Describing DataNumerical MeasuresChapter 3McGraw-.docx

Describing Data:
Numerical Measures
Chapter 3
McGraw-Hill/Irwin
Copyright © 2012 by The McGraw-Hill Companies, Inc.
All rights reserved.
*
Learning Objectives
LO1 Explain the concept of central tendency.
LO2 Identify and compute the arithmetic mean.
LO3 Compute and interpret the weighted mean.
LO4 Determine the median.
LO5 Identify the mode.
LO6 Calculate the geometric mean.
LO7 Explain and apply measures of dispersion.
LO8 Compute and interpret the standard deviation.
LO9 Explain Chebyshev’s Theorem and the Empirical
Rule.
L10 Compute the mean and standard deviation of
grouped data.
3-*

*
1. Central Tendency - Measures of Location
The purpose of a measure of location is to pinpoint the center of
a distribution of data.
There are many measures of location. We will consider five:
The arithmetic mean,
The weighted mean,
The median,
The mode, and
The geometric mean
3-*
LO1 Explain the concept of central tendency
*
1. Central Tendency - Measures of Location
Examples with MS Excel:
The weighted mean,
The median,
The mode
3-*
*

2. The Relative Positions of the Mean, Median and the Mode
3-*
LO2,4,5
*
3. Measures of DispersionMinimum
Maximum
Range
Mean Deviation
3-*
LO7
*
3. Measures of DispersionVariance and Standard Deviation
3-*
LO7
*

3. Measures of DispersionVariance and Standard Deviation
Example: hand calculation, MS formula, Pivot Table
Descriptive Statistics using MS Excel Data Analysis Function
3-*
LO7
*
4. The Empirical Rule
3-*
LO9
*
5. Chebyshev’s Theorem
The arithmetic mean biweekly amount contributed by the
Dupree Paint employees to the company’s profit-sharing plan is
$51.54, and the standard deviation is $7.51. At least what
percent of the contributions lie within plus 3.5 standard
deviations and minus 3.5 standard deviations of the mean?
3-*
LO9 Explain Chebyshev’s Theorem and the Empirical Rule.
*

Central Tendency - Measures of Location
The purpose of a measure of location is to pinpoint the center of
a distribution of data.
There are many measures of location. We will consider five:
The weighted mean,
The median,
The mode, and
The geometric mean
3-*
*

Characteristics of the Mean
The arithmetic mean is the most widely used measure of
location.
Requires the interval scale.
Major characteristics:
All values are used.
It is unique.
The sum of the deviations from the mean is 0.
It is calculated by summing the values and dividing by the
number of values.
3-*
LO2 Identify and compute the arithmetic mean.
*
.
Population Mean
For ungrouped data, the population mean is the sum of all
the population values divided by the total number of population
values:
3-*
LO2
*
EXAMPLE – Population Mean

There are 42 exits on I-75 through the state of Kentucky. Listed
below are the distances between exits (in miles).
Why is this information a population?
What is the mean number of miles between exits?
3-*
LO2
*
EXAMPLE – Population Mean
There are 42 exits on I-75 through the state of Kentucky. Listed
below are the distances between exits (in miles).
Why is this information a population?
This is a population because we are considering all the exits in
Kentucky.
What is the mean number of miles between exits?
3-*

LO2
*
Parameter Versus Statistics
3-*
PARAMETER A measurable characteristic of a population.
STATISTIC A measurable characteristic of a sample.
LO2
.
*
Properties of the Arithmetic Mean
Every set of interval-level and ratio-level data has a mean.
All the values are included in computing the mean.
The mean is unique.
The sum of the deviations of each value from the mean is zero.
3-*
LO2
*

Sample MeanFor ungrouped data, the sample mean is the sum of
all the sample values divided by the number of sample values:
3-*
LO2
*
.
EXAMPLE – Sample Mean
3-*
LO2
*
Weighted MeanThe weighted mean of a set of numbers X1, X2,
..., Xn, with corresponding weights w1, w2, ...,wn, is computed
from the following formula:
3-*
LO3 Compute and interpret the weighted mean
*

EXAMPLE – Weighted Mean
The Carter Construction Company pays its hourly
employees $16.50, $19.00, or $25.00 per hour. There are 26
hourly employees, 14 of which are paid at the $16.50 rate, 10 at
the $19.00 rate, and 2 at the $25.00 rate.
What is the mean hourly rate paid the 26 employees?
3-*
LO3
*
The Median
PROPERTIES OF THE MEDIAN
There is a unique median for each data set.
It is not affected by extremely large or small values and is
therefore a valuable measure of central tendency when such
values occur.
It can be computed for ratio-level, interval-level, and ordinal-
level data.
It can be computed for an open-ended frequency distribution if
the median does not lie in an open-ended class.
3-*

MEDIAN The midpoint of the values after they have been
ordered from the smallest to the largest, or the largest to the
smallest.
LO4 Determine the median.
*
EXAMPLES - Median
The ages for a sample of five college students are:
21, 25, 19, 20, 22
Arranging the data in ascending order gives:
19, 20, 21, 22, 25.
Thus the median is 21.
The heights of four basketball players, in inches, are:
76, 73, 80, 75
Arranging the data in ascending order gives:
73, 75, 76, 80.
Thus the median is 75.5
3-*
LO4

*
The Mode
3-*
MODE The value of the observation that appears most
frequently.
LO5 Identify the mode.
*
Example - Mode
Using the data regarding the distance in miles between exits on
I-75 through Kentucky. The information is repeated below.
What is the modal distance?
Organize the distances into a frequency table.
3-*
LO5
*

