This chapter aims to teach students how to compute and interpret various numerical descriptive measures of data, including measures of central tendency (mean, median, mode), variation (range, variance, standard deviation), and shape (skewness). It covers how to find quartiles and construct box-and-whisker plots. The chapter also discusses population summary measures, rules for describing variation around the mean, and interpreting correlation coefficients.

1. Statistics for Managers
Using Microsoft® Excel
4th Edition
Chapter 3
Numerical Descriptive Measures
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 3-1

2. Chapter Goals
After completing this chapter, you should be able to:

Compute and interpret the mean, median, and mode for a
set of data

Find the range, variance, standard deviation, and
coefficient of variation and know what these values mean

Apply the empirical rule and the Bienaymé - Chebshev rule
to describe the variation of population values around the
mean

Construct and interpret a box-and-whiskers plot

Compute and explain the correlation coefficient
 Use
Statistics numerical measures along with graphs, charts, and
for Managers Using
tables to describe data
Microsoft Excel, 4e © 2004
Chap 3-2
Prentice-Hall, Inc.

3. Chapter Topics

Measures of central tendency, variation, and
shape





Mean, median, mode, geometric mean
Quartiles
Range, inter- quartile range, variance and standard
deviation, coefficient of variation
Symmetric and skewed distributions
Population summary measures
Mean, variance, and standard deviation
Statistics for Managers Using
 The empirical rule and Bienaymé - Chebyshev rule
Microsoft Excel, 4e © 2004
Chap 3-3
Prentice-Hall, Inc.


4. Chapter Topics
(continued)

Five number summary and box-and-whisker
plots

Covariance and coefficient of correlation

Pitfalls in numerical descriptive measures and
ethical considerations
Statistics for Managers Using
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.
Chap 3-4

5. Summary Measures
Describing Data Numerically
Central Tendency
Quartiles
Variation
Shape
Arithmetic Mean
Range
Median
Inter - quartile Range
Mode
Variance
Geometric Mean
Standard Deviation
Statistics for Managers Using
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Skewness
Coefficient of Variation
Chap 3-5

6. Measures of Central Tendency
Overview
Central Tendency
Arithmetic Mean
Median
Mode
n
X=
∑X
i =1
n
Geometric Mean
X G = ( X1 × X 2 ×  × Xn )1/ n
i
Midpoint of
ranked
values
Statistics for Managers Using
Microsoft Excel, 4e © 2004
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Most
frequently
observed
value
Chap 3-6

7. Arithmetic Mean

The arithmetic mean (mean) is the most
common measure of central tendency

For a sample of size n:
n
X=
∑X
i=1
n
i
X1 + X 2 +  + Xn
=
n
Statistics for Managers Using
Sample size
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.
Observed values
Chap 3-7

8. Arithmetic Mean
(continued)



The most common measure of central tendency
Mean = sum of values divided by the number of values
Affected by extreme values (outliers)
0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9 10
Mean = 3
Mean = 4
1 + 2 + 3 + 4 + 5 15
=
=3
Statistics for Managers Using
5
5
1 + 2 + 3 + 4 + 10 20
=
=4
5
5
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.
Chap 3-8

9. Median

In an ordered array, the median is the “middle”
number (50% above, 50% below)
0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9 10
Median = 3
Median = 3

Not affected by extreme values
Statistics for Managers Using
Microsoft Excel, 4e © 2004
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Chap 3-9

10. Finding the Median

The location of the median:
n +1
Median position =
position in the ordered data
2


If the number of values is odd, the median is the middle number
If the number of values is even, the median is the average of
the two middle numbers
n +1
 Note that
is not the value of the median, only the
2
Statistics for Managers Using in the ranked data
position of the median
Microsoft Excel, 4e © 2004
Chap 3-10
Prentice-Hall, Inc.

