1. Analysis of Economic Data
Dr. Ka-fu Wong
ECON1003
Analysis of Economic Data

2. Other Descriptive Measures (dispersion)
Chapter Four
Other Descriptive Measures (dispersion)
GOALS
Compute and interpret the range, the mean deviation, the variance, and the standard deviation of ungrouped data.
Compute and interpret the range, the variance, and the standard deviation from grouped data.
Explain the characteristics, uses, advantages, and disadvantages of each measure of dispersion
Understand Chebyshev’s theorem and the Normal, or Empirical Rule, as they relate to a set of observations.
Compute and interpret quartiles and the interquartile range.
Construct and interpret box plots.
Compute and understand the coefficient of variation and the coefficient of skewness. l

3. Range
The range is the difference between the largest and the smallest value.
Only two values are used in its calculation.
It is influenced by an extreme value.
It is easy to compute and understand.

4. Mean Deviation
The Mean Deviation is the arithmetic mean of the absolute values of the deviations from the arithmetic mean.
All values are used in the calculation.
It is not influenced too much by large or small values.
The absolute values are difficult to manipulate.
Mean deviation is also known as Mean Absolute Deviation (MAD).

5. EXAMPLE 1
The weights of a sample of crates containing books for the bookstore (in pounds ) are:
103, 97, 101, 106, 103
Find the range and the mean deviation.
Range = 106 – 97 = 9

6. Example 1 The first step is to find the mean weight.
The mean deviation is:

7. Population Variance
The population variance is the arithmetic mean of the squared deviations from the population mean.
All values are used in the calculation.
More likely to be influenced by extreme values than mean deviation.
The units are awkward, the square of the original units.

8. Variance The formula for the population variance is:
The formula for the sample variance is:
Note in the sample variance formula the sum of deviation is divided by (n-1) instead of n. Although it is logical to use n instead of (n-1), the division by (n-1) yields an unbiased estimator of the population variance but the division by n yields a biased estimator.

9. EXAMPLE 2 The ages of the Dunn family are: 2, 18, 34, 42
2, 18, 34, 42
What is the population variance?

10. The Population Standard Deviation
The population standard deviation (σ) is the square root of the population variance.
For EXAMPLE 2, the population standard deviation is 15.36, found by

11. EXAMPLE 3 The hourly wages earned by a sample of five students are:
$7, $5, $11, $8, $6.
Find the variance.

12. Sample Standard Deviation
The sample standard deviation is the square root of the sample variance.
In EXAMPLE 3, the sample standard deviation is 2.30

13. Sample Variance For Grouped Data
The formula for the sample variance for grouped data is:

14. Interpretation and Uses of the Standard Deviation
Chebyshev’s theorem: For any set of observations, the minimum proportion of the values that lie within k standard deviations of the mean is at least:
where k2 is any constant greater than 1.

15. Chebyshev’s theorem
Chebyshev’s theorem: For any set of observations, the minimum proportion of the values that lie within k standard deviations of the mean is at least 1- 1/k2 K
Coverage 1 0% 2
75.00% 3
88.89% 4
93.75% 5
96.00% 6
97.22%

16. Interpretation and Uses of the Standard Deviation
Empirical Rule: For any symmetrical, bell-shaped distribution:
About 68% of the observations will lie within 1s the mean,
About 95% of the observations will lie within 2s of the mean
Virtually all the observations will be within 3s of the mean
Empirical rule is also known as normal rule.

17. Bell-shaped Curve showing the relationship between σ and μ
m- 3s
m-2s
m-1s m
m+1s
m+2s
m+ 3s

18. Why are we concern about dispersion?
Dispersion is used as a measure of risk.
Consider two assets of the same expected (mean) returns.
-2%, 0%,+2%
-4%, 0%,+4%
The dispersion of returns of the second asset is larger then the first. Thus, the second asset is more risky.
Thus, the knowledge of dispersion is essential for investment decision. And so is the knowledge of expected (mean) returns.

19. Relative Dispersion
The coefficient of variation is the ratio of the standard deviation to the arithmetic mean, expressed as a percentage:

20. Sharpe Ratio and Relative Dispersion
Sharpe Ratio is often used to measure the performance of investment strategies, with an adjustment for risk.
If X is the return of an investment strategy in excess of the market portfolio, the inverse of the CV is the Sharpe Ratio.
An investment strategy of a higher Sharpe Ratio is preferred.

