This document provides an overview and objectives for Chapter 3 of the textbook "Statistical Techniques in Business and Economics" by Lind. The chapter covers describing data through numerical measures of central tendency (mean, median, mode) and dispersion (range, variance, standard deviation). It includes examples of computing various measures like the weighted mean, median, mode, and interpreting their relationships. The document also lists learning activities for students such as reading the chapter, watching video lectures, completing practice problems in the book, and participating in an online discussion forum.
Business Statistics A Decision Making Approach 8th Edition Groebner Solutions...ForemanForemans
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In the previous lesson we discussed a measure of location known as the measure of central tendency. There are other measures of location which are useful in describing the distribution of the data set. These measures of location include the maximum, minimum, percentiles, deciles and quartiles. How to compute and interpret these measures are also discussed in this lesson.
Presentation of the work on Prime Numbers.
intended for mathematics loving people.
Please send comments and suggestions for improvement to solo.hermelin@gmail.com.
More presentations can be found in my website at http://solohermelin.com.
In the previous lesson we discussed a measure of location known as the measure of central tendency. There are other measures of location which are useful in describing the distribution of the data set. These measures of location include the maximum, minimum, percentiles, deciles and quartiles. How to compute and interpret these measures are also discussed in this lesson.
Presentation of the work on Prime Numbers.
intended for mathematics loving people.
Please send comments and suggestions for improvement to solo.hermelin@gmail.com.
More presentations can be found in my website at http://solohermelin.com.
Statistical Processes
Can descriptive statistical processes be used in determining relationships, differences, or effects in your research question and testable null hypothesis? Why or why not? Also, address the value of descriptive statistics for the forensic psychology research problem that you have identified for your course project. read an article for additional information on descriptive statistics and pictorial data presentations.
300 words APA rules for attributing sources.
Computing Descriptive Statistics
Computing Descriptive Statistics: “Ever Wonder What Secrets They Hold?” The Mean, Mode, Median, Variability, and Standard Deviation
Introduction
Before gaining an appreciation for the value of descriptive statistics in behavioral science environments, one must first become familiar with the type of measurement data these statistical processes use. Knowing the types of measurement data will aid the decision maker in making sure that the chosen statistical method will, indeed, produce the results needed and expected. Using the wrong type of measurement data with a selected statistic tool will result in erroneous results, errors, and ineffective decision making.
Measurement, or numerical, data is divided into four types: nominal, ordinal, interval, and ratio. The businessperson, because of administering questionnaires, taking polls, conducting surveys, administering tests, and counting events, products, and a host of other numerical data instrumentations, garners all the numerical values associated with these four types.
Nominal Data
Nominal data is the simplest of all four forms of numerical data. The mathematical values are assigned to that which is being assessed simply by arbitrarily assigning numerical values to a characteristic, event, occasion, or phenomenon. For example, a human resources (HR) manager wishes to determine the differences in leadership styles between managers who are at different geographical regions. To compute the differences, the HR manager might assign the following values: 1 = West, 2 = Midwest, 3 = North, and so on. The numerical values are not descriptive of anything other than the location and are not indicative of quantity.
Ordinal Data
In terms of ordinal data, the variables contained within the measurement instrument are ranked in order of importance. For example, a product-marketing specialist might be interested in how a consumer group would respond to a new product. To garner the information, the questionnaire administered to a group of consumers would include questions scaled as follows: 1 = Not Likely, 2 = Somewhat Likely, 3 = Likely, 4 = More Than Likely, and 5 = Most Likely. This creates a scale rank order from Not Likely to Most Likely with respect to acceptance of the new consumer product.
Interval Data
Oftentimes, in addition to being ordered, the differences (or intervals) between two adjacent measurement values on a measurement scale are identical. For example, the di ...
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Answer the questions in one paragraph 4-5 sentences.
