# Measure of dispersion by Neeraj Bhandari ( Surkhet.Nepal )

2 de Apr de 2014                                        1 de 40

### Measure of dispersion by Neeraj Bhandari ( Surkhet.Nepal )

• 1. MEASURE OF DISPERSION
• 2. 1- Control the variability : Measuring of variation helps to identify the nature and causes of variation. Such information is useful in controlling the variation. 2- Compare two or more sets of data with respect to their variability: Measures of variation help in the comparison of two or more sets of data with respect to their uniformity or consistency. 3- Facilitate the use of other statistical techniques: Measures of variation facilitate the use of other statistical techniques such as correlation and regression analysis, hypothesis testing, forecasting, quality control and so on. SIGNIFICANCE OF MEASURING VARIATION
• 3. Characteristics of a good measure of dispersion: An ideal measure of dispersion is expected to possess the following properties 1.It should be rigidly defined 2. It should be based on all the items. 3. It should not be unduly affected by extreme items. 4. It should lend itself for algebraic manipulation. 5. It should be simple to understand and easy to calculate. Absolute and Relative Measures : There are two kinds of measures of dispersion, namely 1.Absolute measure of dispersion 2.Relative measure of dispersion.
• 4. Absolute measure of dispersion indicates the amount of variation in a set of values in terms of units of observations. For example, when rainfalls on different days are available in mm, any absolute measure of dispersion gives the variation in rainfall in mm. On the other hand relative measures of dispersion are free from the units of measurements of the observations. They are pure numbers. They are used to compare the variation in two or more sets, which are having different units of measurements of observations. The various absolute and relative measures of dispersion are listed below.
• 5. Absolute measure Relative measure 1. Range 1.Co-efficient of Range 2.Quartile deviation 2.Co-efficient of Quartile deviation 3.Mean deviation 3. Co-efficient of Mean deviation 4.Standard deviation 4.Co-efficient of variation Range and coefficient of Range: This is the simplest possible measure of dispersion and is defined as the difference between the largest and smallest values of the variable. In symbols, Range = L – S. Where L = Largest value. S = Smallest value.
• 6. In individual observations and discrete series, L and S are easily identified. In continuous series, the following two methods are followed. Method 1: L = Upper boundary of the highest class S = Lower boundary of the lowest class. Method 2: L = Mid value of the highest class. S = Mid value of the lowest class. Co-efficient of Range : Co-efficient of Range = Example1: Find the value of range and its co-efficient for the following data.7, 9, 6, 8, 11, 10, 4 L S L S
• 7. Solution: L=11, S = 4. Range = L – S = 11- 4 = 7 Co-efficient of Range = = 0.4667 Example 2: Calculate range and its co efficient from the following distribution. Size: 60-63 63-66 66-69 69-72 72-75 Number: 5 18 42 27 8 Solution: L = Upper boundary of the highest class = 75 S = Lower boundary of the lowest class = 60 Range = L – S = 75 – 60 = 15 11 4 11 4 L S L S
• 8. Co-efficient of Range Merits: 1. It is simple to understand. 2. It is easy to calculate. 3. In certain types of problems like quality control, weather forecasts, share price analysis, et c., range is most widely used. Demerits: 1. It is very much affected by the extreme items. 2. It is based on only two extreme observations. 3. It cannot be calculated from open-end class intervals. 4. It is not suitable for mathematical treatment. 5. It is a very rarely used measure. L S L S 75 60 75 60 75 60 0.1111 75 60 L S L S