11. Mode






A measure of central tendency
Value that occurs most often
Not affected by extreme values
Used for either numerical or categorical data
There may may be no mode
There may be several modes
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Statistics for Managers Using
Microsoft Excel, 4e © 2004
Mode = 9
Prentice-Hall, Inc.
0 1 2 3 4 5 6
No Mode
Chap 3-11

12. Review Example

Five houses on a hill by the beach
$2,000 K
House Prices:
$2,000,000
500,000
300,000
100,000
100,000
$500 K
$300 K
$100 K
Statistics for Managers Using
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.
$100 K
Chap 3-12

13. Review Example:
Summary Statistics
House Prices:
Mean:

Median: middle value of ranked data
= $300,000

$2,000,000
500,000
300,000
100,000
100,000

Mode: most frequent value
= $100,000
Sum 3,000,000
($3,000,000/5)
= $600,000
Statistics for Managers Using
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.
Chap 3-13

14. Which measure of location
is the “best”?

Mean is generally used, unless
extreme values (outliers) exist

Then median is often used, since
the median is not sensitive to
extreme values.

Example: Median home prices may be
reported for a region – less sensitive to
outliers
Statistics for Managers Using
Microsoft Excel, 4e © 2004
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Chap 3-14

15. Geometric Mean

Geometric mean

Used to measure the rate of change of a variable
over time
XG = ( X1 × X 2 ×  × Xn )
1/ n

Geometric mean rate of return

Measures the status of an investment over time
R G = [(1 + R1 ) × (1 + R 2 ) ×  × (1 + Rn )]1/ n − 1
Statistics for Managers Using
 Where R is the rate of return in time period i
Microsoft Excel, 4e i © 2004
Chap 3-15
Prentice-Hall, Inc.

16. Example
An investment of $100,000 declined to $50,000 at the
end of year one and rebounded to $100,000 at end
of year two:
X1 = $100,000
X 2 = $50,000
50% decrease
X3 = $100,000
100% increase
The overall two-year return is zero, since it started and
Statistics for Managers Using
ended at 4e © 2004
Microsoft Excel,the same level.
Chap 3-16
Prentice-Hall, Inc.

17. Example
(continued)
Use the 1-year returns to compute the arithmetic
mean and the geometric mean:
Arithmetic
mean rate
of return:
( −50%) + (100%)
X=
= 25%
2
Misleading result
1/ n
Geometric R G = [(1 + R1 ) × (1 + R 2 ) ×  × (1 + Rn )] − 1
mean rate
= [(1 + ( −50%)) × (1 + (100%))]1/ 2 − 1
More
of return:
Statistics for Managers Using
accurate
1/ 2
1/ 2
= [(. 2004
Microsoft Excel, 4e © 50) × (2)] − 1 = 1 − 1 = 0%
result
Chap 3-17
Prentice-Hall, Inc.

18. Quartiles

Quartiles split the ranked data into 4 segments with
an equal number of values per segment
25%
Q1
25%
25%
Q2
25%
Q3
The first quartile, Q1, is the value for which 25% of the
observations are smaller and 75% are larger
 Q is the same as the median (50% are smaller, 50% are
2
larger)
 Only 25% of the observations are greater than the third
Statistics for Managers Using
quartile
Microsoft Excel, 4e © 2004
Chap 3-18
Prentice-Hall, Inc.


19. Quartile Formulas
Find a quartile by determining the value in the
appropriate position in the ranked data, where
First quartile position:
Q1 = (n+1)/4
Second quartile position: Q2 = (n+1)/2 (the median position)
Third quartile position:
Q3 = 3(n+1)/4
where Using
Statistics for Managersn is the number of observed values
Microsoft Excel, 4e © 2004
Chap 3-19
Prentice-Hall, Inc.

20. Quartiles

Example: Find the first quartile
Sample Data in Ordered Array: 11 12 13 16 16 17 18 21 22
(n = 9)
Q1 = is in the (9+1)/4 = 2.5 position of the ranked data
so use the value half way between the 2nd and 3rd values,
so
Q1 = 12.5
Statistics for Managers Using
Q1 and Q3 are measures of noncentral location
Microsoft Excel,=4e © 2004
Q2 median, a measure of central tendency
Chap 3-20
Prentice-Hall, Inc.