The Relative Positions of the Mean, Median and the Mode
3-*
LO2,4,5
*
The Geometric MeanUseful in finding the average change of
percentages, ratios, indexes, or growth rates over time.It has a
wide application in business and economics because we are
often interested in finding the percentage changes in sales,
salaries, or economic figures, such as the GDP, which
compound or build on each other. The geometric mean will
always be less than or equal to the arithmetic mean. The
formula for the geometric mean is written:
EXAMPLE:
The return on investment earned by Atkins Construction
Company for four successive years was: 30 percent, 20 percent,
-40 percent, and 200 percent. What is the geometric mean rate
of return on investment?
3-*
LO6 Calculate the geometric mean.
*
The Geometric Mean – Finding an Average Percent Change
Over Time
EXAMPLE
During the decade of the 1990s, and into the 2000s, Las

Vegas, Nevada, was the fastest-growing city in the United
States. The population increased from 258,295 in 1990 to
607,876 in 2009. This is an increase of 349,581 people, or a
135.3 percent increase over the period. The population has more
than doubled.
What is the average annual increase?
3-*
LO6
*
Dispersion
A measure of location, such as the mean or the median,
only describes the center of the data. It is valuable from that
standpoint, but it does not tell us anything about the spread of
the data.
For example, if your nature guide told you that the river
ahead averaged 3 feet in depth, would you want to wade across
on foot without additional information? Probably not. You
would want to know something about the variation in the depth.
A second reason for studying the dispersion in a set of data
is to compare the spread in two or more distributions.
3-*
LO7 Explain and apply measures of dispersion.
*

Measures of DispersionRange
Mean Deviation
Variance and Standard Deviation
3-*
LO7
*
EXAMPLE – Range
The number of cappuccinos sold at the Starbucks location
in the Orange Country Airport between 4 and 7 p.m. for a
sample of 5 days last year were 20, 40, 50, 60, and 80.
Determine the range for the number of cappuccinos sold.
Range = Largest – Smallest value
= 80 – 20 = 60
3-*
LO7
*
Mean DeviationA shortcoming of the range is that it is based on
only two values, the highest and the lowest; it does not take into
consideration all of the values. The mean deviation does. It
measures the mean amount by which the values in a population,
or sample, vary from their mean

3-*
MEAN DEVIATION The arithmetic mean of the absolute values
of the deviations from the arithmetic mean.
LO7
*
EXAMPLE – Mean Deviation
The number of cappuccinos sold at the Starbucks location
in the Orange Country Airport between 4 and 7 p.m. for a
sample of 5 days last year were 20, 40, 50, 60, and 80.
Determine the mean deviation for the number of
cappuccinos sold.
Step 1: Compute the mean
3-*
LO7
*
EXAMPLE – Mean Deviation
Step 2: Subtract the mean (50) from each of the
observations, convert to positive if difference is negative
Step 3: Sum the absolute differences found in step 2 then
divide by the number of observations

3-*
LO7
*
Variance and Standard DeviationThe variance and standard
deviations are nonnegative and are zero only if all observations
are the same. For populations whose values are near the mean,
the variance and standard deviation will be small. For
populations whose values are dispersed from the mean, the
population variance and standard deviation will be large.The
variance overcomes the weakness of the range by using all the
values in the population
3-*
VARIANCE The arithmetic mean of the squared deviations
from the mean.
STANDARD DEVIATION The square root of the variance.
LO8 Compute and interpret the standard deviation.
*
Variance – Formula and Computation
Steps in Computing the Variance.
Step 1: Find the mean.
Step 2: Find the difference between each observation and the

mean, and
square that difference.
Step 3: Sum all the squared differences found in step 2
Step 4: Divide the sum of the squared differences by the number
of items in
the population.
3-*
LO8
*
EXAMPLE – Variance and Standard Deviation
The number of traffic citations issued during the last five
months in Beaufort County, South Carolina, is reported below:
What is the population variance?
Step 1: Find the mean.
Step 2: Find the difference between each observation and the
mean, and square that difference.
Step 4: Divide the sum of the squared differences by the number
of items in the population.
3-*
LO8

*
EXAMPLE – Variance and Standard Deviation
The number of traffic citations issued during the last twelve
months in Beaufort County, South Carolina, is reported below:
What is the population variance?
Step 2: Find the difference between each
observation and the mean,
and square that difference.
Step 4: Divide the sum of the squared differences
by the number of items in the population.
3-*
LO8
*
Sample Variance
3-*
LO8

*
EXAMPLE – Sample Variance
The hourly wages for a sample of part-time employees at
Home Depot are: $12, $20, $16, $18, and $19.
What is the sample variance?
3-*
LO8
*
Sample Standard Deviation
3-*
LO8
*
Chebyshev’s Theorem
The arithmetic mean biweekly amount contributed by the
Dupree Paint employees to the company’s profit-sharing plan is

$51.54, and the standard deviation is $7.51. At least what
percent of the contributions lie within plus 3.5 standard
deviations and minus 3.5 standard deviations of the mean?
3-*
LO9 Explain Chebyshev’s Theorem and the Empirical Rule.
*
The Empirical Rule
3-*
LO9
*
The Arithmetic Mean of Grouped Data
3-*
LO10 Compute the mean and standard deviation of grouped
data.
*
The Arithmetic Mean of Grouped Data - Example
Recall in Chapter 2, we constructed a frequency
distribution for Applewood Auto Group profit data for 180

vehicles sold. The information is repeated on the table.
Determine the arithmetic mean profit per vehicle.
3-*
LO10
*
The Arithmetic Mean of Grouped Data - Example
3-*
LO10
*
Standard Deviation of Grouped Data - Example
Refer to the frequency distribution for the Applewood
Auto Group data used earlier. Compute the standard deviation
of the vehicle profits.
3-*
LO10
*
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Describing Data:
Displaying and Exploring Data
Chapter 4
McGraw-Hill/Irwin
*
LO1 Construct and interpret a dot plot.
LO2 Construct and describe a stem-and-leaf display.
LO3 Identify and compute measures of position.
LO4 Construct and analyze a box plot.

