1. Slides by
JOHN
LOUCKS
St. Edward’s
University

2. Chapter 3, Part A Descriptive Statistics: Numerical Measures
Measures of Location
Measures of Variability

3. Measures of Location Mean If the measures are computed
for data from a sample,
they are called sample statistics.
Median
Mode
Percentiles
If the measures are computed
for data from a population,
they are called population parameters.
Quartiles
A sample statistic is referred to
as the point estimator of the
corresponding population parameter.

4. Mean The mean of a data set is the average of all the data values.
The sample mean is the point estimator of the population mean m.

5. Sample Mean Sum of the values of the n observations Number of
in the sample

6. Population Mean m Sum of the values of the N observations Number of
observations in
the population

7. Sample Mean Example: Apartment Rents
Seventy efficiency apartments were randomly
sampled in a small college town. The monthly rent
prices for these apartments are listed below.

8. Sample Mean
Example: Apartment Rents

9. Median The median of a data set is the value in the middle
when the data items are arranged in ascending order.
Whenever a data set has extreme values, the median
is the preferred measure of central location.
The median is the measure of location most often
reported for annual income and property value data.
A few extremely large incomes or property values
can inflate the mean.

10. Median For an odd number of observations: 26 18 27 12 14 27 19
in ascending order
the median is the middle value.
Median = 19

11. Median For an even number of observations: 26 18 27 12 14 27 30 19
in ascending order
the median is the average of the middle two values.
Median = ( )/2 =

12. Averaging the 35th and 36th data values:
Median
Example: Apartment Rents
Averaging the 35th and 36th data values:
Median = ( )/2 =
Note: Data is in ascending order.

13. Mode The mode of a data set is the value that occurs with
greatest frequency.
The greatest frequency can occur at two or more
different values.
If the data have exactly two modes, the data are
bimodal.
If the data have more than two modes, the data are
multimodal.

14. 450 occurred most frequently (7 times)
Mode
Example: Apartment Rents
450 occurred most frequently (7 times)
Mode = 450
Note: Data is in ascending order.

15. Using Excel to Compute the Mean, Median, and Mode
Excel Formula Worksheet
Note: Rows 7-71 are not shown.

16. Using Excel to Compute the Mean, Median, and Mode
Value Worksheet
Note: Rows 7-71 are not shown.

17. Percentiles A percentile provides information about how the
data are spread over the interval from the smallest
value to the largest value.
Admission test scores for colleges and universities
are frequently reported in terms of percentiles.
The pth percentile of a data set is a value such that at least p percent of the items take on this value or less and at least (100 - p) percent of the items take on this value or more.

18. Percentiles Arrange the data in ascending order.
Compute index i, the position of the pth percentile.
i = (p/100)n
If i is not an integer, round up. The p th percentile
is the value in the i th position.
If i is an integer, the p th percentile is the average
of the values in positions i and i +1.

19. Averaging the 56th and 57th data values:
80th Percentile
Example: Apartment Rents
i = (p/100)n = (80/100)70 = 56
Averaging the 56th and 57th data values:
80th Percentile = ( )/2 = 542
Note: Data is in ascending order.

20. 80th Percentile Example: Apartment Rents “At least 80% of the
items take on a
value of 542 or less.”
“At least 20% of the
items take on a
value of 542 or more.”
56/70 = .8 or 80%
14/70 = .2 or 20%

21. Using Excel’s Rank and Percentile Tool
to Compute Percentiles and Quartiles
Using Excel’s Percentile Function
The formula Excel uses to compute the location (Lp)
of the pth percentile is
Lp = (p/100)n + (1 – p/100)
Excel would compute the location of the 80th
percentile for the apartment rent data as follows:
L80 = (80/100)70 + (1 – 80/100) = = 56.2
The 80th percentile would be
( ) = =

22. Using Excel’s Rank and Percentile Tool
to Compute Percentiles and Quartiles
Excel Formula Worksheet
80th percentile
It is not necessary
to put the data
in ascending order.
Note: Rows 7-71 are not shown.

23. Using Excel’s Rank and Percentile Tool
to Compute Percentiles and Quartiles
Excel Value Worksheet
Note: Rows 7-71 are not shown.