21. Measures of Variation
Variation
Range

Interquartile
Range
Variance
Standard
Deviation
Coefficient
of Variation
Measures of variation give
information on the spread or
variability of the data
values.
Statistics for Managers Using
Microsoft Excel, 4e © 2004
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Same center,
different variation
Chap 3-21

22. Range


Simplest measure of variation
Difference between the largest and the smallest
observations:
Range = Xlargest – Xsmallest
Example:
0 1 2 3 4 5 6 7 8 9 10 11 12
Statistics for Managers Using= 14 - 1 = 13
Range
Microsoft Excel, 4e © 2004
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13 14
Chap 3-22

23. Disadvantages of the Range

Ignores the way in which data are distributed
7
8
9
10
11
12
Range = 12 - 7 = 5

7
8
9
10
11
12
Range = 12 - 7 = 5
Sensitive to outliers
1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5
Range = 5 - 1 = 4
1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,120
Statistics for Managers Using = 120 - 1 = 119
Range
Microsoft Excel, 4e © 2004
Chap 3-23
Prentice-Hall, Inc.

24. Inter - quartile Range

Can eliminate some outlier problems by using
the inter - quartile range

Eliminate some high- and low-valued
observations and calculate the range from the
remaining values

Interquartile range = 3rd quartile – 1st quartile
= Q3 – Q1
Statistics for Managers Using
Microsoft Excel, 4e © 2004
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Chap 3-24

25. Interquartile Range
Example:
X
minimum
Q1
25%
12
Median
(Q2)
25%
30
25%
45
X
Q3
maximum
25%
57
Interquartile range
= 57 – 30 = 27
Statistics for Managers Using
Microsoft Excel, 4e © 2004
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70
Chap 3-25

26. Variance

Average (approximately) of squared deviations
of values from the mean
n

Sample variance:
S =
2
Where
∑ (X − X)
i=1
2
i
n -1
X = arithmetic mean
n=
Statistics for Managerssample size
Using
X 2004
Microsoft Excel, 4e ©i = ith value of the variable X
Prentice-Hall, Inc.
Chap 3-26

27. Standard Deviation



Most commonly used measure of variation
Shows variation about the mean
Has the same units as the original data

n
Sample standard deviation:
S=
Statistics for Managers Using
Microsoft Excel, 4e © 2004
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∑ (X − X)
i=1
2
i
n -1
Chap 3-27

28. Calculation Example:
Sample Standard Deviation
Sample
Data (Xi) :
10
12
14
n=8
S =
=
15
17
18
18
24
Mean = X = 16
(10 − X)2 + (12 − X)2 + (14 − X)2 + + (24 − X )2
n −1
(10 −16)2 + (12 −16)2 + (14 −16)2 + + (24 −16)2
8 −1
126
Statistics for Managers Using
=
= 4.2426
Microsoft 7
Excel, 4e © 2004
Prentice-Hall, Inc.
A measure of the “average”
scatter around the mean
Chap 3-28

29. Measuring variation
Small standard deviation
Large standard deviation
Statistics for Managers Using
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Chap 3-29

30. Comparing Standard Deviations
Data A
11
12
13
14
15
16
17
18
19
20 21
Mean = 15.5
S = 3.338
20 21
Mean = 15.5
S = 0.926
20 21
Mean = 15.5
S = 4.570
Data B
11
12
13
14
15
16
17
18
19
Data C
11 12 13 14 15 16 17
Statistics for Managers Using
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.
18
19
Chap 3-30

31. Advantages of Variance and
Standard Deviation

Each value in the data set is used in the
calculation

Values far from the mean are given extra
weight
(because deviations from the mean are squared)
Statistics for Managers Using
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.
Chap 3-31

32. Coefficient of Variation

Measures relative variation

Always in percentage (%)

Shows variation relative to mean

Can be used to compare two or more sets of
data measured in different units
S
CV =
X
Statistics for Managers Using

Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.

 ⋅100%


Chap 3-32

33. Comparing Coefficient
of Variation

Stock A:
 Average price last year = $50
 Standard deviation = $5
S
$5
CVA =   ⋅ 100% =
⋅ 100% = 10%
X
$50
 

Stock B:
 Average price last year = $100
 Standard deviation = $5
S
$5
CVB =   Using
X
Statistics for Managers⋅ 100% = $100 ⋅ 100% = 5%

Microsoft Excel, 4e ©2004
Prentice-Hall, Inc.
Both stocks
have the same
standard
deviation, but
stock B is less
variable relative
to its price
Chap 3-33

