1. Measures of Variability
Range
Interquartile range
Variance
Standard deviation
Coefficient of variation

2. Consider the sample of starting salaries of business grads
Consider the sample of starting salaries of business grads. We would be interested in knowing if there was a low or high degree of variability or dispersion in starting salaries received.

3. Range
Range is simply the difference between the largest and smallest values in the sample
Range is the simplest measure of variability.
Note that range is highly sensitive to the largest and smallest values.

4. Example: Apartment Rents
Seventy studio apartments
were randomly sampled in
a small college town. The
monthly rent prices for
these apartments are listed
in ascending order on the next slide.

5. Range = largest value - smallest value

6. 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.

7. Interquartile Range = Q3 - Q1 = 525 - 445 = 80
3rd Quartile (Q3) = 525
1st Quartile (Q1) = 445
Interquartile Range = Q3 - Q1 = = 80

8. Variance
The variance is a measure of variability that uses all the data
The variance is based on the difference between each observation (xi) and the mean ( for the sample and μ for the population).

9. For the population: For the sample:
The variance is the average of the squared differences between the observations and the mean value
For the population:
For the sample:

10. Standard Deviation
The Standard Deviation of a data set is the square root of the variance.
The standard deviation is measured in the same units as the data, making it easy to interpret.

11. Computing a standard deviation
For the population:
For the sample:

12. Coefficient of Variation
Just divide the standard deviation by the mean and multiply times 100
Computing the coefficient of variation:
For the population
For the sample

13. The heights (in inches) of 25 individuals were recorded and the following statistics were calculated mean = 70range = 20mode = 73variance = 784median = 74 The coefficient of variation equals 10
11.2%
1120%
0.4%
40% 5

14. squared divided by (n - 1) rounded down rounded up
If index i (which is used to determine the location of the pth percentile) is not an integer, its value should be 10
squared
divided by (n - 1)
rounded down
rounded up 5

15. Which of the following symbols represents the variance of the population?10 s2 s m 5

16. Which of the following symbols represents the size of the sample10 5

17. The symbol s is used to represent
the variance of the population
the standard deviation of the sample
the standard deviation of the population
the variance of the sample 10 5

18. The numerical value of the variance
is always larger than the numerical value of the standard deviation
is always smaller than the numerical value of the standard deviation
is negative if the mean is negative
can be larger or smaller than the numerical value of the standard deviation 10 5

19. If the coefficient of variation is 40% and the mean is 70, then the variance is28
2800
1.75
784 10 5

20. Problem 22, page 94

21. Broker-Assisted 100 Shares at $50 per Share
Range
45.05
Interquartile Range
23.98
Variance
190.67
Standard Deviation
13.8
Coefficient of Variation
38.02
25th percentile 6
75th percentile 18
interquart 25
24.995
interquart 75
48.975
Mean
36.32

22. Online 500 Shares at $50 per Share
Range
57.50
Interquartile Range
11.475
Variance
Standard Deviation
11.859
Coefficient of Variation
57.949
25th percentile
75th percentile
interquart 25
13.475
interquart 75
24.95
Mean
20.46

23. The variability of commissions is greater for broker-assisted trades

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

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

26. Using Excel’s Descriptive Statistics Tool
Step 4 When the Descriptive Statistics dialog box
appears:
Enter B1:B71 in the Input Range box
Select Grouped By Columns
Select Labels in First Row
Select Output Range
Enter D1 in the Output Range box
Select Summary Statistics
Click OK

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

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

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

30. Measures of Relative Location and Detecting Outliers
z-scores
Chebyshev’s Theorem
Detecting Outliers
By using the mean and standard deviation together, we can learn more about the relative location of observations in a data set

31. z-score The z-score is compute for each xi : Where
Here we compare the deviation from the mean of a single observation to the standard deviation
The z-score is compute for each xi :
Where
zi is the z-score for xi
is the sample mean
s is the sample standard deviation

32. The z-score can be interpreted as the number of standard deviations xi is from the sample mean

33. Z-scores for the starting salary data
Graduate
Starting Salary
xi - x
z-score 1
2850
-90
-0.543 2
2950 10
0.060 3
3050
110
0.664 4
2880
-60
-0.362 5
2755
-185
-1.117 6
2710
-230
-1.388 7
2890
-50
-0.302 8
3130
190
1.147 9
2940
0.000
3325
385
2.324 11
2920
-20
-0.121 12

34. Chebyshev’s Theorem
At least (1-1/z2) of the data values must be within z standard deviations of the mean, where z is greater than 1.
This theorem enables us to make statements about the proportion of data values that must be within a specified number of standard deviations from the mean

35. Implications of Chebychev’s Theorem
At least .75, or 75 percent of the data values must be within 2 ( z = 2) standard deviations of the mean.
At least .89, or 89 percent, of the data values must be within 3 (z = 3) standard deviations of the mean.
At least .94, or 94percent, of the data values must be within 4 (z = 4) standard deviations from the mean.
Note: z must be greater than one but need not be an integer.

36. Chebyshev’s Theorem For example:
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.)

37. Detecting Outliers
You can use z-scores to detect extreme values in the data set, or “outliers.” In the case of very high z-scores (absolute values) it is a good idea to recheck the data for accuracy.