• 9. Quartile Deviation ( Q.D) : Definition: Quartile Deviation is half of the difference between the first and third quartiles. Hence, it is called Semi Inter Quartile Range. In Symbols, among the quartiles Q1, Q2 and Q3, the range Q3 - Q1 is called inter quartile range and , Semi inter quartile range. Co-efficient of Quartile Deviation : Quartile Deviation and Co-efficient of Quartile Deviation : 3 1 3 1 C o-efficient of Q .D = Q Q Q Q 3 1 2 Q Q 3 1 . 2 Q Q Q D 3 1 3 1 Q Q Coefficient of quartile Deviation Q Q
• 10. Example 3: Find the Quartile Deviation for the following data: 391, 384, 591, 407, 672, 522, 777, 733, 1490, 2488 1 3 3 2.75 8.25 4 thN item
• 11. And for so many problems will be discuss in the class. Merits and Demerits of Quartile Deviation Merits : 1. It is Simple to understand and easy to calculate 2. It is not affected by extreme values. 3. It can be calculated for data with open end classes also. Demerits: 1. It is not based on all the items. It is based on two positional values Q1 and Q3 and ignores the extreme 50% of the items 2. It is not amenable to further mathematical treatment. 3. It is affected by sampling fluctuations. 3 1 955.25 403 . 2 2 276.125 Q Q Q D
• 12. Example 1 Evaluate an quartile range for the following data: Farm size (acre) No. of Farms Farm size (acre) No. of Farms Below 40 394 161 - 200 169 41 - 80 461 201 - 240 113 81 - 120 391 241 and above 148 121 - 160 334
• 13. Farm size (acre) No. of Farms Cumulative frequency (c f) Below 40 394 394 41 – 80 461 855 81 – 120 391 1246 121 – 160 334 1580 161 – 200 169 1749 201 – 240 113 1862 241 and above 148 2010
• 14. 1 3 2010 502.5 , 41 80 4 4 . (502.5 394)4 41 40 50.41 461 3 3* 2010 1507.5 , 121 160 4 4 3 . 4 152.31 . . th th n observation This observation lies n c f Q L h acres f n find observation This observation lies n c f Q L h acres f Q R 3 1 101.9Q Q acres
• 15. Mean Deviation and Coefficient of Mean Deviation: Mean Deviation: The range and quartile deviation are not based on all observations. They are positional measures of dispersion. They do not show any scatter of the observations from an average. The mean deviation is measure of dispersion based on all items in a distribution. Mean deviation is the arithmetic mean of the deviations of a series computed from any measure of central tendency; i.e., the mean, median or mode, all the deviations are taken as positive i.e., signs are ignored. According to Clark and Schekade, “Average deviation is the average amount scatter of the items in a distribution from either the mean or the median, ignoring the signs of the deviations”. x x M ean deviation n f x x M ean deviation N
• 16. Coefficient of mean deviation: Mean deviation calculated by any measure of central tendency is an absolute measure. For the purpose of comparing variation among different series, a relative mean deviation is required. The relative mean deviation is obtained by dividing the mean deviation by the average used for calculating mean deviation. Computation of mean deviation – Individual Series : 1. Calculate the average mean, median or mode of the series. 2. Take the deviations of items from average ignoring signs and denote these deviations by . 100 m ean deviation C oefficient of m ean D eviation m ean If the result is desired in percentage then m ean deviation coefficient of m ean deviation m ean D
• 17. 3. Compute the total of these deviations, i.e., . 4. Divide this total obtained by the number of items. Example 6: Calculate mean deviation from mean and median for the following data: 100,150,200,250,360,490,500,600,671 also calculate coefficients of M.D. Solution: Now arrange the data in ascending order 100, 150, 200, 250, 360, 490, 500, 600, 671 D 1 M edian = Value of ( ) 2 thn item . D Symbolically M D n 3321 369 9 x Mean x n
• 18. = value of = value of = value of the 5th item = 360 X 100 269 260 150 219 210 200 169 160 250 119 110 360 9 0 490 121 130 500 131 140 600 231 240 671 302 311 3321 1570 1561 D X X D X M d M .D from mean 1570 174.44 9 D n . Co-efficient of M .D = M D x M .D from median 1561 173.44 9 M .D 173.44 Co-efficient of M .D.= 0.48 M edian 360 D n 1 ( ) 2 thn 9 1 2 th . D M D n