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SAMPLING MEAN:
DEFINITION:
The term sampling mean is a statistical term used to describe the properties of statistical distributions. In statistical terms, the sample meanfrom a group of observations is an estimate of the population mean. Given a sample of size n, consider n independent random variables X1, X2... Xn, each corresponding to one randomly selected observation. Each of these variables has the distribution of the population, with mean and standard deviation. The sample mean is defined to be
WHAT IT IS USED FOR:
It is also used to measure central tendency of the numbers in a database. It can also be said that it is nothing more than a balance point between the number and the low numbers.
HOW TO CALCULATE IT:
To calculate this, just add up all the numbers, then divide by how many numbers there are.
Example: what is the mean of 2, 7, and 9?
Add the numbers: 2 + 7 + 9 = 18
Divide by how many numbers (i.e., we added 3 numbers): 18 ÷ 3 = 6
So the Mean is 6
SAMPLE VARIANCE:
DEFINITION:
The sample variance, s2, is used to calculate how varied a sample is. A sample is a select number of items taken from a population. For example, if you are measuring American people’s weights, it wouldn’t be feasible (from either a time or a monetary standpoint) for you to measure the weights of every person in the population. The solution is to take a sample of the population, say 1000 people, and use that sample size to estimate the actual weights of the whole population.
WHAT IT IS USED FOR:
The sample variance helps you to figure out the spread out in the data you have collected or are going to analyze. In statistical terminology, it can be defined as the average of the squared differences from the mean.
HOW TO CALCULATE IT:
Given below are steps of how a sample variance is calculated:
· Determine the mean
· Then for each number: subtract the Mean and square the result
· Then work out the mean of those squared differences.
To work out the mean, add up all the values then divide by the number of data points.
First add up all the values from the previous step.
But how do we say "add them all up" in mathematics? We use the Roman letter Sigma: Σ
The handy Sigma Notation says to sum up as many terms as we want.
· Next we need to divide by the number of data points, which is simply done by multiplying by "1/N":
Statistically it can be stated by the following:
·
· This value is the variance
EXAMPLE:
Sam has 20 Rose Bushes.
The number of flowers on each b.
Statistics is a mathematical science including methods of collecting, organizing, and analyzing data in such a way that meaningful conclusions can be drawn from them. In general, its investigations and analyses fall into two broad categories called descriptive and inferential statistics.
A measure of central tendency (also referred to as measures of centre or central location) is a summary measure that attempts to describe a whole set of data with a single value that represents the middle or centre of its distribution.
Basic Business Statistics Chapter 3Numerical Descriptive Measures
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1. MAT 255 Business Statistics I
Chapter 3
Describing Data: Numerical
Measures
Text: Lind, Statistical Techniques in
Business and Economics, 15e
2. Chapter 3 Objectives
Objective Section
Explain the concept of central tendency. 3.1
Identify and compute the arithmetic mean. 3.2, 3.3, 3.4
Compute and interpret the weighted mean. 3.5
Determine the median. 3.6
Identify the mode. 3.7
Describe the shape of a distribution based on the positions of the mean,
median, and mode.
3.9
Calculate the geometric mean. 3.10
Explain and apply measures of dispersion. 3.11, 3.12
Explain Chebyshev’s Theorem and the empirical rule. 3.14
Compute the mean and standard deviation of grouped data. 3.15
3. What should you do?
Learning Activities Details
Read Chapter 3. Pages 22 - 46
View the MAT 255 Chapter 3 Video Lecture
Playlist and take notes as if you were in a lecture
class.
There are videos on the
following topics:
• Measures of Location
Overview
• Page 62, exercise 8
• Page 62, exercise 10
• Page 64, exercise 16
• Page 67, exercise 20
• The Relative Positions of the
Mean, Median, and Mode
• Page 73, exercise 28
• Page 74, exercise 32
• Measures of Dispersion Overview
• Page 77, example
• Page 79, exercise 38
• Page 82, exercise 46
• Page 85, exercise 50
• Page 87, exercise 54
• Page 87, exercise 56
• Page 91, exercise 58
Complete exercise 1 – 11 odd, 15 – 29 odd, 35,
39, 41, 47 – 55 odd 59, 61, 65, 69, 73, 75, 85.
Complete homework for self-assessment purposes. The answers to odd
questions are found in the back of the book.