34. Shape of a Distribution

Describes how data is distributed

Measures of shape

Symmetric or skewed
Left-Skewed
Symmetric
Right-Skewed
Mean < Median
Mean = Median
Median < Mean
Statistics for Managers Using
Microsoft Excel, 4e © 2004
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Chap 3-34

35. Using Microsoft Excel

Descriptive Statistics can be obtained
from Microsoft® Excel

Use menu choice:
tools / data analysis / descriptive statistics

Enter details in dialog box
Statistics for Managers Using
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.
Chap 3-35

36. Using Excel

Use menu choice:
tools / data analysis /
descriptive statistics
Statistics for Managers Using
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.
Chap 3-36

37. Using Excel
(continued)

Enter dialog box
details

Check box for
summary statistics
Statistics OK Managers Using
 Click for
Microsoft Excel, 4e © 2004
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Chap 3-37

38. Excel output
Microsoft Excel
descriptive statistics output,
using the house price data:
House Prices:
$2,000,000
500,000
300,000
100,000
100,000
Statistics for Managers Using
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.
Chap 3-38

39. Population Summary Measures

Population summary measures are called parameters

The population mean is the sum of the values in the
population divided by the population size, N
N
µ=
Where
∑X
i=1
N
i
X1 + X 2 +  + XN
=
N
μ = population mean
N = population
Statistics for Managers Using size
th
Microsoft Excel, 4eX© 2004 of the variable X
i = i value
Prentice-Hall, Inc.
Chap 3-39

40. Population Variance

Average of squared deviations of values from
the mean
N

Population variance:
Where
σ2 =
∑ (X − μ)
i=1
2
i
N
μ = population mean
N = population size
Statistics for Managers Using
X = th value
Microsoft Excel, 4e i © i2004 of the variable X
Prentice-Hall, Inc.
Chap 3-40

41. Population Standard Deviation



Most commonly used measure of variation
Shows variation about the mean
Has the same units as the original data

Population standard deviation:
N
σ=
Statistics for Managers Using
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.
(Xi − μ)2
∑
i=1
N
Chap 3-41

42. The Empirical Rule


If the data distribution is bell-shaped, then
the interval:
μ ± 1σ contains about 68% of the values in
the population or the sample
68%
μ
Statistics for Managers Using
Microsoft Excel, 4e © 2004 μ ± 1σ
Prentice-Hall, Inc.
Chap 3-42

43. The Empirical Rule


μ ± 2σ contains about 95% of the values in
the population or the sample
μ ± 3σ contains about 99.7% of the values
in the population or the sample
95%
μ ± 2σ
Statistics for Managers Using
Microsoft Excel, 4e © 2004
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99.7%
μ ± 3σ
Chap 3-43

44. Bienaymé - Chebyshev Rule

Regardless of how the data are distributed,
at least (1 - 1/k2) of the values will fall within
k standard deviations of the mean (for k > 1)

Examples:
At least
within
(1 - 1/12) = 0% ……..... k=1 (μ ± 1σ)
(1 - 1/22) = 75% …........ k=2 (μ ± 2σ)
(1 - 1/32) = 89% ………. k=3 (μ ± 3σ)
Statistics for Managers Using
Microsoft Excel, 4e © 2004
Chap 3-44
Prentice-Hall, Inc.

45. Exploratory Data Analysis

Box-and-Whisker Plot: A Graphical display of
data using 5-number summary:
Minimum -- Q1 -- Median -- Q3 -- Maximum
Example:
25%
Minimum
Minimum
25%
1st
1st
Quartile
Statistics for ManagersQuartile
Using
Microsoft Excel, 4e © 2004
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25%
Median
Median
25%
3rd
3rd
Quartile
Quartile
Maximum
Maximum
Chap 3-45

46. Shape of Box-and-Whisker Plots

The Box and central line are centered between the
endpoints if data are symmetric around the median
Min

Q1
Median
Q3
Max
A Box-and-Whisker plot can be shown in either vertical
or horizontal format
Statistics for Managers Using
Microsoft Excel, 4e © 2004
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Chap 3-46

47. Distribution Shape and
Box-and-Whisker Plot
Left-Skewed
Q1
Q2 Q3
Symmetric
Q1 Q2 Q3
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Right-Skewed
Q1 Q2 Q3
Chap 3-47

48. Box-and-Whisker Plot Example

Below is a Box-and-Whisker plot for the following
data:
Min
0
Q1
2
2
Q2
2
0 23 5
0 2 3 5
3
3
Q3
4
5
5
Max
10
27
27
27
This for Managers Using
Statisticsdata is right skewed, as the plot depicts
Microsoft Excel, 4e © 2004
Chap 3-48
Prentice-Hall, Inc.