24. Quartiles Quartiles are specific percentiles.
First Quartile = 25th Percentile
Second Quartile = 50th Percentile = Median
Third Quartile = 75th Percentile

25. Third quartile = 75th percentile
Example: Apartment Rents
Third quartile = 75th percentile
i = (p/100)n = (75/100)70 = 52.5 = 53
Third quartile = 525
Note: Data is in ascending order.

26. Third Quartile Using Excel’s Quartile Function
Excel computes the locations of the 1st, 2nd, and 3rd
quartiles by first converting the quartiles to
percentiles and then using the following formula to
compute the location (Lp) of the pth percentile:
Lp = (p/100)n + (1 – p/100)
Excel would compute the location of the 3rd quartile
(75th percentile) for the rent data as follows:
L75 = (75/100)70 + (1 – 75/100) = = 52.75
The 3rd quartile would be
( ) = =

27. Third Quartile Excel Formula Worksheet 3rd quartile
It is not necessary
to put the data
in ascending order.
Note: Rows 7-71 are not shown.

28. Third Quartile
Excel Value Worksheet
Note: Rows 7-71 are not shown.

29. Excel’s Rank and Percentile Tool
Step 1 Click the Data tab on the Ribbon
Step 2 In the Analysis group, click Data Analysis
Step 3 Choose Rank and Percentile from the list of
Analysis Tools
Step 4 When the Rank and Percentile dialog box appears
(see details on next slide)

30. Excel’s Rank and Percentile Tool
Step 4 Complete the Rank and Percentile dialog
box as follows:

31. Excel’s Rank and Percentile Tool
Excel Value Worksheet
Note: Rows are not shown.

32. Averages and Standard Deviation Ch 3
Geometric Mean (GM)
The Geometric Mean is useful in finding the averages of increases in:
Percents
Ratios
Indexes
Growth Rates
The Geometric Mean will always be less than or equal to (never more than) the arithmetic mean
The GM gives a more conservative figure that is not drawn up by large values in the set
Statistics Is Fun! & Statistics Are Fun!

33. Averages and Standard Deviation Ch 3
Geometric Mean
The GM of a set of n positive numbers is defined as the nth root of the product of n values. The formula is:
Statistics Is Fun! & Statistics Are Fun!

34. Geometric Mean Example 1: Percentage Increase
Averages and Standard Deviation Ch 3
Geometric Mean Example 1: Percentage Increase
Statistics Is Fun! & Statistics Are Fun!

35. Verify Geometric Mean Example
Averages and Standard Deviation Ch 3
Verify Geometric Mean Example
Statistics Is Fun! & Statistics Are Fun!

36. Another Use Of GM: Ave. % Increase Over Time
Averages and Standard Deviation Ch 3
Another Use Of GM: Ave. % Increase Over Time
Another use of the geometric mean is to determine the percent increase in sales, production or other business or economic series from one time period to another
Where n = number of periods
Statistics Is Fun! & Statistics Are Fun!

37. Example for GM: Ave. % Increase Over Time
Averages and Standard Deviation Ch 3
Example for GM: Ave. % Increase Over Time
The total number of females enrolled in American colleges increased from 755,000 in 1992 to 835,000 in That is, the geometric mean rate of increase is 1.27%.
The annual rate of increase is 1.27%
For the years 1992 through 2000, the rate of female enrollment growth at American colleges was 1.27% per year
Statistics Is Fun! & Statistics Are Fun!

38. Measures of Variability
It is often desirable to consider measures of variability
(dispersion), as well as measures of location.
For example, in choosing supplier A or supplier B we
might consider not only the average delivery time for
each, but also the variability in delivery time for each.

39. Measures of Variability
Range
Interquartile Range
Variance
Standard Deviation
Coefficient of Variation

40. Range The range of a data set is the difference between the
largest and smallest data values.
It is the simplest measure of variability.
It is very sensitive to the smallest and largest data
values.

41. Range = largest value - smallest value
Example: Apartment Rents
Range = largest value - smallest value
Range = = 190
Note: Data is in ascending order.