Complete the Chapter 3 Practice Worksheet and
view the worked out solutions.
The practice worksheet is for self-assessment purposes. You can view the
worked out solutions to the worksheet after you complete it.
Post a relevant comment, insight or question to
the MAT 255-4H1 Fall 2014 Google+ Community.
The Google+ Community allows you to communicate with classmates in an
online open forum.
Complete MAT 255 Chapter 3 Survey. Communicate to the instructor the areas where you still need help.
4. Measures of Location Overview
Text: Lind, Statistical Techniques in
Business and Economics, 15e
Delaware Technical Community College
5. • The purpose of a measure of location is to
pinpoint the center of a distribution of data.
• In addition to measures of locations, we
should consider the dispersion – often called
the variation or spread – in the data.
• Five measures of location:
1. The arithmetic mean
2. The weighted mean
3. The median
4. The mode
5. The geometric mean
6. Population Mean: 𝝁 =
𝑿
𝑵
• 𝜇 represents the population mean. It is the
Greek lower case letter “mu.”
• 𝑁 is the number of values in the population.
• 𝑋 represents any particular value.
• is the Greek capital letter “sigma” and
indicates the operation of adding.
• 𝑋 is the sum of the 𝑋 values in a population.
A parameter is a characteristic of a population.
7. Sample Mean: 𝑿 =
𝑿
𝒏
• 𝑋 represents the sample mean. It is read “X bar.”
• 𝑛 is the number of values in the sample.
• 𝑋 represents any particular value.
• is the Greek capital letter “sigma” and
indicates the operation of adding.
• 𝑋 is the sum of the 𝑋 values in a population.
A statistic is a characteristic of a sample.
8. Important properties of the arithmetic mean:
1. Every set of interval- or ratio- level has a mean.
2. All the values are included in computing the mean.
3. The mean is unique.
4. The sum of the deviations of each value from the mean is
0.
𝑋 − 𝑋 = 0
One disadvantage: if one or two values are either extremely
high or extremely low compared to the majority of the data,
then the mean might not be an appropriate average to
represent the data.
10. Median: The midpoint of the values after they
have been ordered from the smallest to the
largest, or the largest to the smallest.
The median is not affected by extremely large
of small values.
The median can be computed for ordinal-level
data or higher.
11. Mode: The value of the observation that appears most
frequently.
The mode is especially useful in summarizing nominal-
level data.
The mode can be determined for all levels of data –
nominal, ordinal, interval, and ratio. The mode is not
affected by extremely high or low values. For many
data sets, though, there is no mode, causing it to be
used less often than the mean and median.
There can be one mode, multiple modes, or no mode for
a set of dat.
12. Page 62, exercise 8
Text: Lind, Statistical Techniques in
Business and Economics, 15e
Delaware Technical Community College
13. Page 62, exercise 8:
The accounting department at a mail-order
company counted the following numbers of
incoming calls per day to the company’s toll-free
number during the first 7 days in May:
14, 24, 19, 31, 36, 26, 17.
a. Compute the arithmetic mean.
b. Indicate whether it is a statistic or a parameter.
14. Page 62, exercise 10
Text: Lind, Statistical Techniques in
Business and Economics, 15e
Delaware Technical Community College
15. Page 62, exercise 10:
The Human Relations Director at Ford began a study
of the overtime hours in the Inspection
Department. A sample of 15 showed they worked
the following number of overtime hours last month:
13, 13, 12, 15, 7, 15, 5, 12, 6, 7, 12, 10, 9, 13, 12
a. Compute the arithmetic mean.
b. Indicate whether it is a statistic or a parameter.
16. Page 64, exercise 16
Text: Lind, Statistical Techniques in
Business and Economics, 15e
Delaware Technical Community College
17. Page 64, exercise 16:
Andrews and Associates specializes in corporate
law. They charge $100 an hour for researching a
case, $75 an hour for consultations, and $200 an
hour for writing a brief. Last week one of the
associates spent 10 hours consulting with her
client, 10 hours researching the case, and 20
hours writing the brief. What was the weighted
mean hourly charge for her legal services?
