1. LESSON 4: MEASURES OF VARIABILITY AND PROPORTION
Outline
The range, variance, standard deviation and coefficient of variation
Interpretation of standard deviation
Population and sample variance
Approximation from the grouped data
Skewness
Interquartile range and box plots
The proportion

2. MEASURES OF VARIABILITY: EXAMPLE
Heights of players of two teams in inches are as follows:
Team I: 72,73,76,76,78, so mean=75, median=mode=76
Team II: 67,72,76,76,84, so mean=75, median=mode=76
How about the variation?

3. MEASURES OF VARIABILITY RANGE
The first and simplest measure of variability is the range.
The range of a set of measurements is the numerical difference between the largest and smallest measurements.
Range = Largest value - Smallest value

4. MEASURES OF VARIABILITY RANGE
Team I Range
= 78-72
= inches
Team II Range
= 84-67
= inches
So, Team I variation is
a. less
b. more

5. MEASURES OF VARIABILITY VARIANCE, STANDARD DEVIATION, CV
A major drawback of the range is that it uses only two extreme values, ignores all the intermediate values, and provides no information on the dispersion of the values between the smallest and largest observations.
On the other hand, variance / standard deviation / CV, uses all the values and provides information on the dispersion of the intermediate values
Computation of variance / standard deviation / CV requires computation of deviation from the mean

6. MEASURES OF VARIABILITY VARIANCE, STANDARD DEVIATION, CV
Team I deviations from the mean:
(72-75)=-3, (73-75)=-2, (76-75)=1, ( )= , ( )=

7. MEASURES OF VARIABILITY VARIANCE, STANDARD DEVIATION, CV
Team I deviations from the mean:
-3, -2, 1, ,
From the property of mean (see Lesson 3, Slides 10-11), sum of deviations from the mean is zero. Check =

8. MEASURES OF VARIABILITY VARIANCE, STANDARD DEVIATION, CV
Sum of squared deviations from the mean is not necessarily 0 e.g., sum of squared deviations
Although sum of squared deviations increases if the dispersion increases, the sum depends on the number of measurements. So, mean squared deviations is a preferred measure of dispersion.

9. MEASURES OF VARIABILITY VARIANCE, STANDARD DEVIATION, CV
Variance is the mean squared deviation
For example, Team I variance

10. MEASURES OF VARIABILITY VARIANCE, STANDARD DEVIATION, CV
Standard deviation is the root mean squared deviation i.e., square root of variance.
So, Team I standard deviation

11. MEASURES OF VARIABILITY VARIANCE, STANDARD DEVIATION, CV
Coefficient of variation is the standard deviation divided by the mean.
So, Team I coefficient of variation

12. MEASURES OF VARIABILITY VARIANCE, STANDARD DEVIATION, CV
Why there are three similar terms?
In the above example, variance has unit inch2
But, standard deviation has unit inch - the unit of the original data. So, standard deviation may sometimes be preferred over variance.
Coefficient of variation is dimension less. Hence, coefficient of variation is a useful quantity for comparing the variability in data sets having different standard deviations and different means.

13. MEASURES OF VARIABILITY VARIANCE, STANDARD DEVIATION, CV
Interpretation of standard deviation
It’s difficult to interpret
A higher standard deviation implies a greater variability
Standard deviation is widely used to approximate the proportion of measurements that fall into various intervals of values. This is specially true if the data has a bell-shaped distribution.

14. MEASURES OF VARIABILITY VARIANCE, STANDARD DEVIATION, CV
Interpretation of standard deviation
An empiricial rule states that if the data has a bell-shaped distribution,
approximately 68% measurements fall within one standard deviation of the mean i.e., between (mean-standard deviation) and (mean+standard deviation)
approximately 95% measurements fall within two standard deviations of the mean, and
virtually all the measurements fall within three standard deviations of the mean

15. MEASURES OF VARIABILITY VARIANCE, STANDARD DEVIATION, CV
-3 -2 -1  +2 +3
Mean
68.26%
95.44%
99.74%

16. MEASURES OF VARIABILITY VARIANCE, STANDARD DEVIATION, CV
Interpretation of standard deviation
Example: suppose that the final marks has a bell-shaped distribution, with a mean of 75 and a standard deviation of 7. Then,
approximately 68% marks fall between (75-7)=68 and (75+7)=82.
approximately 95% marks fall between (75-27)=61 and (75+27)=89, and
virtually all the measurements fall between (75-37) =54 and (75+37)=96

17. POPULATION VARIANCE
The population variance is the mean squared deviation from the population mean:
Where 2 stands for the population variance
 is the population mean
N is the total number of values in the population
is the value of the i-th observation.
represents a summation

18. SAMPLE VARIANCE The sample variance is defined as follows:
Where s2 stands for the sample variance
is the sample mean
n is the total number of values in the sample
is the value of the i-th observation.
represents a summation

19. SAMPLE VARIANCE
Notice that the sample variance is defined as the sum of the squared deviations divided by n-1.
Sample variance is computed to estimate the population variance.
An unbiased estimate of the population variance may be obtained by defining the sample variance as the sum of the squared deviations divided by n-1 rather than by n.
Defining sample variance as the mean squared deviation from the sample mean tends to underestimate the population variance.