LO5 Compute and describe the coefficient of skewness.
LO6 Create and interpret a scatterplot.
LO7 Develop and explain a contingency table.
Learning Objectives
4-*
*
1. Dot Plot –Example
4-*
If you want to learn it, please go to
http://www.youtube.com/watch?v=N7HHmTpccZI
LO1
*
2. Stem-and-Leaf
Stem-and-leaf display is 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.
Advantage of the stem-and-leaf display over a frequency
distribution - the identity of each observation is not lost.
4-*
LO2

*
3. Measures of Position
The standard deviation is the most widely used measure of
dispersion.
Alternative ways of describing spread of data include
determining the location of values that divide a set of
observations into equal parts.
These measures include quartiles, deciles, and percentiles.
4-*
*
4. Boxplot Example
Step1: Create an appropriate scale along the horizontal axis.
Step 2: 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).

Step 3: Extend horizontal lines from the box out to the
minimum value (13
minutes) and the maximum value (30 minutes).
4-*
LO4
*
.
5. Commonly Observed Shapes
4-*
The coefficient of skewness can range from -3 up to 3.
LO5
*
.
6. Describing Relationship between Two Variables – Scatter
Diagram
4-*
LO6
*

6. Contingency Tables
There are four dealerships in the Applewood Auto group.
Suppose we want to compare 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? The table below is the cross-tabulation of the raw
data of the two variables.
From the contingency table, we observe the following:From the
Total column on the right, 90 of the 180 cars sold 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 percent, of the cars sold were sold for a profit more
than the median.The percent profits above the median for the
other dealerships are 50 percent for Olean, 42 percent for
Sheffield, and 60 percent for Tionesta.
4-*
LO7
*
Dot Plots
A dot plot groups the data as little as possible and the identity

of an individual observation is not lost.
To develop a dot plot, each observation is simply displayed as a
dot 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.
4-*
LO1 Construct and interpret a dot plot.
*
Dot Plots - Examples
The Service Departments at Tionesta Ford Lincoln
Mercury 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 during the 24
working at the two Dealerships. Construct dot plots and report
summary statistics to compare the two dealerships.
4-*
LO1
*
Dot Plot – Minitab Example
4-*
LO1

*
Stem-and-Leaf
In Chapter 2, frequency distribution was used to organize data
into a meaningful form.
A major advantage to organizing the data into a frequency
distribution is that we get a quick visual picture of the shape of
the distribution.
There are two disadvantages, however, to organizing the data
into a frequency distribution:
The exact identity of each value is lost
Difficult to tell how the values within each class are
distributed.
One technique that is used to display quantitative information in
a condensed form is the stem-and-leaf display.
4-*
LO2 Construct and interpret a stem and leaf plot.
*
Stem-and-leaf Plot Example
Listed in Table 4–1 is the number of 30-second radio
advertising spots purchased by each of the 45 members of the
Greater Buffalo Automobile Dealers Association last year.
Organize the data into a stem-and-leaf display. Around
what values do the number of advertising spots tend to cluster?
What is the fewest number of spots purchased by a dealer? The

largest number purchased?
4-*
LO2
*
Stem-and-Leaf
Stem-and-leaf display is 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.
Advantage of the stem-and-leaf display over a frequency
distribution - the identity of each observation is not lost.
4-*
LO2
*
Stem-and-leaf Plot Example
The usual procedure is to sort the leaf values from the smallest
to largest.
4-*
LO2

*
Stem-and-leaf: Another Example (Minitab)
4-*
LO2
*
Measures of Position
The standard deviation is the most widely used measure of
dispersion.
Alternative ways of describing spread of data include
determining the location of values that divide a set of
observations into equal parts.
These measures include quartiles, deciles, and percentiles.
4-*
*
Percentile Computation

To formalize the computational procedure, let Lp refer to the
location of a desired percentile. So if we wanted to find the
33rd percentile we would use L33 and if we wanted the median,
the 50th percentile, then L50.
The number of observations is n, so if we want to locate the
median, its position is at (n + 1)/2, or we could write this as
(n + 1)(P/100), where P is the desired percentile.
4-*
LO3
*
Percentiles - Example
Listed below are the commissions earned last month by a
sample of 15 brokers at Salomon Smith Barney’s Oakland,
California, office.
$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.
4-*
LO3

*
.
Percentiles – Example (cont.)
Step 1: Organize the data from lowest to largest value
$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
4-*
LO3
*
Step 2: Compute the first and third quartiles. Locate L25
and L75 using:
4-*
LO3
*

