1. Measures of Dispersion
DR ABHIJIT GANGULY

2. Learning Outcome-2 AC 2.3 and AC2.4
To understand the measures of dispersion

3. Summary Measures Describing Data Numerically Central Tendency
Quartiles
Variation
Shape
Arithmetic Mean
Range
Skewness
Median
Interquartile Range
Mode
Variance
Geometric Mean
Standard Deviation
Coefficient of Variation

4. Quartiles
Quartiles split the ranked data into 4 segments with an equal number of values per segment
25%
25%
25%
25% Q1 Q2 Q3
The first quartile, Q1, is the value for which 25% of the observations are smaller and 75% are larger
Q2 is the same as the median (50% are smaller, 50% are larger)
Only 25% of the observations are greater than the third quartile

5. Quartile Formulas
Find a quartile by determining the value in the appropriate position in the ranked data, where
First quartile position: Q1 = n/4
Second quartile position: Q2 = n/2 (the median position)
Third quartile position: Q3 = 3n/4
where n is the number of observed values

6. Quartiles Example: Find the first quartile (n = 9)
Sample Data in Ordered Array:
(n = 9)
Q1 is in the 9/4 = 2.25 position of the ranked data
so Q1 = 12.25
Q1 and Q3 are measures of noncentral location
Q2 = median, a measure of central tendency

7. Quartiles
(continued)
Example:
Overtime Hours
10-15
15-20
20-25
25-30
30-35
35-40
Total
No. of employees 11 20 35 8 6
100
Calculate Median, First Quartile, 7th Decile & Interquartile Range

8. Coefficient of Variation
Measures of Variation
Variation
Range
Interquartile
Range
Variance
Standard Deviation
Coefficient of Variation
Measures of variation give information on the spread or variability of the data values.
Same center,
different variation

9. Range = Xlargest – Xsmallest
Simplest measure of variation
Difference between the largest and the smallest values in a set of data:
Range = Xlargest – Xsmallest
Example:
Range = = 13

10. Disadvantages of the Range
Ignores the way in which data are distributed
Sensitive to outliers
Range = = 5
Range = = 5
1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5
Range = = 4
1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,120
Range = = 119

11. Coefficient of Range Coefficient of Range = [L-S]/[L+S] Where,
L = Largest value in the data set
S = Smallest value in the data set

12. Interquartile Range
Can eliminate some outlier problems by using the interquartile range
Eliminate some high- and low-valued observations and calculate the range from the remaining values
Interquartile range = 3rd quartile – 1st quartile
= Q3 – Q1

13. Interquartile Range Example: Median (Q2) X X Q1 Q3 Interquartile range
maximum
minimum
25% % % %
Interquartile range
= 57 – 30 = 27

14. Coefficient of QD
Coefficient of QD= [Q3-Q1]/[Q3+Q1]

15. Average Deviation
Coefficient of AD = (Average Deviation)/(Mean or Median)

16. Calculate the Coefficient of AD (mean)
Sales (in thousand Rs)
No. of days
10-20 3
20-30 6
30-40 11
40-50
50-60 2

17. Variance
Average (approximately) of squared deviations of values from the mean
Sample variance:
Where
= mean
n = sample size
Xi = ith value of the variable X

18. Standard Deviation Most commonly used measure of variation
Shows variation about the mean
Is the square root of the variance
Has the same units as the original data
Sample standard deviation:

19. Calculation Example: Sample Standard Deviation
Sample Data (Xi) :
n = Mean = X = 16
A measure of the “average” scatter around the mean

20. Population vs. Sample Standard Deviation

21. Population vs. Sample Standard Deviation (Grouped Data)

22. Calculate the Standard Deviation for the following sample
Number of pins knocked down in ten-pin bowling matches

23. Solution
Number of pins knocked down in ten-pin bowling matches

24. Calculate the Standard Deviation for the following sample
Heights of students

25. Solution

26. Calculate the Standard Deviation for the following sample
Number of typing errors

27. Comparing Standard Deviations
Data A
Mean = 15.5
S = 3.338
Data B
Mean = 15.5
S = 0.926
Data C
Mean = 15.5
S = 4.567

28. Measuring variation
Small standard deviation
Large standard deviation

29. Advantages of Variance and Standard Deviation
Each value in the data set is used in the calculation
Values far from the mean are given extra weight (because deviations from the mean are squared)

30. Coefficient of Variation
Measures relative variation
Always in percentage (%)
Shows variation relative to mean
Can be used to compare two or more sets of data measured in different units

31. Comparing Coefficient of Variation
Stock A:
Average price last year = $50
Standard deviation = $5
Stock B:
Average price last year = $100
Both stocks have the same standard deviation, but stock B is less variable relative to its price

32. Sample vs. Population CV

33. Calculate!
Calculate the coefficient of variation for the following data set.
The price, in cents, of a stock over five trading days was 52, 58, 55, 57, 59.

