Definition of dispersion

Introducing to measure of dispersion

1. Welcome

2. Measures of Dispersion

3. Contents Introduction Of measures of dispersion. Definition of Dispersion. Purpose of Dispersion Properties of a Good Measures of Dispersion Measures of Dispersion Range Quartile deviation. Mean deviation. Variance Standard deviation. Coefficient of variance. Conclusion References .

4. Definition of Dispersion Dispersion measures the variability of a set of observations among themselves or about some central values. According to Brooks & Dicks, “Dispersion or Spread is the degree of the scatter or variations of the variables about some central values.”

5. Purposes of Dispersion To determine the reliability of average; To serve as a basis for the control of the variability; To compare to or more series with regard to their variability; and To facilitate the computation of other statistical measures.

6. Properties of a Good Measures of Dispersion It should be simple to understand. It is should be easy to compute. It is should rigidly defined. It should be based on all observations. It should have sampling fluctuation. It should be suitable for further algebraic treatment. It should be not be affected by extreme observations.

7. Measures of Dispersion The numerical values by which we measure the dispersion or variability of a set of data or a frequency distribution are called measures of dispersion. There are two kinds of Measures of Dispersion: 1 . Absolute measures of dispersion 2 . Relative measures of dispersion

8. The Absolute measures of dispersion are: 1 . Range 2 . Quartile deviation 3 . Mean deviation 4 . Variance & standard deviation The Relative measures of dispersion are: 1 . Coefficient of range 2 . Coefficient of quartile deviation 3 . Coefficient of mean deviation 4 . Coefficient of variation

9. 1. Range   Range is the difference between the largest & smallest observation in set of data. In symbols, Range = L – S. Where , L = Largest value. S = Smallest value. In individual observations and discrete series, L and S are easily identified.

10. 2. Coefficient of Range The percentage ratio of range & sum of maximum & minimum observation is known as Coefficient of Range.   Coefficient of Range ×100  

11. The monthly incomes in taka of seven employees of a firm are 5500,5750,6500,67 50,7000 & 8500. Compute Range & Coefficient of Range. Solution The range of the income of the employees is   Range = 8500-5500 = TK 3000 Example

12. Coefficient of Range ×100           Example

13. Merits and Demerits of Range   Merits The range measure the total spread in the set of data. It is rigidly defined. It is the simplest measure of dispersion. It is easiest to compute. It takes the minimum time to compute. It is based on only maximum and minimum values.

14. Demerits It is not based on all the observations of a set of data. It is affected by sampling fluctuation. It is cannot be computed in case of open-end distribution. It is highly affected by extreme values.

15. When To Use the Range The range is used when you have ordinal data or you are presenting your results to people with little or no knowledge of statistics. The range is rarely used in scientific work as it is fairly insensitive. It depends on only two scores in the set of data, X L and X S Two very different sets of data can have the same range: 1 1 1 1 9 vs 1 3 5 7 9

16. 3 . Quartile Deviation The average difference of 3 rd & 1 st Quartile is known as Quartile Deviation. Quartile Deviation  

17. 4 . Coefficient of Quartile Deviation The relative measure corresponding to this measure, called the coefficient of quartile deviation, is defied by Coefficient of Quartile Deviation=  

18. Advantage of Quartile Deviation It is superior to range as a measure of variation. It is useful in case of open-end distribution. It is not affected by the presence of extreme values. It is useful in case of highly skewed distribution.

19. Disadvantage of Quartile Deviation It is not based on all observations. It ignores the first 25% and last 25% of the observation. It is not capable of mathematical manipulation. It is very much affected by sampling fluctuation. It is not a good measure of dispersion since it depends on two position measure.

20. 3. Mean Deviation   The difference of mean from their observation & their mean is known as mean deviation  

21. Mean Deviation for ungrouped data Suppose X1, X2,………., Xn are n values of variable, and is the mean and the mean deviation (M.D.) about mean is defined by

22. Mean Deviation for grouped data

23. 4. Coefficient of Mean Deviation The Percentage ratio of mean deviation & mean is as known as Coefficient of Mean Deviation(C.M.D .).

24. For Grouped & Ungrouped Data C.M.D.

25. Merits of Mean Deviation : It is easy to understand and to compute. It is less affected by the extreme values. It is based on all observations . Limitations of Mean Deviation: This method may not give us accurate results. It is not capable of further algebraic treatment. It is rarely used in sociological and business studies.

26.   5. Variance The square deviation of mean from their observation and their mean is as known as variance.

27. For ungrouped data:

28. For grouped data:

29. 6.Standard Deviation The square deviation of mean from their observation & the square root variance is as known as Standard Deviation

30. Merits of Standard Deviation It is rigidly defied. It is based on all observations of the distribution. It is amenable to algebraic treatment. It is less affected by the sampling fluctuation. It is possible to calculate the combined standard deviation

31. Demerits of Standard Deviation As compared to other measures it is difficult to compute. It is affected by the extreme values. It is not useful to compare two sets of data when the observations are measured in different ways.

32. 6. Coefficient of Variation The percentage ratio of Standard deviation and mean is as known as coefficient of variation

33. For grouped & Ungrouped data

34. Example Consider the measurement on yield and plant height of a paddy variety. The mean and standard deviation for yield are 50 kg and 10 kg respectively. The mean and standard deviation for plant height are 55 am and 5 cm respectively. Here the measurements for yield and plant height are in different units. Hence the variabilities can be compared only by using coefficient of variation.   For yield, CV=  

35. Conclusion The measures of variations are useful for further treatment of the Data collected during the study. The study of Measures of Dispersion can serve as the foundation for comparison between two or more frequency distributions. Standard deviation or variance is never negative. When all observations are equal, standard deviation is zero. when all observations in the data are increased or decreased by constant, standard deviation remains the same.

36. Reference Business Statistics, by Manindra Kumar Roy & Jiban Chandra Paul, first edition,2012, page no 162-195. https:// www.slideshare.net/sachinshekde9/measures-of-dispersion-38120163.

37. Thank You