In the previous example the location formula yielded a
whole number. What if there were 6 observations in the sample
with the following ordered observations: 43, 61, 75, 91, 101,
and 104 , that is n=6, and we wanted to locate the first quartile?
Locate the first value in the ordered array and then move
.75 of the distance between the first and second 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.
The 1st and 2nd values are 43 and 61. Moving 0.75 of the
distance between these numbers, the 25th percentile is 56.5,
obtained as 43 + 0.75*(61- 43)
4-*
LO3
*
Percentiles – Example (Minitab)
4-*
LO3

*
Box Plot
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.
4-*
LO4 Construct and analyze a box plot.
*
Boxplot - Example
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?
4-*
LO4
*
Boxplot Example
Step1: Create an appropriate scale along the horizontal axis.
Step 2: 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).
Step 3: Extend horizontal lines from the box out to the
minimum value (13
minutes) and the maximum value (30 minutes).
4-*
LO4
*
.
Boxplot – Using Minitab
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?
The MINITAB statistical software system was used to
develop the following chart and summary statistics. What can
we conclude about the distribution of the age of the buyers?
The median age of purchaser
was 46 yrs.
25 percent were more than
52.75 years of age
50 percent of the purchasers
were between the ages of 40
and 52.75 years
The distribution of age is
symmetric
4-*
LO4
*
Referring back to the Applewood Auto Group data, we use
Minitab to generate a box plot on the variable age of the buyers.
What can we conclude about the distribution of the variable age
of the buyer?
The box plot tells us that the median age of the purchasers was
46 years. About 25% were older than 52.75 years of age, about
half of them were between the ages of 40 and 52.75 years of
age. The distribution of age is symmetric.
Skewness

In Chapter 3, measures of central location (the mean, median,
and mode) for a set of observations and measures of data
dispersion (e.g. range and the standard deviation) were
introduced
Another characteristic of a set of data is the shape.
There are four shapes commonly observed:
symmetric,
positively skewed,
negatively skewed,
bimodal.
4-*
LO5 Compute and understand the coefficient of skewness.
*
Skewness - Formulas for Computing
The coefficient of skewness can range from -3 up to 3.
A value near -3, 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 that there is
no skewness present.
4-*
LO5
*
Commonly Observed Shapes

4-*
LO5
*
.
Skewness – An Example
Following are the earnings per share for a sample of 15
software companies for the year 2010. 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.
What is your conclusion regarding the shape of the distribution?
4-*
LO5
*
.
Skewness – An Example Using Pearson’s Coefficient
4-*
LO5
*

Skewness – A Minitab Example
4-*
LO5
*
Describing Relationship between Two Variables
When we study the relationship between two variables we refer
to the data as bivariate.
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).
4-*
LO6 Create and interpret a scatterplot.
*
Describing Relationship between Two Variables – Scatter

Diagram Examples
4-*
LO6
*
Diagram Excel Example
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 purchaser?
Would it be reasonable to conclude that the more
expensive vehicles are purchased by older Buyers?
4-*
LO6
*
Diagram Excel Example

4-*
LO6
*
Contingency Tables
A scatter diagram requires that both of the variables be at least
interval scale.
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.
4-*
LO7 Develop and explain a contingency table.
*
Contingency Tables
A contingency table is a cross-tabulation that simultaneously
summarizes two variables of interest.
Examples:
Students at a university are classified by gender and class rank.
A product is classified as acceptable or unacceptable and by the
shift (day, afternoon, 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.).
4-*
LO7
*
Contingency Tables – An Example
There are four dealerships in the Applewood Auto group.
Suppose we want to compare 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? The table below is the cross-tabulation of the raw
data of the two variables.
From the contingency table, we observe the following:From the
Total column on the right, 90 of the 180 cars sold 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 percent, of the cars sold were sold for a profit more
than the median.The percent profits above the median for the
other dealerships are 50 percent for Olean, 42 percent for
Sheffield, and 60 percent for Tionesta.
4-*
LO7
*
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Describing Data:
Frequency Tables, Frequency
Distributions, and Graphic Presentation
Chapter 2
McGraw-Hill/Irwin
*
LEARNING OBJECTIVES
LO1 Make a frequency table for a set of data.
LO2 Organize data into a bar chart.
LO3 Present a set of data in a pie chart.
LO4 Create a frequency distribution for a
data set.
LO5 Understand a relative frequency
distribution.
LO6 Present data from a frequency
distribution in a histogram or frequency

polygon.
2-*
*
1. Frequency Table/Distribution
TABLE 2–1 Frequency Table for Vehicles Sold Last Month at
Applewood Auto Group by Location
2-*
FREQUENCY TABLE A grouping of qualitative data into
mutually exclusive classes showing the number of observations
in each class.
LO2
*
.
2. Bar Charts
2-*
BAR CHART A graph in which the classes are reported on the
horizontal axis and the class frequencies on the vertical axis.
The class frequencies are proportional to the heights of the bars.
LO2 Organize data into a bar chart
*

3. Pie Charts
2-*
PIE CHART A chart that shows the proportion or percent that
each class represents of the total number of frequencies.
*
4. Histogram
2-*
HISTOGRAM A graph in which the classes are marked on the
The class frequencies are represented by the heights of the bars
and the bars are drawn adjacent to each other.Frequency Table
(class #, interval, frequency)Bar Chart3. Get rid of the gaps
LO6
*

Describing Data with Charts, Tables and Graphs - Example
The Applewood Auto Group (AAG)sells a wide range of
vehicles through its four dealerships. Ms. Kathryn Ball, a
member of the senior management team at AAG, 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 the
buyers. What are their ages? How many vehicles have they
previously purchased from one of the Applewood dealerships?
What type of vehicle did they purchase? Partial data for 180
customers are shown on the table on the right.
2-*
LO1 Make a frequency table for a set of data
*
Frequency Table

2-*
FREQUENCY TABLE A grouping of qualitative data into
in each class.
LO2
*
.
Bar Charts
2-*
BAR CHART A graph in which the classes are reported on the
The class frequencies are proportional to the heights of the bars.
LO2 Organize data into a bar chart
*
Pie Charts
2-*
PIE CHART A chart that shows the proportion or percent that
each class represents of the total number of frequencies.