20. SAMPLE VARIANCE
A sample of monthly advertising expenses (in 000$) is taken. The data for five months are as follows: 2.5, 1.3, 1.4, 1.0 and 2.0. Compute the sample variance.

21. SAMPLE VARIANCE An alternate formula for the sample variance:
Where s2 stands for the sample variance
is the sample mean
n is the total number of values in the sample
is the value of the i-th observation.
represents a summation

22. SAMPLE VARIANCE
A sample of monthly sales expenses (in 000 units) is taken. The data for five months are as follows: 264, 116, 165, 101 and 209. Compute the sample variance using the alternate formula.

23. POPULATION/SAMPLE STANDARD DEVIATION
The standard deviation is the positive square root of the variance:
Population standard deviation:
Sample standard deviation:

24. POPULATION/SAMPLE STANDARD DEVIATION
Compute the sample standard deviation of advertising data: 2.5, 1.3, 1.4, 1.0 and 2.0
Compute the sample standard deviation of sales data: 264, 116, 165, 101 and 209

25. POPULATION/SAMPLE CV
The coefficient of variation is the standard deviation divided by the means
Population coefficient of variation:
Sample coefficient of variation:

26. POPULATION/SAMPLE CV
Compute the sample coefficient of variation of advertising data: 2.5, 1.3, 1.4, 1.0 and 2.0
Compute the sample coefficient of variation of sales data: 264, 116, 165, 101 and 209

27. SAMPLE VARIANCE APPROXIMATED FROM GROUPED DATA
Sample variance from grouped data:
Where s2 stands for the sample variance
is the sample mean
n is the total number of observations
is the midpoint of the k-th class
is the frequency of the k-th class
represents a summation over all classes

28. SAMPLE VARIANCE APPROXIMATED FROM GROUPED DATA
Compute the sample variance of days to maturity of 40 investments from the following grouped data:

29. SAMPLE COEFFICIENT OF SKEWNESS
The sample coefficient of skewness:
Where SK stands for the coefficient of skewness
s is the sample standard deviation
is the sample mean
m is the sample median

30. SAMPLE COEFFICIENT OF SKEWNESS
Compute the sample coefficient of skewness of the advertising data: 2.5, 1.3, 1.4, 1.0 and 2.0
Mean, = 1.64 (see slide 20)
Sample standard deviation, s = (see slides 20, 24)
Median, m =

31. INTERQUARTILE RANGE AND BOX PLOTS
The interquartile range represents the range of the middle 50% observations and is the difference between the third quartile and the first.
The interquartile range
The range and interquartile range are combined in a box plot.

32. INTERQUARTILE RANGE AND BOX PLOTS
A box plot is used to graphically represent the data set. These plots involve five values:
the minimum value, S
the first quartile,
the second quartile or median,
the third quartile,
and the maximum value, L

33. INTERQUARTILE RANGE AND BOX PLOTS
Example: Construct a box plot with the following data which shows the assets of the 15 largest North American banks, rounded off to the nearest hundred million dollars: 111, 135, 217, 108, 51 , 98, 65, 85, 75, 75, 93, 64, 57, 56, 98

34. INTERQUARTILE RANGE AND BOX PLOTS
Sort the data in the ascending order (low to high): 51, 56, 57, 64, 65, 75, 75, 85, 93, 98, 98, 108, 111, 135, 217
Find

36. INTERQUARTILE RANGE AND BOX PLOTS
If the median is near the center of the box, the distribution is approximately symmetric.
If the median falls to the left of the center of the box, the distribution is positively skewed.
If the median falls to the right of the center of the box, the distribution is negatively skewed.
If the lines are about the same length, the distribution is approximately symmetric.
If the line segment to the right of the box is larger than the one to the left, the distribution is positively skewed.
If the line segment to the left of the box is larger than the one to the right, the distribution is positively skewed.

40. THE PROPORTION Population proportion is denoted by
The parameter is a number between 0 and 1
Sample proportion is denoted by P
P serves as an estimator of and calculated as follows:

41. READING AND EXERCISES Lesson 4 Reading: Section 2-3, pp. 50-61
2-30, 2-37, 2-41