Measures of Dispersion .pptx

statistics

1. MEASURES OF DISPERSION

2. Dispersion/Variation Measures of variation give information on the spread or variability of the data values.

3. Measures of Variation Variation Variance Standard Deviation Coefficient of Variation Population Variance Sample Variance Population Standard Deviation Sample Standard Deviation Range Interquartile Range

4. Measure of Dispersion Measure of Dispersion indicate the spread of scores It is necessary to know the amount of variation and the degree of variation absolute measures are used to know the amount of variation relative measures are used to know the degree of variation

5. Absolute measures It can be divided into positional measures Based on some items of the series such as Range Quartile deviation or semi – interquartile range Based on all items in series such as Mean deviation, Standard deviation or Variance

6. Relative measures It is measuring or estimating things proportionally to one another. It is used for the comparison between two or more series with varying size or number of items, varying central values or units of calculation. Based on some items of the series such as Coefficient of Range Coefficient of Quartile deviation or semi – interquartile range Based on all items in series such as Coefficient of Mean deviation, Coefficient of Standard deviation or Variance

7. Range Difference between the largest and the smallest observations. Range = x maximum – x minimum Example: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Range = 14 - 1 = 13

8. Coefficient of range /Relative range Absolute range = —————————————— Sum of the two extremes = x maximum _ x minimum x maximum + x minimum

9. Example SALES FIGURE RANGE = H-L = 92-80 = 12 COEFFICIENT OF RANGE = H-L / H+L = 92 -80 / 92+80 = 12/172 = 0.069 MONTHLY SALES 1 2 3 4 5 6 Rs (000) 80 82 82 84 84 86 MONTHLY SALES 7 8 9 10 11 12 Rs (000) 86 88 88 90 90 92

10. Disadvantages of the Range Ignores the way in which data are distributed 7 8 9 10 11 12 Range = 12 - 7 = 5 7 8 9 10 11 12 Range = 12 - 7 = 5 Sensitive to outliers Range = 5 - 1 = 4 Range = 120 - 1 = 119 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 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

11. Mean Deviation Mean Deviation can be calculated from any value of Central Tendency, viz. Mean, Median, Mode Mean Deviation about Mean =

12. Coefficient of Mean Deviation Coefficient of Mean Deviation (about mean) = Mean Deviation Mean Coefficient of Mean Deviation (about median) = Mean Deviation Median

13. MEAN DEVIATION FOR GROUPED DATA FORMULA

14. Example Calculate the mean deviation from arithmetic mean median mode with respect of the marks obtained by nine students gives below and show that the mean deviation from median is minimum. Marks (out of 25): 7, 4, 10, 9, 15, 12, 7, 9, 7

15. Solution

16. FIND AVERAGE DEVIATION OF GIVEN DATA DIVIDEND YIELD NO OF COMPANIES (f) MID POINT ( m) fm X-µ I X - µ I f . I X - µ I 0-3 2 3-6 7 6-9 10 9-12 12 12-15 9 15-18 6 18-21 4

17. Variance Average of squared deviations of values from the mean

18. Standard Deviation Most commonly used measure of variation Shows variation about the mean Has the same units as the original data

19. Actual Mean Method

20. Assumed Mean Method (short cut method)

21. Calculation Example: Sample Standard Deviation Sample Data (X i ) : 10 12 14 15 17 18 18 24 n = 8 Mean = x = 16

22. Mean = 15.5 s = 3.338 11 12 13 14 15 16 17 18 19 20 21 Data B Data A Mean = 15.5 s = .9258 Mean = 15.5 s = 4.57 Data C 11 12 13 14 15 16 17 18 19 20 21 11 12 13 14 15 16 17 18 19 20 21 Comparing Standard Deviations

23. Calculate standard deviation from the following distribution of marks

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

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

26. SKEWNESS and Kurtosis

27. Skewness Skewness refers to lack of symmetry or departure from symmetry. If the value of mean is greater than mode then distribution is positively skewed. If the value of mean is less than mode then distribution is negatively skewed

28. Measure of Skewness Skweness = Where S = standard deviation N = number of data points Mean Y = mean of the data

29. Kurtosis Kurtosis means : “bulginess” Kurtosis is the degree of peakedness of a distribution usually taken relative to a normal distribution. A distribution having a relatively high peak is called ‘leptokurtic’ . A distribution which plat topped is called ‘ platykurtic ’ . A normal distribution which is neither very peaked nor very flat-topped is also called ‘ mesokurtic ’ .

30. Kurtosis is measured by a quantity denoted by β2 where: β2 = Where S = standard deviation N = number of data points Mean Y = mean of the data If β2 = 3, it is mesokurtic or normal, If β2 > 3, it is leptokurtic, and If β2 < 3, it platykurtic . Measures of Kurtosis

31. Difference between Variation and Skewness Variation tells about the amount of the variation or dispersion in the data. Skewness tells us about the direction of variation. In business and economic series, measures of variation have greater practical applications than measures of skewness .