*
Pie Chart Using Excel
2-*
LO3
*
.
Frequency Distribution
2-*
FREQUENCY DISTRIBUTION A grouping of data into
in each class.
LO4 Create a frequency distribution for a data set.
*
Bar Chart and Pie Chart Example
SkiLodges.com is test marketing its new website and is
interested in how easy its Web page design is to navigate. It
randomly selected 200 regular Internet users and asked them to
perform a search task on the Web page. Each person was asked

to rate the relative ease of navigation as poor, good, excellent,
or awesome. The results are shown in the table on the right.
1. What type of measurement scale is used for ease of
navigation?
2. Draw a bar chart for the survey results.
3. Draw a pie chart for the survey results.
2-*
LO2 and LO3
*
Bar Chart and Pie Chart Example – Ski Lodges.Com
1. What type of measurement scale is used for ease of
navigation?
2. Draw a bar chart for the survey results.
3. Draw a pie chart for the survey results.
2-*
LO2 and LO3
*
Relative Class FrequenciesClass frequencies can be converted
to relative class frequencies to show the fraction of the total
number of observations in each class. A relative frequency
captures the relationship between a class total and the total
number of observations.

2-*
LO5 Understand a relative frequency distribution.
*
Frequency Distribution
Class interval: The class interval is obtained by
subtracting the lower limit of a class from the lower limit of the
next class.
Class frequency: The number of observations in each
class.
Class midpoint: A point that divides a class into two equal
parts. This is the average of the upper and lower class limits.
2-*
LO5
*
EXAMPLE – Creating a Frequency Distribution Table
Kathryn Ball of the Applewood Auto Group wants to develop
tables, charts, and graphs to show the typical profit for each
sale. Table 2–4 reports the profit on each of the 180 vehicles
sold last month at the four Applewood locations.
What is the typical profit on each sale?

What is the largest profit on any sale?
What is the lowest profit on any sale?
Around what value did the profits tend to cluster?
2-*
LO6 Present data from a frequency distribution in a histogram
or frequency polygon.
*
Constructing a Frequency Table - ExampleStep 1: Decide on the
number of classes.
A useful recipe to determine the number of classes (k) is
the “2 to the k rule.” such that 2k > n.
There were 180 vehicles sold, so n = 180. If we try k = 7,
then 27 = 128, somewhat less than 180. Hence, 7 is not enough
classes. If we let k = 8, then 28 = 256, which is greater than
180. So the recommended number of classes is 8.
Step 2: Determine the class interval or width.
-L)/k where i is the class interval, H
is the highest observed value, L is the lowest observed value,
and k is the number of classes.
Round up to some convenient number, such as a multiple
of 10 or 100. Use a class width of $400
2-*
LO6

*
Constructing a Frequency Table - ExampleStep 1: Decide on the
number of classes.
A useful recipe to determine the number of classes (k) is
the “2 to the k rule.” such that 2k > n.
There were 180 vehicles sold, so n = 180. If we try k = 7,
then 27 = 128, somewhat less than 180. Hence, 7 is not enough
classes. If we let k = 8, then 28 = 256, which is greater than
180. So the recommended number of classes is 8.
Step 2: Determine the class interval or width.
-L)/k where i is the class interval, H
is the highest observed value, L is the lowest observed value,
and k is the number of classes.
Round up to some convenient number, such as a multiple
of 10 or 100. Use a class width of $400
2-*Class # kSample
#122438416532664712882569512101024=2^k
LO6

*
Constructing a Frequency Table - ExampleStep 3: Set the
individual class limits
2-*
LO6
*
Step 4: Tally the vehicle profits into the classes.
Step 5: Count the number of items in each class.
Constructing a Frequency Table
2-*
LO6
*

Relative Frequency Distribution
To convert a frequency distribution to a relative frequency
distribution, each of the class frequencies is divided by the total
number of observations.
TABLE 2–8 Relative Frequency Distribution of Profit for
Vehicles Sold Last Month at Applewood Auto Group
2-*
LO6
*
Graphic Presentation of a Frequency Distribution
The three commonly used graphic forms
are:HistogramsFrequency polygonsCumulative frequency
distributions
2-*
LO6
*

Histogram
2-*
HISTOGRAM A graph in which the classes are marked on the
The class frequencies are represented by the heights of the bars
and the bars are drawn adjacent to each other.
LO6
*
Histogram Using Excel
2-*
LO6
*
Frequency
PolygonA frequency polygon, similar to a histogram, also shows
the shape of a distribution
It consists of line segments connecting the class midpoints of
the class frequencies.
2-*
LO6
*

