Measures of Dispersion

Techniques to measure dispersion in a data and relative measures of dispersion.

1. MEASURES OF DISPERSION

2. W HAT I S M EASURE O F D ISPERSION? It has two terms: □ Measure: It means a “specific method of estimation” □ Dispersion: (also known as scatter, spread, variation) the term means “ difference or deviation of a certain values from their central value”

3. D EFINITION “ The measurement of the degree of variation or the extent to which items vary from their central value in a population or sample”

4. P URPOSE ● To compare the variability of two or more data set. ● To serve as the basis for the control of variability. ● To determine the reliability of an average. ● To facilitate the use of other statistical measures .

5. C HARACTERISTICS o F I DEAL M EASURE O F D ISPERSION ◦ It should be rigidly defined. ◦ It should be easy to understand & calculate. ◦ It should be based on all observations of a data. ◦ It should be easily subjected for further mathematical operations. ◦ It must be least affected by the sampling fluctuation .

6. Classification of measures of Dispersion

7. Absolute Measures of Dispersion

8. Range Range is defined as the difference between the maximum and the minimum observation of the given data. If Xm the maximum observation, X0 the minimum observation then Range = X m – X0

9. Range of series Individual Series In case of individual Series, the difference between largest value and smallest value can be determined and it is called range. Discrete series To find the range; first’ order the data from least to greatest. Then subtract the smallest value from the largest value in the set.

10. Continuous series In case of continuous frequency distribution, range, according to the definition, is calculated as the difference between the lower limit of the minimum interval and upper limit of the maximum interval of the grouped data. Example, Range of following series is 40-0=40. Class Boundaries Frequency 0-10 12 10-20 8 20-30 10 30-40 5 40-50 7

11. Quartile Deviation “ One half of the inter quartile range is called quartile deviation” A simple way to estimate the spread of a distribution about a measure of its central tendency . The difference Q3−Q1 is called the inter quartile range .

12. Quartiles Quartiles are used to divide a given dataset into four equal halves.

13. The first quartile or the lower quartile is the 25th percentile, also denoted by Q1. The third quartile or the upper quartile is the 75th percentile, also denoted by Q3.

14. Sorted Data – 5, 10, 15, 17, 18, 19, 20, 21, 25, 28 n(number of data) = 10 First Quartile Q 1   = (n+1/4) th term =   10+1/4th term = 2.75th term = 2nd term + 0.75 × (3rd term – 2nd term) = 10 + 0.75 × (15 – 10) = 10 + 3.75 = 13.75 Third Quartile Q 3   =  3 (n+1/4) th term . =   3(10+1)4th term = 8.25th term = 8th term + 0.25 × (9th term – 8th term) = 21 + 0.25 × (25 – 21) = 21 + 1 = 22

15. Quartile Deviation = Semi-Inter Quartile Range =  Q 3 –Q 1 × 2 =  22–13.752 × 2 = 8.252 × 2 = 4.125

16. Mean deviation -   The average of the absolute values of deviation from the mean is called mean deviation. Formula M.D from mean = ∑ ∣ X−mean ∣ /n Where, X = Given values n = Total no. of values

17. Example Set of values is ( 1, 2, 3, 4,5 ) x̅ is Mean = (15 ÷ 5) = 3 The difference between this x̅ and the values in the set is (2, 1, 0, -1,-2) and sum of set values = 6 Mean Deviation = (6 ÷ 5) = 1.2

18. Variance -   The variance is the average of the squared difference between each data value and the mean. The variance is computed as follows :

19. Standard deviation - s Standard deviation is calculated as the square root of average of squared deviations taken from actual mean . It is also called Root mean square deviation .

20. Merits of standard deviations This measure is most suitable for making comparisons among two or more series about variability . It takes into account all the items and is capable of future algebraic treatment and statistical analysis .

21. Demerits of standard deviations It is difficult to complete It assigns more weights to extreme item and less weights to items that are nearer to mean .

22. Relative measures of Dispersions

23. Coefficient of Range “ The relative measure of the distribution based on range is known as the coefficient range.’’ Where, The difference between the maximum and minimum values of a given set of data known as the range.

24. Formula Coefficient of Range = ( x m - x o ) / ( x m + x o ) Where, x m = Maximum Value x o = Minimum Value

25. Example Data set = 8, 5, 6, 7, 3, 2, 4 Step 1: Find Range Range = Maximum Value - Minimum Value Step 2: Find Range Coefficient Coefficient of Range = (Maximum Value - Minimum Value) / (Maximum Value + Minimum Value)

26. C oefficient of the Mean Deviation A relative measure of dispersion based on the mean deviation is called the coefficient of the mean deviation or the coefficient of dispersion. C oefficient of M.D. = Mean Deviation about A * 100 A Where, A can be mean,mode or median

27. Coefficient of Variance Also known as  relative standard deviation  ( RSD ) It is defined as the ratio of standard deviation to mean. Formula CV = s / µ where, s = standard deviation µ = mean

28. Example The coefficient of variation can also be used to compare  variability  between different measures. Regular Test Randomized Answers SD 10.2 12.7 Mean 59.9 44.8 CV % 17.03 28.35

29. Merits of CV widely used in analytical chemistry to express the precision and repeatability of an experiment. used in fields such as engineering or physics when doing quality assurance studies utilized by economists and investors in economic models

30. Coefficient of Quartile Deviation A relative measure of dispersion based on the quartile deviation is called the coefficient of quartile deviation.  Also called quartile coefficient of dispersion. Coefficient of Quartile Deviation Q3–Q1 Q3+Q1 = ×100

31. From Example of Quartile Deviation First Quartile Q 1 = 13.75 Third Quartile Q 3   = 22 Coefficient of QV = 22–13.752 = 8.25 = 0.23 *100 = 23 22 + 13.75 35.75

32.   Difference between Absolute and Relative Measure of Dispersion Absolute measures An absolute measure is one that uses numerical variations to determine the degree of error. measure the extent of dispersion of the item values from a measure of central tendency. Relative measures use statistical variations based on percentages to determine how far from reality a figure is within context. are known as ‘Coefficient of dispersion’- obtained as ratios or percentages.

33. Absolute measures They are expressed in terms of the original units of the series. useful for understanding the dispersion within the context of experiment and measurements Comparatively easy to compute and comprehend. Relative measures They are pure numbers independent of the units of measurement. useful for making comparisons between separate data sets or different experiments Comparatively difficult to compute and comprehend

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