42. Interquartile Range
The interquartile range of a data set is the difference
between the third quartile and the first quartile.
It is the range for the middle 50% of the data.
It overcomes the sensitivity to extreme data values.

43. Interquartile Range = Q3 - Q1 = 525 - 445 = 80
Example: Apartment Rents
3rd Quartile (Q3) = 525
1st Quartile (Q1) = 445
Interquartile Range = Q3 - Q1 = = 80
Note: Data is in ascending order.

44. Variance The variance is a measure of variability that utilizes
all the data.
It is based on the difference between the value of
each observation (xi) and the mean ( for a sample,
m for a population).

45. Variance The variance is the average of the squared
differences between each data value and the mean.
The variance is computed as follows:
for a
sample
for a
population

46. Standard Deviation
The standard deviation of a data set is the positive
square root of the variance.
It is measured in the same units as the data, making
it more easily interpreted than the variance.

47. Standard Deviation The standard deviation is computed as follows:
for a
sample
for a
population

48. Coefficient of Variation
The coefficient of variation indicates how large the
standard deviation is in relation to the mean.
The coefficient of variation is computed as follows:
for a
sample
for a
population

49. Sample Variance, Standard Deviation, And Coefficient of Variation
Example: Apartment Rents
Variance
Standard Deviation
the standard
deviation is
about 11%
of the mean
Coefficient of Variation

50. Using Excel to Compute the Sample Variance, Standard Deviation, and Coefficient of Variation
Formula Worksheet
Note: Rows 8-71 are not shown.

51. Using Excel to Compute the Sample Variance, Standard Deviation, and Coefficient of Variation
Value Worksheet
Note: Rows 8-71 are not shown.

52. Using Excel’s Descriptive Statistics Tool
Step 1 Click the Data tab on the Ribbon
Step 2 In the Analysis group, click Data Analysis
Step 3 Choose Descriptive Statistics from the list of
Analysis Tools
Step 4 When the Descriptive Statistics dialog box appears:
(see details on next slide)

53. Using Excel’s Descriptive Statistics Tool
Excel’s Descriptive Statistics Dialog Box

54. Using Excel’s Descriptive Statistics Tool
Excel Value Worksheet (Partial)
Note: Rows 9-71 are not shown.

55. Using Excel’s Descriptive Statistics Tool
Excel Value Worksheet (Partial)
Note: Rows 1-8 and are not shown.

56. End of Chapter 3, Part A

57. Chapter 3, Part B Descriptive Statistics: Numerical Measures
Measures of Distribution Shape, Relative Location, and Detecting Outliers
Exploratory Data Analysis

58. Measures of Distribution Shape, Relative Location, and Detecting Outliers
z-Scores
Chebyshev’s Theorem
Empirical Rule
Detecting Outliers

59. Distribution Shape: Skewness
An important measure of the shape of a distribution is called skewness.
The formula for computing skewness for a data set is somewhat complex.
Skewness can be easily computed using statistical software.
Excel’s SKEW function can be used to compute the
skewness of a data set.

60. Distribution Shape: Skewness
Symmetric (not skewed)
Skewness is zero.
Mean and median are equal.
Skewness = 0
.05
.10
.15
.20
.25
.30
.35
Relative Frequency

61. Distribution Shape: Skewness
Moderately Skewed Left
Skewness is negative.
Mean will usually be less than the median.
Skewness =
Relative Frequency
.05
.10
.15
.20
.25
.30
.35

62. Distribution Shape: Skewness
Moderately Skewed Right
Skewness is positive.
Mean will usually be more than the median.
Skewness = .31
Relative Frequency
.05
.10
.15
.20
.25
.30
.35

63. Distribution Shape: Skewness
Highly Skewed Right
Skewness is positive (often above 1.0).
Mean will usually be more than the median.
Skewness = 1.25
Relative Frequency
.05
.10
.15
.20
.25
.30
.35

64. Distribution Shape: Skewness
Example: Apartment Rents
Seventy efficiency apartments were randomly
sampled in a college town. The monthly rent prices
for the apartments are listed below in ascending order.