Histogram Versus Frequency PolygonBoth provide a quick
picture of the main characteristics of the data (highs, lows,
points of concentration, etc.)Advantage of the histogram – it
depicts each class as a rectangle, with the height of the
rectangular bar representing the number in each class.
Advantage of the frequency polygon - it allows us to compare
directly two or more frequency distributions.
2-*
LO6
.
*
Cumulative Frequency Distribution
2-*
LO6
*
.
Cumulative Frequency Distribution
2-*
LO6
*

What is Statistics?
Chapter 1
McGraw-Hill/Irwin
*
Learning Objectives
LO1 List ways statistics is used.
LO2 Know the differences between descriptive and inferential
statistics.
LO3 Understand the differences between a sample and a
population.
LO4 Explain the difference between qualitative and quantitative
variables.
LO5 Compare the differences between discrete and continuous
variables.
LO6 Recognize the levels of measurement in data.
1-*
*
1. The Big Picture and the Use of Statistics

McGraw-Hill/Irwin
Data collection and analysis for decision making
Use in Business
Confidence level: 95%, 98%, 99%
Significance level (α: 5%, 2%, 1%
“I keep saying that the sexy job in the next 10 years will
be statisticians,” said Hal Varian, chief economist at
Google. “And I’m not kidding.” - For Today’s Graduate, Just
One Word: Statistics, NY Times, 2009
*
2. Types of Statistics – Descriptive Statistics and Inferential
Statistics
Descriptive Statistics - methods of organizing, summarizing,
and presenting data in an informative way.
1-*
Inferential Statistics: A decision, estimate, prediction, or
generalization about a population, based on a sample.
*
3. Population versus Sample

1-*
*
4. Types of Variables
A. Qualitative or Attribute variable - the characteristic being
studied is nonnumeric.
EXAMPLES: Gender, religious affiliation, type of
automobile owned, state of birth, eye color are examples.
B. Quantitative variable - information is reported numerically.
EXAMPLES: balance in your checking account, minutes
remaining in class, or number of children in a family.
1-*
*
5. Quantitative Variables - Classifications
Quantitative variables can be classified as either discrete
or continuous.
A. Discrete variables: can only assume certain values and there
are usually “gaps” between values.
EXAMPLE: the number of bedrooms in a house, or the

number of hammers sold at the local Home Depot (1,2,3,…,etc).
B. Continuous variable can assume any value within a specified
range.
EXAMPLE: The pressure in a tire, the weight of a pork chop, or
the height of students in a class.
1-*
*
Summary of Types of Variables
1-*
*
.
6. Four Levels of Measurement
Nominal level - data that is classified into categories and
cannot be arranged in any particular order.
EXAMPLES: eye color, gender, religious affiliation.

Ordinal level – data arranged in some order, but the
differences between data values cannot be determined or are
meaningless.
EXAMPLE: During a taste test of 4 soft drinks, Mellow
Yellow was ranked number 1, Sprite number 2, Seven-up
number 3, and Orange Crush number 4.
Interval level - similar to the ordinal level, with the
additional property that meaningful amounts of differences
between data values can be determined. There is no natural zero
point.
EXAMPLE: Temperature on the Fahrenheit scale.
Ratio level - the interval level with an inherent zero
starting point. Differences and ratios are meaningful for this
level of measurement.
EXAMPLES: Monthly income of surgeons, or distance
traveled by manufacturer’s representatives per month.
1-*
*

Uses of StatisticsStatistics is one of the tools used to make
decisions in businessWe apply statistical concepts in our
livesAs a student of business or economics, basic knowledge
and skills to organize, analyze, and transform data and to
present the information.
1-*
*
Why Study Statistics?
Numerical information is everywhere
Statistical techniques are used to make decisions that affect our
daily lives
The knowledge of statistical methods will help you understand
how decisions are made and give you a better understanding of
how they affect you.
No matter what line of work you select, you will find
yourself faced with decisions where an understanding of data
analysis is helpful.
1-*
*

Who Uses Statistics?
Statistical techniques are used extensively by marketing,
accounting, quality control, consumers, professional sports
people, hospital administrators, educators, politicians,
physicians, etc...
1-*
*
Types of Statistics – Descriptive Statistics and Inferential
Statistics
Descriptive Statistics - methods of organizing, summarizing,
and presenting data in an informative way.
EXAMPLE 1: The United States government reports the
population of the United States was 179,323,000 in 1960;
203,302,000 in 1970; 226,542,000 in 1980; 248,709,000 in
1990, and 265,000,000 in 2000.
EXAMPLE 2: According to the Bureau of Labor Statistics,
the average hourly earnings of production workers was $17.90
for April 2008.
1-*
*

Types of Statistics – Descriptive Statistics and Inferential
Statistics
Inferential Statistics: A decision, estimate, prediction, or
generalization about a population, based on a sample.
Note: In statistics the word population and sample have a
broader meaning. A population or sample may consist of
individuals or objects
1-*
*
Population versus Sample
1-*
*
Why take a sample instead of studying every member of the
population?
Prohibitive cost of census