34. Solution

35. Pooled Standard Deviation

36. Pooled Standard Deviation

37. Z Scores
A measure of distance from the mean (for example, a Z-score of 2.0 means that a value is 2.0 standard deviations from the mean)
The difference between a value and the mean, divided by the standard deviation
A Z score above 3.0 or below -3.0 is considered an outlier

38. Z Scores
(continued)
Example:
If the mean is 14.0 and the standard deviation is 3.0, what is the Z score for the value 18.5?
The value 18.5 is 1.5 standard deviations above the mean
(A negative Z-score would mean that a value is less than the mean)

39. Shape of a Distribution
Describes how data are distributed
Measures of shape
Symmetric or skewed
Left-Skewed
Symmetric
Right-Skewed
Mean < Median
Mean = Median
Median < Mean

40. The Empirical Rule
If the data distribution is approximately bell-shaped, then the interval:
contains about 68% of the values in the population or the sample
68%

41. The Empirical Rule contains about 95% of the values in
the population or the sample
contains about 99.7% of the values in the population or the sample
95%
99.7%

43. Chebyshev Rule
Regardless of how the data are distributed, at least (1 - 1/k2) x 100% of the values will fall within k standard deviations of the mean (for k > 1)
Examples:
(1 - 1/12) x 100% = 0% ……..... k=1 (μ ± 1σ)
(1 - 1/22) x 100% = 75% … k=2 (μ ± 2σ)
(1 - 1/32) x 100% = 89% ………. k=3 (μ ± 3σ)
At least
within

44. Calculate! Calculate the mean & the standard dev.
Variable
Frequency
44-46 3
46-48 24
48-50 27
50-52 21
52-54 5
Calculate the mean & the standard dev.
Calculate the percentage of observations between the mean &

45. Find the correct Mean & Std. dev.
The Mean & Standard Deviation of a set of 100 observations were worked out as 40 & 5 respectively. It was later learnt that a mistake had been made in data entry – in place of 40 (the correct value), 50 was entered.

46. Skewness
A fundamental task in many statistical analyses is to characterize the location and variability of a data set (Measures of central tendency vs. measures of dispersion)
Both measures tell us nothing about the shape of the distribution
It is possible to have frequency distributions which differ widely in their nature and composition and yet may have same central tendency and dispersion.
Therefore, a further characterization of the data includes skewness

47. Positive & Negative Skew
Positive skewness
There are more observations below the mean than above it
When the mean is greater than the median
Negative skewness
There are a small number of low observations and a large number of high ones
When the median is greater than the mean

48. Measures of Skew
Skew is a measure of symmetry in the distribution of scores
Normal (skew = 0)
Positive Skew
Negative Skew

49. Measures of Skew

50. The Rules
Rule One. If the mean is less than the median, the data are skewed to the left.
Rule Two. If the mean is greater than the median, the data are skewed to the right.

51. Karl Pearson Coefficient of Skewness

52. Karl Pearson Coefficient - Properties
Advantage – Uses the data completely
Disadvantage – Is sensitive to extreme values

53. Calculate!
Calculate the coefficient of skewness & comment on its value

54. Calculate!
Profits (lakhs)
No. of Companies 17 53
199
194
327
208 2
Calculate the coefficient of skewness & comment on its value

55. Bowley’s Coefficient of Skewness

56. Bowley’s Coefficient - Properties
Advantage – Is not sensitive to extreme values
Disadvantage – Does not use the data completely

57. Calculate!

58. Kelly’s Coefficient of Skewness

59. Calculate!
Wages
No. of Workers
Below 200 10 25
145
220 70
Above 400 30
Obtain the limits of daily wages of the central 50% of the workers
Calculate Bowley’s & Kelly’s Coefficients of Skewness

60. Moments about the Mean

61. Thanks