65. Distribution Shape: Skewness
Example: Apartment Rents
.05
.10
.15
.20
.25
.30
.35
Skewness = .92
Relative Frequency

66. z-Scores The z-score is often called the standardized value.
It denotes the number of standard deviations a data
value xi is from the mean.

67. z-Scores An observation’s z-score is a measure of the relative
location of the observation in a data set.
A data value less than the sample mean will have a
z-score less than zero.
A data value greater than the sample mean will have
a z-score greater than zero.
A data value equal to the sample mean will have a
z-score of zero.

68. Standardized Values for Apartment Rents
z-Scores
Example: Apartment Rents
z-Score of Smallest Value (425)
Standardized Values for Apartment Rents

69. Chebyshev’s Theorem
At least (1 - 1/z2) of the items in any data set will be
within z standard deviations of the mean, where z is
any value greater than 1.

70. Chebyshev’s Theorem At least of the data values must be 75%
within of the mean.
75%
z = 2 standard deviations
At least of the data values must be
within of the mean.
89%
z = 3 standard deviations
At least of the data values must be
within of the mean.
94%
z = 4 standard deviations

71. Chebyshev’s Theorem Example: Apartment Rents
Let z = 1.5 with = and s = 54.74
At least (1 - 1/(1.5)2) = = 0.56 or 56%
of the rent values must be between
- z(s) = (54.74) = 409
and
+ z(s) = (54.74) = 573
(Actually, 86% of the rent values
are between 409 and 573.)

72. +/- 1 standard deviation
Empirical Rule
For data having a bell-shaped distribution:
of the values of a normal random variable
are within of its mean.
68.26%
+/- 1 standard deviation
of the values of a normal random variable
are within of its mean.
95.44%
+/- 2 standard deviations
of the values of a normal random variable
are within of its mean.
99.72%
+/- 3 standard deviations

73. Empirical Rule x 99.72% 95.44% 68.26% m m – 3s m – 1s m + 1s m + 3s
99.72%
95.44%
68.26% x m
m – 3s
m – 1s
m + 1s
m + 3s
m – 2s
m + 2s

74. Detecting Outliers An outlier is an unusually small or unusually large
value in a data set.
A data value with a z-score less than -3 or greater
than +3 might be considered an outlier.
It might be:
an incorrectly recorded data value
a data value that was incorrectly included in the
data set
a correctly recorded data value that belongs in
the data set

75. Standardized Values for Apartment Rents
Detecting Outliers
Example: Apartment Rents
The most extreme z-scores are and 2.27
Using |z| > 3 as the criterion for an outlier, there
are no outliers in this data set.
Standardized Values for Apartment Rents

76. Exploratory Data Analysis
Five-Number Summary
Box Plot

77. Five-Number Summary 1 Smallest Value 2 First Quartile 3 Median 4
Third Quartile 5
Largest Value

78. Five-Number Summary Example: Apartment Rents Lowest Value = 425
First Quartile = 445
Median = 475
Third Quartile = 525
Largest Value = 615

79. Box Plot Example: Apartment Rents
A box is drawn with its ends located at the first and
third quartiles.
A vertical line is drawn in the box at the location of
the median (second quartile).
400
425
450
475
500
525
550
575
600
625
Q1 = 445
Q3 = 525
Q2 = 475

80. Box Plot
Limits are located (not drawn) using the interquartile range (IQR).
Data outside these limits are considered outliers.
The locations of each outlier is shown with the symbol * .
continued

81. Box Plot Example: Apartment Rents
The lower limit is located 1.5(IQR) below Q1.
Lower Limit: Q (IQR) = (80) = 325
The upper limit is located 1.5(IQR) above Q3.
Upper Limit: Q (IQR) = (80) = 645
There are no outliers (values less than 325 or
greater than 645) in the apartment rent data.

82. Box Plot Example: Apartment Rents
Whiskers (dashed lines) are drawn from the ends
of the box to the smallest and largest data values
inside the limits.
400
425
450
475
500
525
550
575
600
625
Smallest value
inside limits = 425
Largest value
inside limits = 615

83. End of Chapter 3, Part B