Destruction of item being studied may be required
Not possible to test or inspect all members of a population
being studied
1-*
*
Usefulness of a Sample in Learning about a Population
Using a sample to learn something about a population is
done extensively in business, agriculture, politics, and
government.
EXAMPLE: Television networks constantly monitor the
popularity of their programs by hiring Nielsen and other
organizations to sample the preferences of TV viewers.
1-*
*
Types of Variables
A. Qualitative or Attribute variable - the characteristic being
studied is nonnumeric.
EXAMPLES: Gender, religious affiliation, type of
automobile owned, state of birth, eye color are examples.
B. Quantitative variable - information is reported numerically.
EXAMPLES: balance in your checking account, minutes

remaining in class, or number of children in a family.
1-*
*
Quantitative Variables - Classifications
Quantitative variables can be classified as either discrete
or continuous.
A. Discrete variables: can only assume certain values and there
are usually “gaps” between values.
EXAMPLE: the number of bedrooms in a house, or the
number of hammers sold at the local Home Depot (1,2,3,…,etc).
B. Continuous variable can assume any value within a specified
range.
EXAMPLE: The pressure in a tire, the weight of a pork chop, or
the height of students in a class.
1-*
*

Summary of Types of Variables
1-*
*
.
Four Levels of Measurement
Nominal level - data that is classified into categories and
cannot be arranged in any particular order.
EXAMPLES: eye color, gender, religious affiliation.
Ordinal level – data arranged in some order, but the
differences between data values cannot be determined or are
meaningless.
EXAMPLE: During a taste test of 4 soft drinks, Mellow
Yellow was ranked number 1, Sprite number 2, Seven-up
number 3, and Orange Crush number 4.
Interval level - similar to the ordinal level, with the
additional property that meaningful amounts of differences
between data values can be determined. There is no natural zero
point.
EXAMPLE: Temperature on the Fahrenheit scale.

Ratio level - the interval level with an inherent zero
starting point. Differences and ratios are meaningful for this
level of measurement.
EXAMPLES: Monthly income of surgeons, or distance
traveled by manufacturer’s representatives per month.
1-*
*
Nominal-Level Data
Properties:
Observations of a qualitative variable can only be classified and
counted.
There is no particular order to the labels.
1-*
*
Ordinal-Level Data
Properties:
Data classifications are represented by sets of labels or names
(high, medium, low) that have relative values.
Because of the relative values, the data classified can be ranked
or ordered.
1-*

*
Interval-Level Data
Properties:
Data classifications are ordered according to the amount of the
characteristic they possess.
Equal differences in the characteristic are represented by equal
differences in the measurements.
Example: Women’s dress sizes listed on the table.
1-*
*
Ratio-Level DataPractically all quantitative data is recorded on
the ratio level of measurement.Ratio level is the “highest” level
of measurement.
Properties:
Data classifications are ordered according to the amount of the
characteristics they possess.
Equal differences in the characteristic are represented by equal
differences in the numbers assigned to the classifications.
The zero point is the absence of the characteristic and the ratio
between two numbers is meaningful.
1-*
.

*
Why Know the Level of Measurement of a Data?The level of
measurement of the data dictates the calculations that can be
done to summarize and present the data.To determine the
statistical tests that should be performed on the data
1-*
*
Summary of the Characteristics for Levels of Measurement
1-*
*
*
*
*
*
*
*
*
*

.
*
*
*
*
*
*
*
*
*
*
*
*
.
*
*
*
*
.
*

*
*
dataID_responseQ3. Our company has utilized
digital technologies to do the following in response to digital
trends - 1.create new adventure or products/servicesQ3. Our
company has utilized
trends - 2.have innovationQ3. Our company has utilized
trends - 3.Transform existing businessQ3. Our company has
utilized
trends - 4.Minimize the costQ3. Our company has utilized
trends - 5.Increase the revenueQ3. Our company has utilized
trends - 6.helps business operations and logisticsQ4. Our
company’s primarily strategy to strengthen its digital innovation
capabilities
through - 1. Developing existing employees' digital
capabilitiesQ4. Our
capabilities
through - 2. Contractors and consultantsQ4. Our
capabilities
through - 3. External relationships (e.g., partnerships and
other external collaboration)Q4. Our
capabilities
through - 4. Recruiting employees with digital talentQ4. Our
capabilities
through - 5. Recruiting leaders with digital talentQ4. Our

capabilities
through - 6. Mergers and acquisitionsQ5. Our company clear
digital strategic objectives on how to - 1.optimize business
operationsQ5. Our company clear digital strategic objectives on
how to - 2. improve efficiencyQ5. Our company clear
digital strategic objectives on how to - 3.increase
innovationsQ5. Our company clear digital strategic objectives
on how to - 4. manage external customer relationshipQ5. Our
company clear digital strategic objectives on how to - 5. manage
internal employee relationshipQ5. Our company clear digital
strategic objectives on how to - 6. manage external relationship
with partners and suppliers.Q6. I think - 1. It is important for
me to work for a company that is digitally enabled or is a
digital leader.Q6. I think - 2. Our company should help me
develop skills to thrive in a digital environment.Q6. I think - 3.
Our company provides me with the resources or opportunities to
develop skills to thrive in a digital environmentQ6. I think - 4.
In our company, I can develop the skills to thrive in a
more digital business environmentQ6. I think - 5. I often think
about quitting my job in our companyQ6. I think - 6. If I was
completely free to choose, I would prefer to working for another
this company.Q7. Operations - 1. Increase innovationQ7.
Operations - 2. Develop new core business lines or
products/servicesQ7. Operations - 3. Improve customer
experience and engagementQ7. Operations - 4. Increase the
market share and salesQ7. Operations - 5. Minimize the cost
and Increase efficiencyQ7. Operations - 6. Transform and
optimize business processQ8. Marketing - 1. Digital technology
helps market growthQ8. Marketing - 2. Digital technology
helps customer serviceQ8. Marketing - 3. Digital technology
helps fast market accessQ8. Marketing - 4. Digital
technology helps to predict the future market trends.Q8.
Marketing - 5. Digital technology helps influence and engage
customers.Q8. Marketing - 6. Help raise the search ranking/web
traffic and reach huge customer basesQ9. Employee and

