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Session 7 Nagesh S J C E

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Session 7 Nagesh S J C E

1. 1. Session – 7 Measures of Dispersions The measures of Central Tendency alone will not exhibit various characteristics of the frequency distribution having the same total frequency. Two distribution can have the same mean but can differ significantly. We need to know the extent of variation or deviation of the values in comparison with the central value or average referred to as the measures of central tendency. Measures of dispassion are the ‘average of second order’. The are based on the average of deviations of the values obtained from central tendencies x , Me or z. The variability is the basic feature of the values of variables. Such type of variation or dispersion refers to the ‘lack of uniformity’. Definition: A measure of dispersion may be defined as a statistics signifying the extent of the scatteredness of items around a measure of central tendency. Absolute and Relative Measures of Dispersion: A measure of dispersion may be expressed in an absolute form, or in a relative form. It is said to be in absolute form when it states the actual amount by which the value of item on an average deviates from a measure of central tendency. Absolute measures are expressed in concrete units i.e., units in terms of which the data have been expressed e.g.: Rupees, Centimetres, Kilogram etc. and are used to describe frequency distribution. A relative measures of dispersion is a quotient by dividing the absolute measures by a quality in respect to which absolute deviation has been computed. It is as such a pure number and is usually expressed in a percentage form. Relative measures are used for making comparisons between two or more distribution. Thus, absolute measures are expressed in terms of original units and they are not suitable for comparative studies. The relative measures are expressed in ratios or percentage and they are suitable for comparative studies. Measures of Dispersion Types Following are the common measures of dispersions. a. The Range b. The Quartile Deviation (QD) c. The Mean Deviation (MD) d. The Standard Deviation (SD)
2. 2. Range ‘Range’ represents the differences between the values of the extremes’. The range of any such is the difference between the highest and the lowest values in the series. The values in between two extremes are not all taken into consideration. The range is an simple indicator of the variability of a set of observations. It is denoted by ‘R’. In a frequency distribution, the range is taken to be the difference between the lower limit of the class at the lower extreme of the distribution and the upper limit of the distribution and the upper limit of the class at the upper extreme. Range can be computed using following equation. Range = Large value – Small value L arg e value − Small value Coefficient of Range = L arg e value + Small value Problems 1. Compute the range and also the co-efficient of range of the given series of state which one is more dispersed and which is more uniform. Series – I – 9, 10, 15, 19, 21 Series – II – 1, 15, 24, 28, 29 R = LV – SV = 21 – 9 = 12 R = LV – SV = 29 – 1 = 28 12 12 R 28 CR = = = 0.4 CR = = = 0.933 21 + 9 30 LV + SV 30 Series I is les dispersed and more uniform Series II is more dispersed and less uniform Evaluating Criteria i. Less the CR is less dispersion ii. More the CR is less uniform Range Merits i. It is very simplest to measure. ii. It is defined rigidly iii. It is very much useful in Statistical Quality Control (SBC). iv. It is useful in studying variation in price of shars and stocks.
3. 3. Limitations i. It is not stable measure of dispersion affected by extreme values. ii. It does not considers class intervals and is not suitable for C.I. problems. iii. It considers only extreme values. 2. Find range of Co-efficient of range from following data. A: 10 11 12 13 14 B: 40 41 42 43 44 C: 100 101 102 103 104 Series - I Series – II Series – III R =LV – 3m = 14 – 10 R = 44 - 40 R = 104 - 100 = 4 = 4 = 4 R R R CR = CR = CR = LV + SV LV + SV LV + SV 4 4 4 = = = 24 84 204 = 0.166 = 0.0476 = 0.0196 Series III is less dispersed and more uniform Series I is more dispersed and less uniform 3. Compute range and coefficient of range for the following data. x: 6 12 18 24 30 36 42 f: 20 130 16 14 20 15 40 Range = LV – SV = 42 – 6 = 36 R 36 CR = = = 0.75 LV + SV 48
4. 4. Quartile Deviation Quartile divides the total frequency in to four equal parts. The lower quartile Q1 refers to the values of variates corresponding to the cumulative frequency N/4. Upper quartile Q3 refers the value of variants corresponding to cumulative frequency ¾ N. 1 Quartile deviation is defined as QD = (Q3 – Q1). In this quartile Q2 as it 2 N corresponds to the value of variate with cumulative frequency is equal to c.f. = . 2 1 a) QD = (Q3 – Q1) 2 Q 3 − Q1 b) Relative measure of dispersion coefficient of QD = Q 3 + Q1 Problems 1. Find quartile deviation and coefficient of quartile deviation for the given grouped data also compute middle quartile. Class f 1 – 10 3 11 – 20 16 21 – 30 26 31 – 40 31 41 – 50 16 51 – 60 8 Σf = N = 100 Class f Cf 1 – 10 3 3 11 – 20 16 19 21 – 30 26 45  Q1 Class 31 – 40 31 76  Q2 & Q3 Class 41 – 50 16 92 51 – 60 8 100 N = 100
5. 5. N 100 Q1  = 25 4 4 h N  Q1 =  +  4 − C f   10 Q1 = 20.5 + [ 25 − 19] 26 Q1 = 22.80 h N  Q2 =  +  2 − C f   10 Q2 = 30.5 + [ 50 − 45] 31 Q2 = 32.11 h 3  Q3 =  +  4 N − C f   10 Q3 = 30.5 + [ 75 − 45] 31 Q3 = 40.17 1 QD = (Q3 – Q1) = 0.5 (Q3 – Q1) 2 1 = (40.17 – 22.80) 2 = 8.685 Q 3 − Q1 Coef. QD = Q 3 + Q1 40.17 − 22.80 = 40.17 + 22.80 17.37 = 62.97 = 0.275
6. 6. 2. Find quartile deviation from the following marks of 12 students and also co-efficient of quartile deviation. Sl. No. Marks 1. 25 2. 30 3. 37 4. 43 5. 48 6. 54 7. 61 8. 67 9. 72 10. 80 11. 84 12. 89 Q1 = 3.25th item = 3rd item + 0.25 of item = 37 + 0.25 (43 - 37) Q1 = 38.5 Q3 =9.75th item = 9 + 0.75rd item = 72 + 0.75 (80- 72) Q3 = 78 1 QD = (Q3 – Q1) 2 1 = (78 – 38.3) 2 QD = 19.75 Q 3 − Q1 Coef. QD = Q 3 + Q1 78 − 38.5 = = 0.339 78 + 38.5 3. Compute quartile deviation. and its Coefficient for the data given below:
7. 7. x f Cf 58 15 15 59 20 35 60 32 67  Q1 Class 61 35 102 62 33 135 63 22 157  Q3 Class 64 20 177 65 10 187 65 8 195 N = 195 n + 1th Q1 = size 4 195 + 1th = size 4 Q1 = 48.78th size and corresponding to cf 67, which gives Q1 = 60 3 Q3 = ( n + 1) th size 4 3 = (196) th = 146.33 th size . 4 It lies in 157, cf. Against cf 157 Q3 = 63 1 QD = (Q3 – Q1) 2 1 = (63 – 60) 2 QD = 1.5 Q 3 − Q1 Coef. QD = Q 3 + Q1 63 − 60 3 = = 63 + 60 123 = 0.024 Merits of Quartile Deviation
8. 8. • It is very easy to compute • It is not affected by extreme values of variable. • It is not at all affected by open and class intervals. Demerits of Quartile Deviation • It ignores completely the portions below the lower quartile and above the upper of quartile. • It is not capable for further mathematical treatment. • It is greatly affected by fluctuations in the sampling. • It is only the positional average but not mathematical average.