52 2,555 Premium cut
Round 1.08 8,775 Ultra ideal cut Princess 0.51 1,337 Ideal cut
Round 1.02 4,282 Premium cut Round 0.51 1,558 Premium cut
Round 1.02 6,943 Ideal cut Round 0.45 1,191 Premium cut
Marquise 1.01 7,038 Good cut Princess 0.44 1,319 Average cut
Princess 1.00 4,868 Premium cut Marquise 0.44 1,319 Premium
cut
Round 0.91 5,106 Premium cut Round 0.40 1,133 Premium cut
Round 0.90 3,921 Good cut Round 0.35 1,354 Good cut
Round 0.90 3,733 Premium cut Round 0.32 896 Premium cut
Round 0.84 2,621 Premium cut
a. Develop a box plot of the variable price and comment on the
result. Are there any
outliers? What is the median price? What are the values of the
first and the third
quartiles?
b. Develop a box plot of the variable size and comment on the
result. Are there any
79. outliers? What is the median price? What are the values of the
first and the third
quartiles?
c. Develop a scatter diagram between the variables price and
size. Be sure to put price
on the vertical axis and size on the horizontal axis. Does there
seem to be an associ-
ation between the two variables? Is the association direct or
indirect? Does any point
seem to be different from the others?
d. Develop a contingency table for the variables shape and cut
grade. What is the most
common cut grade? What is the most common shape? What is
the most common
combination of cut grade and shape?
38. Listed below is the amount of commissions earned last
month for the eight mem-
bers of the sales staff at Best Electronics. Calculate the
coefficient of skewness using
both methods. Hint: Use of a spreadsheet will expedite the
calculations.
980.9 1,036.5 1,099.5 1,153.9 1,409.0 1,456.4 1,718.4 1,721.2
39. Listed below is the number of car thefts in a large city
over the last week. Calculate
the coefficient of skewness using both methods. Hint: Use of a
spreadsheet will expe-
dite the calculations.
3 12 13 7 8 3 8
80. DESCRIBING DATA: DISPLAYING AND EXPLORING DATA
125
40. The manager of Information Services at Wilkin
Investigations, a private investigation firm,
is studying the relationship between the age (in months) of a
combination printer, copier,
and fax machine and its monthly maintenance cost. For a sample
of 15 machines, the
manager developed the following chart. What can the manager
conclude about the re-
lationship between the variables?
34 39 44
Months
$130
120
110
100
90
80
M
on
th
ly
81. M
ai
nt
en
an
ce
C
os
t
49
41. An auto insurance company reported the following
information regarding the age
of a driver and the number of accidents reported last year.
Develop a scatter diagram for
the data and write a brief summary.
Age Accidents Age Accidents
16 4 23 0
24 2 27 1
18 5 32 1
17 4 22 3
42. Wendy’s offers eight different condiments (mustard,
catsup, onion, mayonnaise, pickle,
lettuce, tomato, and relish) on hamburgers. A store manager
collected the following in-
formation on the number of condiments ordered and the age
group of the customer.
What can you conclude regarding the information? Who tends to
order the most or least
82. number of condiments?
Age
Number of Condiments Under 18 18 up to 40 40 up to 60 60 or
older
0 12 18 24 52
1 21 76 50 30
2 39 52 40 12
3 or more 71 87 47 28
43. Here is a table showing the number of employed and
unemployed workers 20 years or
older by gender in the United States.
Number of Workers (000)
Gender Employed Unemployed
Men 70,415 4,209
Women 61,402 3,314
a. How many workers were studied?
b. What percent of the workers were unemployed?
c. Compare the percent unemployed for the men and the
women.
126 A REVIEW OF CHAPTERS 1–4
D A T A A N A L Y T I C S
44. Refer to the North Valley real estate data recorded on
83. homes sold during the last
year. Prepare a report on the selling prices of the homes based
on the answers to the
following questions.
a. Compute the minimum, maximum, median, and the first and
the third quartiles of
price. Create a box plot. Comment on the distribution of home
prices.
b. Develop a scatter diagram with price on the vertical axis and
the size of the home on
the horizontal. Is there a relationship between these variables?
Is the relationship
direct or indirect?
c. For homes without a pool, develop a scatter diagram with
price on the vertical axis
and the size of the home on the horizontal. Do the same for
homes with a pool. How
do the relationships between price and size for homes without a
pool and homes
with a pool compare?
