1. St. Edward’s University
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 Perhaps the most important measure of location is the mean.
The mean provides a measure of central location.
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
Fifty efficiency apartments were randomly
sampled in a small college town. The monthly rent
prices for these apartments are listed below.

8. Sample Mean
Xbar = SUM(X)/n

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 25 and 26th data values:
Median
Example: Apartment Rents
Averaging the 25 and 26th 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.
Caution: If the data are bimodal or multimodal,
Excel’s MODE function will incorrectly identify a
single mode.

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. 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.

16. Measures of Variability
Range
Variance
Standard Deviation
Coefficient of Variation

17. 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.

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

19. 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).
The variance is useful in comparing the variability
of two or more variables.

20. 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

21. 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.
The standard deviation is computed as follows:
for a
sample
for a
population

22. 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

23. 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.

24. Using Excel to Compute the Sample Variance, Standard Deviation, and Coefficient of Variation
Formula Worksheet A B C D E 1
Apart-
ment
Monthly
Rent ($) 2
525
Mean
=AVERAGE(B2:B71) 3
440
Median
=MEDIAN(B2:B71) 4
450
Mode
=MODE.SNGL(B2:B71) 5
615
Variance
=VAR.S(B2:B71) 6
480
Std. Dev.
=STDEV.S(B2:B71) 7
510
C.V.
=E6/E2*100
Note: Rows 8-71 are not shown.

25. Using Excel to Compute the Sample Variance, Standard Deviation, and Coefficient of Variation
Value Worksheet A B C D E 1
Apart-
ment
Monthly
Rent ($) 2
525
Mean
490.80 3
440
Median
475.00 4
450
Mode
450.00 5
615
Variance 6
480
Std. Dev.
54.74 7
510
C.V.
11.15
Note: Rows 8-71 are not shown.

26. 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)

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

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

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

30. 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.

31. 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 = pn + (1 – p)
Excel would compute the location of the 80th
percentile for the apartment rent data as follows:
L80 = (0.8)70 + (1 – 0.8) = = 56.2
The 80th percentile would be
( ) = =
Excel interpolates over the interval from 0 to n.

32. Using Excel’s Rank and Percentile Tool
to Compute Percentiles and Quartiles
Excel Formula Worksheet
80th percentile A B C D 1
Apart-
ment
Monthly
Rent ($) 80 th
Percentile 2
525
=PERCENTILE.INC(B2:B71,.8) 3
440 4
450 5
615 6
480
It is not necessary
to put the data
in ascending order.
Note: Rows 7-71 are not shown.

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

34. Third Quartile Using Excel’s QUARTILE.INC 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
( ) = =

35. Third Quartile Excel Formula Worksheet 3rd quartileB C D 1
Apart-
ment
Monthly
Rent ($)
Third Quartile 2
525
=QUARTILE.INC(B2:B71,3) 3
440 4
450 5
615 6
480
It is not necessary
to put the data
in ascending order.
Note: Rows 7-71 are not shown.

36. Using Excel’s QUARTILE.INC Function
If the value of 1 in the QUARTILE.INC function is
changed to 0, Excel computes the minimum value in
the data set.
If the value of 1 is changed to 4, Excel computes the
maximum value in the data set.

37. 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)

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

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

40. 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.

41. 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.

42. Trimmed Mean Another measure, sometimes used when extreme
values are present, is the trimmed mean.
It is obtained by deleting a percentage of the
smallest and largest values from a data set and then
computing the mean of the remaining values.
For example, the 5% trimmed mean is obtained by
removing the smallest 5% and the largest 5% of the
data values and then computing the mean of the
remaining values.

44. End of Chapter 3, Part A