18. Page 67, exercise 20
Text: Lind, Statistical Techniques in
Business and Economics, 15e
Delaware Technical Community College
19. Page 67, exercise 20:
The following are the ages of the 10 people in
the video arcade at the Southwyck Shopping
Mall at 10 A.M.
12, 8, 17, 6, 11, 14, 8, 17, 10, 8
Determine the median age. Determine the
mode.
20. The Relative Positions of the
Mean, Median, and Mode
Text: Lind, Statistical Techniques in
Business and Economics, 15e
Delaware Technical Community College
21. The Relative Positions of the Mean,
Median, and Mode
A histogram is a graphical display of a frequency
distribution for quantitative data. That
distribution can take various shapes. Here we
will discuss characteristics for a symmetric
distribution, a positively skewed distribution,
and a negatively skewed distribution.
23. A Positively Skewed Distribution
0
1
2
3
4
5
6
7
8
9
0 up to 5 5 up to 10 10 up to 15 15 up to 20 20 up to 25 25 up to 30 30 up to 35
24. A Negatively Skewed Distribution
0
1
2
3
4
5
6
7
8
9
0 up to 5 5 up to 10 10 up to 15 15 up to 20 20 up to 25 25 up to 30 30 up to 35
25. Page 73, exercise 28
Text: Lind, Statistical Techniques in
Business and Economics, 15e
Delaware Technical Community College
26. Page 73, exercise 28:
Compute the geometric mean of the following
percent increases: 2, 8, 6, 4, 10, 6, 8
27. Geometric Mean: 𝐺𝑀 =
𝑛
𝑋1 𝑋2 ⋯ 𝑋 𝑛
The geometric mean is useful in finding the
average change of percentages, ratios, indexes,
or growth rates over time. The geometric mean
will always be less than or equal to the
arithmetic mean.
28. Page 74, exercise 32
Text: Lind, Statistical Techniques in
Business and Economics, 15e
Delaware Technical Community College
29. Page 74, exercise 32:
JetBlue Airways is an American low-cost airline
headquartered in New York City. Its main base is
John F. Kennedy International Airport. JetBlue’s
revenue in 2002 was $635.2 million. By 2009,
revenue had increased to $3,290.0 million.
What was the geometric mean annual increase
for the period?
30. Rate of Increase Over Time:
𝐺𝑀 =
𝑛 Value at end of period
Value at start of period
− 1
32. A measure of location only describes the center of
data. You would also want to know about the
variation (or dispersion) of the data as well in order
to have a more complete picture.
A small value for a measure of dispersion indicates
that the data are clustered closely. The mean is
therefore considered representative of the data.
Conversely, a large measure of dispersion indicates
that he mean is not reliable.
Comparing the measures of dispersions of multiple
distributions is also helpful.
33. Range: The difference between the largest and
the smallest values in a data set.
Range = Largest Value – Smallest Value
34. Mean Deviation: The arithmetic mean of the
absolute values of the deviations from the
arithmetic mean. It measures the mean
amount by which the values in a population, or
sample vary from their mean.
Mean Deviation: 𝑴𝑫 =
𝑿− 𝑿
𝒏
35. Variance: The arithmetic mean of the squared
deviations from the mean.
Population Variance: 𝝈 𝟐
=
𝑿−𝝁 𝟐
𝑵
Read as “sigma squared”
Standard Deviation: The square root of the
variance.
Population Standard Deviation: 𝛔 =
𝑿−𝝁 𝟐
𝑵
36. Sample Variance: 𝒔 𝟐
=
𝑿− 𝑿 𝟐
𝒏−𝟏
Sample Standard Deviation: 𝒔 =
𝑿− 𝑿 𝟐
𝒏−𝟏
Although the use of 𝑛 is logical since 𝑋 is used to
estimate 𝜇, it tends to underestimate the population
variance, 𝜎2. The use of 𝑛 − 1 in the denominator
provides the appropriate correction for this tendency.