45. Refer to the Baseball 2016 data that report information on
the 30 Major League
Baseball teams for the 2016 season.
a. In the data set, the year opened, is the first year of operation
for that stadium. For
each team, use this variable to create a new variable, stadium
age, by subtracting
the value of the variable, year opened, from the current year.
Develop a box plot
with the new variable, age. Are there any outliers? If so, which
of the stadiums are
84. outliers?
b. Using the variable, salary, create a box plot. Are there any
outliers? Compute the
quartiles using formula (4–1). Write a brief summary of your
analysis.
c. Draw a scatter diagram with the variable, wins, on the
vertical axis and salary on the
horizontal axis. What are your conclusions?
d. Using the variable, wins, draw a dot plot. What can you
conclude from this plot?
46. Refer to the Lincolnville School District bus data.
a. Referring to the maintenance cost variable, develop a box
plot. What are the mini-
mum, first quartile, median, third quartile, and maximum
values? Are there any
outliers?
b. Using the median maintenance cost, develop a contingency
table with bus manufac-
turer as one variable and whether the maintenance cost was
above or below the
median as the other variable. What are your conclusions?
A REVIEW OF CHAPTERS 1–4
This section is a review of the major concepts and terms
introduced in Chapters 1–4. Chapter 1 began by describing the
meaning and purpose of statistics. Next we described the
different types of variables and the four levels of measurement.
Chapter 2 was concerned with describing a set of observations
by organizing it into a frequency distribution and then
portraying the frequency distribution as a histogram or a
frequency polygon. Chapter 3 began by describing measures of
85. location, such as the mean, weighted mean, median, geometric
mean, and mode. This chapter also included measures of
dispersion, or spread. Discussed in this section were the range,
variance, and standard deviation. Chapter 4 included
several graphing techniques such as dot plots, box plots, and
scatter diagrams. We also discussed the coefficient of skew-
ness, which reports the lack of symmetry in a set of data.
Throughout this section we stressed the importance of statistical
software, such as Excel and Minitab. Many computer
outputs in these chapters demonstrated how quickly and
effectively a large data set can be organized into a frequency
distribution, several of the measures of location or measures of
variation calculated, and the information presented in
graphical form.
A REVIEW OF CHAPTERS 1–4 127
124 14 150 289 52 156 203 82 27 248
39 52 103 58 136 249 110 298 251 157
186 107 142 185 75 202 119 219 156 78
116 152 206 117 52 299 58 153 219 148
145 187 165 147 158 146 185 186 149 140
Use a statistical software package such as Excel or Minitab to
help answer the following
questions.
a. Determine the mean, median, and standard deviation.
b. Determine the first and third quartiles.
c. Develop a box plot. Are there any outliers? Do the amounts
follow a symmetric distri-
bution or are they skewed? Justify your answer.
86. d. Organize the distribution of funds into a frequency
distribution.
e. Write a brief summary of the results in parts a to d.
2. Listed below are the 45 U.S. presidents and their age as
they began their terms in
office.
Number Name Age
1 Washington 57
2 J. Adams 61
3 Jefferson 57
4 Madison 57
5 Monroe 58
6 J. Q. Adams 57
7 Jackson 61
8 Van Buren 54
9 W. H. Harrison 68
10 Tyler 51
11 Polk 49
12 Taylor 64
13 Fillmore 50
14 Pierce 48
15 Buchanan 65
16 Lincoln 52
17 A. Johnson 56
18 Grant 46
19 Hayes 54
20 Garfield 49
21 Arthur 50
22 Cleveland 47
23 B. Harrison 55
Number Name Age
87. 24 Cleveland 55
25 McKinley 54
26 T. Roosevelt 42
27 Taft 51
28 Wilson 56
29 Harding 55
30 Coolidge 51
31 Hoover 54
32 F. D. Roosevelt 51
33 Truman 60
34 Eisenhower 62
35 Kennedy 43
36 L. B. Johnson 55
37 Nixon 56
38 Ford 61
39 Carter 52
40 Reagan 69
41 G. H. W. Bush 64
42 Clinton 46
43 G. W. Bush 54
44 Obama 47
45 Trump 70
Use a statistical software package such as Excel or Minitab to
help answer the following
questions.
a. Determine the mean, median, and standard deviation.
b. Determine the first and third quartiles.
c. Develop a box plot. Are there any outliers? Do the amounts
follow a symmetric distri-
bution or are they skewed? Justify your answer.
d. Organize the distribution of ages into a frequency
distribution.
e. Write a brief summary of the results in parts a to d.