21. Skewness
Skewness is the measurement of the lack of symmetry of the distribution.
The coefficient of skewness can range from 3.00 up to 3.00.
A value of 0 indicates a symmetric distribution.
It is computed as follows: Or

22. Why are we concerned about skewness?
Skewness measures the degree of asymmetry in risk.
Upside risk
Downside risk
Consider the distribution of asset returns:
Right skewed implies higher upside risk than downside risk.
Left skewed implies higher downside risk than upside risk.

23. Interquartile Range
The Interquartile range is the distance between the third quartile Q3 and the first quartile Q1.
This distance will include the middle 50 percent of the observations.
Interquartile range = Q3 - Q1

24. EXAMPLE 5
For a set of observations the third quartile is 24 and the first quartile is 10. What is the quartile deviation?
The interquartile range is = Fifty percent of the observations will occur between 10 and 24.

25. Box Plots
A box plot is a graphical display, based on quartiles, that helps to picture a set of data.
Five pieces of data are needed to construct a box plot: the Minimum Value, the First Quartile, the Median, the Third Quartile, and the Maximum Value.

26. EXAMPLE 6
Based on a sample of 20 deliveries, Buddy’s Pizza determined the following information. The minimum delivery time was 13 minutes and the maximum 30 minutes. The first quartile was 15 minutes, the median 18 minutes, and the third quartile 22 minutes. Develop a box plot for the delivery times.

27. EXAMPLE 6 continued median min Q1 Q3 max

28. Working with mean and Standard Deviation
Set
Data
Mean
St Dev
(1) 19 20 21
20.00
0.82
(2) -1 1
0.00
(3)
0.71
(4) 38 40 42
40.00
1.63
(5) 57 60 63
60.00
2.45
(6)
(7) 3 5 8
5.33
2.05
(8) 4 7 9
6.67
(9) 12 17
12.00
4.08
(10) 27 32 35 45 56 72
35.56
18.04

29. Working with mean and Standard Deviation
Set
Data
Mean
St Dev
(1) 19 20 21
20.00
0.82
(2) -1 1
0.00
(3)
0.71
(4) 38 40 42
40.00
1.63
(5) 57 60 63
60.00
2.45
(2) = (1) – mean(1):
Mean(2)=0; Stdev(2)=Stdev(1)
(3) = (1) + mean(1)
Mean(3)=Mean(1); Stdev(3)<Stdev(1).
(4) = (1)*2; (5) = (1)*3
Mean(4)=mean(1)*2; mean(5)=mean(1)*3
Stdev(4)=stdev(1)*2; stdev(5)=stdev(1)*3

30. Working with mean and Standard Deviation
Set
Data
Mean
St Dev
(1) 19 20 21
20.00
0.82
(6)
(7) 3 5 8
5.33
2.05
(8) 4 7 9
6.67
(9) 12 17
12.00
4.08
(10) 27 32 35 45 56 72
35.56
18.04
(6)=(1) multiplied by some frequency
Mean(6)=Mean(1); Stdev(6)=Stdev(1).
(9) = (7)+(8)
Mean(9)=mean(7)+mean(8)
(10) = (7) *(8)
Mean(10)=mean(7)*mean(8)

31. Further results about mean and variance of transformed variables
E(X) = mean or expected values
V(X) = E[(X-E(X))2]=E(X2) – E(X)2
E(a+bX) = a+bE(X)
E(X+Y) = E(X) + E(Y)
V(X+Y) = V(X) + V(Y) if X and Y are independent.

32. Further results about mean and variance of transformed variables
E(a+bX) = a+bE(X)
E(X+Y) = E(X) + E(Y)
Suppose we invest $1 in two assets. $a in asset X and $(1-a) in asset Y. Their expected returns are respectively E(X) and E(Y). We will expect a return of E(aX+(1-a)Y) = aE(X) + (1-a)E(Y) for this investment portfolio.
If these two assets are independent or uncorrelated so that C(X,Y) =0, then the variance is V(aX+(1-a)Y) = a2V(X) + (1-a)2V(Y)

33. Chapter Four
Other Descriptive Measures (dispersion)
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