Because the primary use of the sample statistic 𝑠2
is to
estimate population parameters like 𝜎2, 𝑛 − 1 is
preferred to 𝑛 in defining the sample variance. This
convention is also used when computing the sample
standard deviation.
37. The variance and standard deviation are also
based on the deviations from the mean.
However, instead of using the absolute value of
the deviations, the variance and the standard
deviation square the deviations.
38. Page 77, example
Text: Lind, Statistical Techniques in
Business and Economics, 15e
Delaware Technical Community College
39. Page 77, example:
The chart below shows the
number of cappuccinos sold
at Starbucks in the Orange
County airport and the
Ontario, California, airport
between 4 and 5 P.M. for a
sample of five days last
month. Determine the
mean, median, range, and
mean deviation for each
location. Comment on the
similarities and differences
in these measures.
Orange
County
Ontario
20 20
40 49
50 50
60 51
80 80
40. Page 79, exercise 38
Text: Lind, Statistical Techniques in
Business and Economics, 15e
Delaware Technical Community College
41. Page 79, exercise 38:
A sample of eight companies in the aerospace
industry was surveyed as to their return on
investment last year. The results are (in
percent):
10.6, 12.6, 14.8, 18.2, 12.0, 14.8, 12.2, and 15.6
Calculate the range, arithmetic mean, mean
deviation, and interpret the values.
42. Page 82, exercise 46
Text: Lind, Statistical Techniques in
Business and Economics, 15e
Delaware Technical Community College
43. Page 82, exercise 46:
The annual incomes of the five vice presidents of
TMV industries are $125,000; $128,000; $122,000;
$133,000; and $140,000. Consider this population.
a. What is the range?
b. What is the arithmetic mean?
c. What is the population variance? The standard
deviation?
d. The annual incomes of officers of another firm
similar to TMV industries were also studied. The
mean was $129,000 and the standard deviation
$8,612. Compare the means and dispersions in
the two firms.
44. Page 85, exercise 50
Text: Lind, Statistical Techniques in
Business and Economics, 15e
Delaware Technical Community College
45. Page 85, exercise 50:
Compute the sample variance and the sample
standard deviation.
The sample of eight companies in the aerospace
industry was surveyed as to their return on
investment last year. The results are: 10.6, 12.6,
14.8, 18.2, 12.0, 14.8, 12.2, and 15.6
46. Page 87, exercise 54
Text: Lind, Statistical Techniques in
Business and Economics, 15e
Delaware Technical Community College
47. Page 87, exercise 54:
The mean income of a group of sample
observations is $500; the standard deviation is
$40. According to Chebyshev’s theorem, at least
what percent of incomes will lie between $400
and $600.
48. Chebyshev’s Theorem: For any set of
observations (sample or population), the
proportion of the values that lie within k
standard deviations of the mean is at least 𝟏 −
𝟏
𝒌 𝟐, where k is any constant greater than 1.
49. Page 87, exercise 56
Text: Lind, Statistical Techniques in
Business and Economics, 15e
Delaware Technical Community College
50. Page 87, exercise 56:
The distribution of a sample of the number of
drinks sold per day at a nearby Wendy’s is
symmetric and bell-shaped. The mean number
of drinks sold per day is 91.9 with a standard
deviation of 4.67. Using the empirical rule, sales
will be between what two values on 68 percent
of the days? Sales will be between what two
values on 95 percent of the days?
51. Empirical Rule: For a symmetrical, bell-shaped
frequency distribution, approximately 68
percent of the observations will lie within plus
and minus one standard deviation of the mean;
about 95 percent of the observations will lie
within plus and minus two standard deviations
of the mean; and 99.7 percent will lie within
plus or minus three standard deviations of the
mean.
53. Page 91, exercise 58
Text: Lind, Statistical Techniques in
Business and Economics, 15e
Delaware Technical Community College
54. Page 91, exercise 58:
Determine the mean and standard deviation of
the following frequency distribution.
Class Frequency
0 up to 5 2
5 up to 10 7
10 up to 15 12
15 up to 20 6
20 up to 25 3