88. P R O B L E M S
1. The duration in minutes of a sample of 50 power
outages last year in the state of
South Carolina is listed below.
128 A REVIEW OF CHAPTERS 1–4
3. Listed below is the 2014 median household income for the
50 states and the
District of Columbia.
https://www.census.gov/hhes/www/income/data/historical/
household/
State Amount
Alabama 42,278
Alaska 67,629
Arizona 49,254
Arkansas 44,922
California 60,487
Colorado 60,940
Connecticut 70,161
Delaware 57,522
D.C. 68,277
Florida 46,140
Georgia 49,555
Hawaii 71,223
Idaho 53,438
Illinois 54,916
Indiana 48,060
Iowa 57,810
Kansas 53,444
89. Kentucky 42,786
Louisiana 42,406
Maine 51,710
Maryland 76,165
Massachusetts 63,151
Michigan 52,005
Minnesota 67,244
Mississippi 35,521
Missouri 56,630
State Amount
Montana 51,102
Nebraska 56,870
Nevada 49,875
New Hampshire 73,397
New Jersey 65,243
New Mexico 46,686
New York 54,310
North Carolina 46,784
North Dakota 60,730
Ohio 49,644
Oklahoma 47,199
Oregon 58,875
Pennsylvania 55,173
Rhode Island 58,633
South Carolina 44,929
South Dakota 53,053
Tennessee 43,716
Texas 53,875
Utah 63,383
Vermont 60,708
Virginia 66,155
Washington 59,068
West Virginia 39,552
Wisconsin 58,080
90. Wyoming 55,690
Use a statistical software package such as Excel or Minitab to
help answer the following
questions.
a. Determine the mean, median, and standard deviation.
b. Determine the first and third quartiles.
c. Develop a box plot. Are there any outliers? Do the amounts
follow a symmetric distri-
bution or are they skewed? Justify your answer.
d. Organize the distribution of funds into a frequency
distribution.
e. Write a brief summary of the results in parts a to d.
4. A sample of 12 homes sold last week in St. Paul, Minnesota,
revealed the following
information. Draw a scatter diagram. Can we conclude that, as
the size of the home
(reported below in thousands of square feet) increases, the
selling price (reported in
$ thousands) also increases?
Home Size Home Size
(thousands of Selling Price (thousands of Selling Price
square feet) ($ thousands) square feet) ($ thousands)
1.4 100 1.3 110
1.3 110 0.8 85
1.2 105 1.2 105
1.1 120 0.9 75
1.4 80 1.1 70
1.0 105 1.1 95
91. 5. Refer to the following diagram.
0 40 80 120 160 200
* *
a. What is the graph called?
b. What are the median, and first and third quartile values?
c. Is the distribution positively skewed? Tell how you know.
d. Are there any outliers? If yes, estimate these values.
e. Can you determine the number of observations in the study?
A REVIEW OF CHAPTERS 1–4 129
C A S E S
A. Century National Bank
The following case will appear in subsequent review sec-
tions. Assume that you work in the Planning Department of
the Century National Bank and report to Ms. Lamberg. You
will need to do some data analysis and prepare a short writ-
ten report. Remember, Mr. Selig is the president of the bank,
so you will want to ensure that your report is complete and
accurate. A copy of the data appears in Appendix A.6.
Century National Bank has offices in several cities in
the Midwest and the southeastern part of the United
States. Mr. Dan Selig, president and CEO, would like to
know the characteristics of his checking account custom-
ers. What is the balance of a typical customer?
How many other bank services do the checking ac-
count customers use? Do the customers use the ATM ser-
vice and, if so, how often? What about debit cards? Who
uses them, and how often are they used?
To better understand the customers, Mr. Selig asked
Ms. Wendy Lamberg, director of planning, to select a sam-
92. ple of customers and prepare a report. To begin, she has
appointed a team from her staff. You are the head of the
team and responsible for preparing the report. You select a
random sample of 60 customers. In addition to the balance
in each account at the end of last month, you determine
(1) the number of ATM (automatic teller machine) transac-
tions in the last month; (2) the number of other bank ser-
vices (a savings account, a certificate of deposit, etc.) the
customer uses; (3) whether the customer has a debit card
(this is a bank service in which charges are made directly to
the customer’s account); and (4) whether or not interest is
paid on the checking account. The sample includes cus-
tomers from the branches in Cincinnati, Ohio; Atlanta,
Georgia; Louisville, Kentucky; and Erie, Pennsylvania.