Partners - 1. Increase employment engagement and
commitmentQ9. Employee and Partners - 2. Facilitate in-time
and transparent internal and external communicationsQ9.
Employee and Partners - 3. Improve/Strengthen relationships
with partners/suppliersQ9. Employee and Partners - 4. Improve
teamwork and organizational collaborationsQ10. Digital
technology usage
is necessary for our company due to pressure from customers -
1. Our customers are migrating to use digital technologyQ10.
Digital technology usage
2. Our customer prefers to use digital technologyQ10. Digital
technology usage
3. Our customers make purchasing decisions using digital
technologyQ11. Digital technology
usage is necessary for our company due to pressure from
competitors and partners - 1. Our competitors have adopted
digital technologyQ11. Digital technology
competitors and partners - 2. We aim to keep up with our
competitorsQ11. Digital technology
competitors and partners - 3. Using digital technology is
necessary to establish competitive advantagesQ11. Digital
technology
competitors and partners - 4. Out partners prefer to
communicate with us using digital technologyQ11. Digital
technology
competitors and partners - 5. Our suppliers or contractors prefer
us to use digital technologiesQ12. To compete and thrive in a
more
digital business environment, our company has difficulty in - 1.
building a significantly new or different talent base to

compete effectively in the digital economy.Q12. To compete
and thrive in a more
Ensuring employees to gain knowledge and skillsets about
how to effectively use different digital technological toolsQ12.
To compete and thrive in a more
Finding the right employees to engage with different digital
technological toolsQ13. I am satisfied with the company in the
following aspects - 1. Quality of product/serviceQ13. I am
satisfied with the company in the following aspects - 2.
Customer serviceQ13. I am satisfied with the company in
the following aspects - 3. ProfitabilityQ13. I am satisfied with
the company in the following aspects - 4. Pace of product
developmentQ13. I am satisfied with the company in the
following aspects - 5. Customer satisfactionQ14. We have
observed the following outcomes
with the use of digital technologies in our organization - 1.
Reduce marketing costQ14. We have observed the
following outcomes
Reduce communication/operation costQ14. We have
observed the following outcomes
Increase in revenue and profitQ14. We have observed the
following outcomes
Increase sales and/or market shareQ14. We have observed
the following outcomes
Better customer satisfactionQ14. We have observed the
following outcomes
Better relationships with external partners/suppliersQ14.
We have observed the following outcomes

More or higher employee satisfaction and engagementQ30. Your
gender_Male_YQ31.Age.OriginalAgeQ32. Your
race/ethnicityQ33. Education_Year_OriginalQ34. Years of
Work ExperienceQ35. Are you married_Yes_1Q36. Have you
founded a business
or directly helped in founding a business before?Q37. Your job
position in the company that you own or currently work the
mostQ38. Your company size (number of employees)Q39. Your
company industryQ40. Your company's year of
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EXAM #1
NAME_______________________________
BUS500C GRADES
_____________________________
Based on the MS Excel file on the Entrepreneurship Survey,
please see the data in the Excel file and provide the answers to
the following (The final submission can either in Excel or in
this MS Word file; graphs can be formatted using computer
programs or images drawn by hand. The original questionnaire
is in the attached pdf file. Only some of the questions of the
original survey are included in the Excel file (Q3-14 and Q30-
40. Original Age and Education Year are given):
1. Please list all the variables under two categories: (1)
Quantitative variables; (2) Qualitative variables ( 10 points,
Review PPT page 5)
2. Please list all the variables under three categories: (1)
Nominal; (2) Ordinal; (3) Interval/Ratio (10 Points, Review PPT
Page 8).

3. Choose any reasonable variables, please produce (1) one
Frequency table; and (2) one Bar/Pie chart (10 Points, Review
PPT Page 10 and page 12)
4. Choose any reasonable variable, please produce one
Histogram graph and write a paragraph of your observations
based on the graph (10 Points, Review PPT Page 13; hint: only
Interval/Ratio are meaningfully appropriate for histogram )
5. Please choose any reasonable variable and calculate its (1)
Mean, (2) Median, (3) Mode, (4) Range (5) and Variance (20
Points, Review PPT Page 15-17; hint: only Interval/Ratio are
meaningfully appropriate for such descriptive statistics)
6. Choose any reasonable variable, please calculate its Standard
Deviation and draw a graph for its potential population
distribution using the Empirical Rule (10 Points, Review PPT
Page 17-18; hint: only Interval/Ratio are meaningfully
appropriate for such descriptive statistics)
7. Choose any reasonable variable, please calculate its quartiles
and draw a Box plot (10 Points, Review PPT Page 21-22; hint:
only Interval/Ratio are meaningfully appropriate for such
descriptive statistics)
8. Choose any reasonable variable, please draw a Scatter plot
and write a paragraph of your observations based on the graph
(10 Points, Review PPT Page 24; hint: only Interval/Ratio are
meaningfully appropriate for such descriptive statistics. For
ordinal variables with 5 or more scales, such as Q3-14, you can
treat them as interval variables for this question).

9. Choose any reasonable variable, please draw a Contingency
table and write a paragraph of your observations based on the
table (10 Points, Review PPT Page 25)

Describing DataNumerical MeasuresChapter 3McGraw-.docx

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