• 19. • Q2. Find the mean absolute deviation from mean for the following frequency distribution of sales (Rs. In thousand) in a co- operative store. • Ans= 47.05 Sales 50-100 100-150 150-200 200-250 250-300 300-350 No. of days 11 23 44 19 8 7
• 20. Variance and Standard deviation 2 2 2 2 2 2 2 2 i i Ungrouped data x x Variance n x x n d d where d x A n n
• 21. 2 2 2 2 2 2 2 2 i i For grouped data f x x Variance n f x x n f d f d where d x A n n
• 22. Standard deviation Standard deviation is a measure of the spread or dispersion of a set of data. It is calculated by taking the square root of the variance and is symbolized by S.D., In other words Where variance is denoted by σ2 2 . .S D
• 23. Interpretation and Uses of the Standard Deviation • Empirical Rule: • For any symmetrical, bell shaped distribution, approximately 68% of the observations will lie within 1 of the mean ( ); approximately 95.44% within 2 of the mean ( ); and approximately 99.7% within 3 of the mean ( ). -3 -2 -1 0 +1 +2 +3
• 24. EXAMPLE • The ages of the Dunn family are: 2, 18, 34, 42 What is the variance? 96 24 4 X 2 24 -22 484 18 24 -6 36 34 24 10 100 42 24 18 324 96 0 944 944 / 4 = 236 X x i X X 2 i X X
• 25. EXAMPLE The hourly wages earned by a sample of five students are: 7, 5, 11, 8, 6. Find the variance. 4.7 5 37 X 5 7.4 -2.4 5.76 6 7.4 -1.4 1.96 7 7.4 -0.4 0.16 8 7.4 0.6 0.36 11 7.4 3.6 12.96 0 21.2 X X XX 2 XX 21.2 (5) 4.24
• 26. Example-1: Find Standard Deviation of Ungroup Data Family No. 1 2 3 4 5 6 7 8 9 10 Size (xi) 3 3 4 4 5 5 6 6 7 7
• 27. i x xxi 2 xxi Family No. 1 2 3 4 5 6 7 8 9 10 Total 3 3 4 4 5 5 6 6 7 7 50 -2 -2 -1 -1 0 0 1 1 2 2 0 4 4 1 1 0 0 1 1 4 4 20 5 10 50 n x x i 2 2 20 2 10 10 i x x 2 1.414 Here,
• 28. EXAMPLE: Find the standard deviation Price frequency 15--18 8 18—21 23 21—24 17 24—27 18 27—30 8 30—33 4 33—36 2 Total 80
• 29. Now we calculate standard deviation where =23.06 • Price frequency Midpoint 15--18 8 16.5 -6.56 43.03 344.24 18—21 23 19.5 -3.56 12.67 291.41 21—24 17 22.5 0.56 0.31 5.27 24—27 18 25.5 2.44 5.95 107.1 27—30 8 28.5 5.44 29.59 236.72 30—33 4 31.5 8.44 71.23 284.92 33—36 2 34.5 11.44 130.87 261.74 Total 80 1531.4 i X X 2 i X X 2 i i f X X x
• 30. 2 1531.4 . . 80 19.1425 4.3752 i i f x x S D N 2 1531.4i i f x x
• 31. Example-2: Find Standard Deviation of Group Data Mid point frequency 3 2 5 3 7 2 8 2 9 1 Total 10
• 32. Solution 3 2 6 -3 9 18 5 3 15 -1 1 3 7 2 14 1 1 2 8 2 16 2 4 8 9 1 9 3 9 9 Total 10 60 - - 40 2 xxf ii 2 xxixxiii xfi fi x 2 2 40 4 10 i i f x x n 6 10 60 i ii f xf x . . 4 2S D
• 33. Assignment-1:Find standard deviation Marks 40-50 50-60 60-70 70-80 80-90 90-100 Total students 6 11 19 17 13 4 70
• 34. Data for Assignment-2 Marks No. of students 0-5 3 5-10 11 10-15 9 15-20 17 20-25 3 25-30 4 Total
• 35. Data for Assignment-3 Marks No. of students 4—8 16 8—12 11 12—16 19 16—20 17 20—24 13 24--28 14 Total 90
• 36. Coefficient of Variation Standard deviation . *100C V mean The set of data for which the coefficient of variation is low is said to be more consistent or more stable. Q1. The weekly sales of two products A and B were recorded as given below: Product A 59 75 27 63 27 28 56 Product B 150 200 125 310 330 250 225 Find out which of the two shows greater fluctuation in sales.
• 37. 1 2 1 2 1 2 2 2 2 2 1 1 1 2 2 2 12 1 Combined standard deviation : The combined standard deviation of two sets of data containing n mean standard deviation n n n and n observation with x and x and and respectively is given by d d 2 12 1 12 1 2 12 2 1 1 2 2 12 1 2 n Combined standard deviation , n n n n where d x x d x x x x and x
• 38. Q1. For a group of 50 male workers, the mean and standard deviation of their monthly wages are Rs 6,300 and Rs 900 respectively . For a group of 40 female workers, these are Rs 5400 and Rs 600 respectively. Find the standard deviation of monthly wages for the combined group of workers.
• 39. Relation Between Different Measures of Variation 2 3 4 5 5 . 6 6 . 5 5 3 standard . . 4 2 Quartile deviation M ean Absolute deviation Quartile deviation M D M ean Absolute deviation Q D deviation M D or Q D