1. Develop a graph or table that portrays the checking
balances. What is the balance of a typical customer?
Do many customers have more than $2,000 in their
accounts? Does it appear that there is a difference in
the distribution of the accounts among the four
branches? Around what value do the account bal-
ances tend to cluster?
2. Determine the mean and median of the checking ac-
count balances. Compare the mean and the median
balances for the four branches. Is there a difference
among the branches? Be sure to explain the difference
between the mean and the median in your report.
3. Determine the range and the standard deviation of
the checking account balances. What do the first and
third quartiles show? Determine the coefficient of
skewness and indicate what it shows. Because
Mr. Selig does not deal with statistics daily, include a
brief description and interpretation of the standard
deviation and other measures.
93. B. Wildcat Plumbing Supply Inc.:
Do We Have Gender Differences?
Wildcat Plumbing Supply has served the plumbing
needs of Southwest Arizona for more than 40 years.
The company was founded by Mr. Terrence St. Julian
and is run today by his son Cory. The company has
grown from a handful of employees to more than 500
today. Cory is concerned about several positions within
the company where he has men and women doing es-
sentially the same job but at different pay. To investi-
gate, he collected the information below. Suppose you
are a student intern in the Accounting Department and
have been given the task to write a report summarizing
the situation.
Yearly Salary ($000) Women Men
Less than 30 2 0
30 up to 40 3 1
40 up to 50 17 4
50 up to 60 17 24
60 up to 70 8 21
70 up to 80 3 7
80 or more 0 3
To kick off the project, Mr. Cory St. Julian held a meeting
with his staff and you were invited. At this meeting, it was
suggested that you calculate several measures of
130 A REVIEW OF CHAPTERS 1–4
94. location, create charts or draw graphs such as a cumula-
tive frequency distribution, and determine the quartiles
for both men and women. Develop the charts and write
the report summarizing the yearly salaries of employees
at Wildcat Plumbing Supply. Does it appear that there are
pay differences based on gender?
C. Kimble Products: Is There a Difference
In the Commissions?
At the January national sales meeting, the CEO of Kimble
Products was questioned extensively regarding the com-
pany policy for paying commissions to its sales represen-
tatives. The company sells sporting goods to two major
markets. There are 40 sales representatives who call di-
rectly on large-volume customers, such as the athletic de-
partments at major colleges and universities and
professional sports franchises. There are 30 sales repre-
sentatives who represent the company to retail stores lo-
cated in shopping malls and large discounters such as
Kmart and Target.
Upon his return to corporate headquarters, the CEO
asked the sales manager for a report comparing the com-
missions earned last year by the two parts of the sales
team. The information is reported below. Write a brief re-
port. Would you conclude that there is a difference? Be
sure to include information in the report on both the cen-
tral tendency and dispersion of the two groups.
Commissions Earned by Sales Representatives
Calling on Large Retailers ($)
1,116 681 1,294 12 754 1,206 1,448 870 944 1,255
1,213 1,291 719 934 1,313 1,083 899 850 886 1,556
886 1,315 1,858 1,262 1,338 1,066 807 1,244 758 918
95. Commissions Earned by Sales Representatives
Calling on Athletic Departments ($)
354 87 1,676 1,187 69 3,202 680 39 1,683 1,106
883 3,140 299 2,197 175 159 1,105 434 615 149
1,168 278 579 7 357 252 1,602 2,321 4 392
416 427 1,738 526 13 1,604 249 557 635 527
P R A C T I C E T E S T
There is a practice test at the end of each review section. The
tests are in two parts. The first part contains several objec-
tive questions, usually in a fill-in-the-blank format. The second
part is problems. In most cases, it should take 30 to 45
minutes to complete the test. The problems require a calculator.
Check the answers in the Answer Section in the back of
the book.
Part 1—Objective
1. The science of collecting, organizing, presenting, analyzing,
and interpreting data to assist in
making effective decisions is called . 1.
2. Methods of organizing, summarizing, and presenting data in
an informative way are
called . 2.
3. The entire set of individuals or objects of interest or the
measurements obtained from all
individuals or objects of interest are called the . 3.