49. The Sample Covariance

The sample covariance measures the strength of the
linear relationship between two variables (called
bivariate data)

The sample covariance:
n
cov ( X , Y ) =
∑ ( X − X)( Y − Y )
i=1
i
i
n −1
 Only concerned
Statistics for Managerswith the strength of the relationship
Using
 No causal effect is implied
Microsoft Excel, 4e © 2004
Chap 3-49
Prentice-Hall, Inc.

50. Interpreting Covariance

Covariance between two random variables:
cov(X,Y) > 0
X and Y tend to move in the same direction
cov(X,Y) < 0
X and Y tend to move in opposite directions
cov(X,Y) = 0
X and Y are independent
Statistics for Managers Using
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Chap 3-50

51. Coefficient of Correlation

Measures the relative strength of the linear
relationship between two variables

Sample coefficient of correlation:
n
r=
∑ ( X − X)( Y − Y )
i=1
n
i
( X i − X )2
∑
i
n
( Yi − Y )2
∑
=1
Statistics for iManagers Using i=1
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.
cov ( X , Y )
=
SX SY
Chap 3-51

52. Features of
Correlation Coefficient, r

Unit free

Ranges between –1 and 1

The closer to –1, the stronger the negative linear
relationship

The closer to 1, the stronger the positive linear
relationship

The closer to 0, the weaker any positive linear
relationship
Statistics for Managers Using
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Chap 3-52

53. Scatter Plots of Data with Various
Correlation Coefficients
Y
Y
r = -1
Y
X
Y
r = -.6
X
Y
Y
Statistics for Managers Using
X
Microsoft Excel, 4e © 2004 r = +.3
r = +1
Prentice-Hall, Inc.
r=0
X
X
r=0
Chap 3-53
X

54. Using Excel to Find
the Correlation Coefficient


Choose Correlation from
the selection menu

Statistics for Managers Using
Microsoft Excel, 4e © 2004
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Select
Tools/Data Analysis
Click OK . . .
Chap 3-54

55. Using Excel to Find
the Correlation Coefficient
(continued)
Input data range and select
appropriate options
 Click OK to get output
Statistics for Managers Using
Microsoft Excel, 4e © 2004
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
Chap 3-55

56. Interpreting the Result

Scatter Plot of Test Scores
r = .733
100

There is a relatively
strong positive linear
relationship between
test score #1
and test score #2
Test #2 Score
95
90
85
80
75
70
70
75
80
85
90
95
Test #1 Score
Students who scored high on the first test tended
to score high on second test
Statistics for Managers Using
Microsoft Excel, 4e © 2004
Chap 3-56
Prentice-Hall, Inc.

100

57. Pitfalls in Numerical
Descriptive Measures

Data analysis is objective


Should report the summary measures that best meet
the assumptions about the data set
Data interpretation is subjective

Should be done in fair, neutral and clear manner
Statistics for Managers Using
Microsoft Excel, 4e © 2004
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Chap 3-57

58. Ethical Considerations
Numerical descriptive measures:



Should document both good and bad results
Should be presented in a fair, objective and
neutral manner
Should not use inappropriate summary
measures to distort facts
Statistics for Managers Using
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.
Chap 3-58

59. Chapter Summary

Described measures of central tendency

Mean, median, mode, geometric mean

Discussed quartiles

Described measures of variation


Range, interquartile range, variance and standard
deviation, coefficient of variation
Illustrated shape of distribution
Symmetric, skewed, box-and-whisker plots
Statistics for Managers Using
Microsoft Excel, 4e © 2004
Chap 3-59
Prentice-Hall, Inc.


60. Chapter Summary
(continued)

Discussed covariance and correlation
coefficient

Addressed pitfalls in numerical descriptive
measures and ethical considerations
Statistics for Managers Using
Microsoft Excel, 4e © 2004
Prentice-Hall, Inc.
Chap 3-60