4. List the two types of variables. 4.
5. The number of bedrooms in a house is an example of a .
(discrete variable,
96. continuous variable, qualitative variable—pick one) 5.
6. The jersey numbers of Major League Baseball players are an
example of what level of
measurement? 6.
7. The classification of students by eye color is an example of
what level of measurement? 7.
8. The sum of the differences between each value and the mean
is always equal to what value? 8.
9. A set of data contained 70 observations. How many classes
would the 2k method suggest to
construct a frequency distribution? 9.
10. What percent of the values in a data set are always larger
than the median? 10.
11. The square of the standard deviation is the . 11.
12. The standard deviation assumes a negative value when . (all
the values are negative,
at least half the values are negative, or never—pick one.) 12.
13. Which of the following is least affected by an outlier?
(mean, median, or range—pick one) 13.
Part 2—Problems
1. The Russell 2000 index of stock prices increased by the
following amounts over the last 3 years.
18% 4% 2%
What is the geometric mean increase for the 3 years?
2. The information below refers to the selling prices ($000) of
homes sold in Warren, Pennsylvania, during 2016.
97. Selling Price ($000) Frequency
120.0 up to 150.0 4
150.0 up to 180.0 18
180.0 up to 210.0 30
210.0 up to 240.0 20
240.0 up to 270.0 17
270.0 up to 300.0 10
300.0 up to 330.0 6
a. What is the class interval?
b. How many homes were sold in 2016?
c. How many homes sold for less than $210,000?
d. What is the relative frequency of the 210 up to 240 class?
e. What is the midpoint of the 150 up to 180 class?
f. The selling prices range between what two amounts?
3. A sample of eight college students revealed they owned the
following number of CDs.
52 76 64 79 80 74 66 69
a. What is the mean number of CDs owned?
b. What is the median number of CDs owned?
c. What is the 40th percentile?
d. What is the range of the number of CDs owned?
e. What is the standard deviation of the number of CDs owned?
4. An investor purchased 200 shares of the Blair Company for
$36 each in July of 2013, 300 shares
at $40 each in September 2015, and 500 shares at $50 each in
January 2016. What is the
investor’s weighted mean price per share?
5. During the 50th Super Bowl, 30 million pounds of snack
98. food were eaten. The chart below depicts
this information.
Snack Nuts
8%
Potato Chips
37%
Tortilla Chips
28%
Pretzels
14%
Popcorn
13%
a. What is the name given to this graph?
b. Estimate, in millions of pounds, the amount of potato chips
eaten during the game.
c. Estimate the relationship of potato chips to popcorn. (twice
as much, half as much, three
times, none of these—pick one)
d. What percent of the total do potato chips and tortilla chips
comprise?
A REVIEW OF CHAPTERS 1–4 131
LEARNING OBJECTIVES
When you have completed this chapter, you will be able to:
100. Many of the top dealerships
are publicly owned with shares traded on the New York Stock
Exchange
or NASDAQ. In 2014, the largest megadealership was
AutoNation (ticker
symbol AN), followed by Penske Auto Group (PAG), Group 1
Automotive,
Inc. (ticker symbol GPI), and the privately owned Van Tuyl
Group.
These large corporations use statistics and analytics to
summarize
and analyze data and information to support their decisions. As
an ex-
ample, we will look at the Applewood Auto group. It owns four
dealer-
ships and sells a wide range of vehicles. These include the
popular
Korean brands Kia and Hyundai, BMW and Volvo sedans and
luxury
SUVs, and a full line of Ford and Chevrolet cars and trucks.
Ms. Kathryn Ball is a member of the senior management team at
Applewood Auto Group, which has its corporate offices
adjacent to Kane
Motors. She is responsible for tracking and analyzing vehicle
sales and
the profitability of those vehicles. Kathryn would like to
summarize the profit earned on
the vehicles sold with tables, charts, and graphs that she would
review monthly. She
wants to know the profit per vehicle sold, as well as the lowest
and highest amount of
profit. She is also interested in describing the demographics of
101. the buyers. What are
their ages? How many vehicles have they previously purchased
from one of the Apple-
wood dealerships? What type of vehicle did they purchase?
The Applewood Auto Group operates four dealerships:
• Tionesta Ford Lincoln sells Ford and Lincoln cars and trucks.
• Olean Automotive Inc. has the Nissan franchise as well as the
General Motors
brands of Chevrolet, Cadillac, and GMC Trucks.
• Sheffield Motors Inc. sells Buick, GMC trucks, Hyundai, and
Kia.
• Kane Motors offers the Chrysler, Dodge, and Jeep line as well
as BMW and Volvo.
Every month, Ms. Ball collects data from each of the four
dealerships
and enters them into an Excel spreadsheet. Last month the
Applewood
Auto Group sold 180 vehicles at the four dealerships. A copy of
the first
few observations appears to the left. The variables collected
include:
• Age—the age of the buyer at the time of the purchase.
• Profit—the amount earned by the dealership on the sale of
each
vehicle.
• Location—the dealership where the vehicle was purchased.
• Vehicle type—SUV, sedan, compact, hybrid, or truck.
• Previous—the number of vehicles previously purchased at any
of the
103. be classified into distinct
categories. Examples of qualitative data include political
affiliation (Republican, Demo-
crat, Independent, or other), state of birth (Alabama, . . . ,
Wyoming), and method of
payment for a purchase at Barnes & Noble (cash, digital wallet,
debit, or credit). On the
other hand, quantitative variables are numerical in nature.
Examples of quantitative data
relating to college students include the price of their textbooks,
their age, and the num-
ber of credit hours they are registered for this semester.
In the Applewood Auto Group data set, there are five variables
for each vehicle
sale: age of the buyer, amount of profit, dealer that made the
sale, type of vehicle sold,
and number of previous purchases by the buyer. The dealer and
the type of vehicle are
qualitative variables. The amount of profit, the age of the buyer,
and the number of pre-
vious purchases are quantitative variables.
Suppose Ms. Ball wants to summarize last month’s sales by
location. The
first step is to sort the vehicles sold last month according to
their location and
then tally, or count, the number sold at each location of the four
locations:
Tionesta, Olean, Sheffield, or Kane. The four locations are
used to develop a
frequency table with four mutually exclusive (distinctive)
classes. Mutually exclu-
sive classes means that a particular vehicle can be assigned to
only one class. In
addition, the frequency table must be collectively exhaustive.
105. of the class frequencies is
divided by the total number of observations. Again, this is
easily accomplished using Excel.
The fraction of vehicles sold last month at the Kane location is
0.289, found by 52 divided
by 180. The relative frequency for each location is shown in
Table 2–2.
TABLE 2–2 Relative Frequency Table of Vehicles Sold by
Location Last Month at Applewood Auto Group
Location Number of Cars Relative Frequency Found by
Kane 52 .289 52/180
Olean 40 .222 40/180
Sheffield 45 .250 45/180
Tionesta 43 .239 43/180
Total 180 1.000
DESCRIBING DATA: FREQUENCY TABLES, FREQUENCY
DISTRIBUTIONS, AND GRAPHIC PRESENTATION 21
GRAPHIC PRESENTATION
OF QUALITATIVE DATA
The most common graphic form to present a qualitative variable
is a bar chart. In most
cases, the horizontal axis shows the variable of interest. The
vertical axis shows the
frequency or fraction of each of the possible outcomes. A
distinguishing feature of a bar
chart is there is distance or a gap between the bars. That is,
because the variable of in-
106. terest is qualitative, the bars are not adjacent to each other.
Thus, a bar chart graphically
describes a frequency table using a series of uniformly wide
rectangles, where the
height of each rectangle is the class frequency.
LO2-2
Display a frequency table
using a bar or pie chart.
BAR CHART A graph that shows qualitative classes on the
horizontal axis and the
class frequencies on the vertical axis. The class frequencies are
proportional to the
heights of the bars.
PIE CHART A chart that shows the proportion or percentage
that each class
represents of the total number of frequencies.
We use the Applewood Auto Group data as an example (Chart
2–1). The variables
of interest are the location where the vehicle was sold and the
number of vehicles sold
at each location. We label the horizontal axis with the four
locations and scale the verti-
cal axis with the number sold. The variable location is of
nominal scale, so the order of
the locations on the horizontal axis does not matter. In Chart 2–
1, the locations are
listed alphabetically. The locations could also be in order of
decreasing or increasing
frequencies.
The height of the bars, or rectangles, corresponds to the number
of vehicles at
107. each location. There were 52 vehicles sold last month at the
Kane location, so the
height of the Kane bar is 52; the height of the bar for the Olean
location is 40.
Nu
m
be
r o
f V
eh
ic
le
s
So
ld
50
40
30
20
10
0
Kane Olean
Location