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Measures of Dispersion Measures of Dispersion Tariq Mahmood Bajwa UVAS, LHR. 2 Measures of Dispersion The scatter of the values about their centre is called dispersion and any measure indicating the amount of scatter about the centre is called a Measure of Dispersion. The individual observations of a variable tend to scatter about their centre. The highest degree of concentration is that all the observations are of same size. The scatter in this case would be zero and mean will be exactly same as the individual values of the variable. Tariq Mahmood Bajwa UVAS, LHR. 3 Types of Measures of Dispersion There are two main types of measures of dispersion: 1. Absolute Measure of Dispersion 2. Relative Measure of Dispersion Absolute Measure of Dispersion The absolute measure of dispersion measures the variation present among the observations in the unit of the variable or square of the unit of the variable. Tariq Mahmood Bajwa UVAS, LHR. 4 Types of Measures of Dispersion Relative Measure of Dispersion The relative measure of dispersion measures the variation present among the observations relative to their average. It is expressed in the form of a ratio, coefficient or percentage. It is independent of the unit of measurement. Tariq Mahmood Bajwa UVAS, LHR. 5 Types of Measures of Dispersion The commonly used measures of absolute dispersion are: 1. Range 2. Quartile Deviation 3. Mean (Average) Deviation 4. Variance and Standard Deviation Tariq Mahmood Bajwa UVAS, LHR. 6 Types of Measures of Dispersion Their corresponding measures of relative dispersion are: 1. Coefficient of Range 2. Coefficient of Quartile Deviation 3. Coefficient of Mean (Average) Deviation 4. Coefficient of Variation (CV) Tariq Mahmood Bajwa UVAS, LHR. 7 Range If X1, X2, …, Xn are n observations of a variable X, with X1 and Xn as the smallest and largest observations respectively. Then its range is defined as: Range = Xn - X1 Tariq Mahmood Bajwa UVAS, LHR. 8 Range Example: The following data set shows the weekly TV viewing times, in hours. Calculate range and range coefficient of variation. 25, 41, 27, 32, 43, 66, 35, 31, 15, 5, 34, 26, 32, 38, 16, 30, 38, 30, 20, 21. Range X n X 1 66 5 61hours Range X n X 1 66 5 Semi Range 30.5hours 2 2 2 X n X 1 Mid Range 35.5 2 66 - 5 Co - efficient of Range 0.859 66 5 Tariq Mahmood Bajwa UVAS, LHR. 9 Quartile Deviation If X1, X2, …, Xn are n observations of a variable X, with Q1 and Q3 as their first and third quartiles respectively, then their Quartile Deviation (QD) is as: IQR Q3 Q1 IQR Q3 Q1 SIQR QD 2 2 Q3 Q1 MIQR 2 Tariq Mahmood Bajwa UVAS, LHR. 10 Quartile Deviation Example: Calculate QD and coefficient of QD of above data set shows the weekly TV viewing times, in hours. Q1 22.0 h Q 3 36.5 h IQR 36.5 - 22.0 14.5 h IQR Q 3 Q1 SIQR QD 7.25 h 2 2 Q3 Q1 MIQR 29.25 h 2 SIQR Q3 Q1 Co - Efficient of Q.D 0.248 MIQR Q3 Q1 Tariq Mahmood Bajwa UVAS, LHR. 11 Mean Deviation If X1, X2, …, Xn are n observations of a variable X, with m as their average (mean, median or mode), then their mean deviation, denoted by MD, is defined as: MD X i m n Tariq Mahmood Bajwa UVAS, LHR. 12 Mean Deviation Example: Calculate MD and coefficient of MD. X X 605 30.25 h n 20 MD X i m 175 8.75 h n 20 M.D. 8.75 Coefficient of MD 0.289 Average Used 30.25 Tariq Mahmood Bajwa UVAS, LHR. 13 X X-Mean |X-Mean| X X-Mean |X-Mean| 25 -5.25 5.25 34 3.75 3.75 41 10.75 10.75 26 -4.25 4.25 27 -3.25 3.25 32 1.75 1.75 32 1.75 1.75 38 7.75 7.75 43 12.75 12.75 16 -14.25 14.25 66 35.75 35.75 30 -0.25 0.25 35 4.75 4.75 38 7.75 7.75 31 0.75 0.75 30 -0.25 0.25 15 15.25 15.25 20 -10.25 10.25 5 -25.25 25.25 21 -9.25 9.25 Continue 605 0 175.00 Tariq Mahmood Bajwa UVAS, LHR. 14 Variance and Standard Deviation The Variance is defined as the mean of the squared deviations from mean. The population variance is denoted by σ2 where as sample variance is denoted by S2 and defined as For ungrouped data (x - ) 2 = 2 For population N 2 (x - x )2 S = For sample n Tariq Mahmood Bajwa UVAS, LHR. 15 Variance and Standard Deviation For grouped data f (x - ) 2 = 2 For population N 2 f (x - x )2 S = For sample n Standard deviation The positive square root of the variance is called Standard Deviation. It is denoted by σ (S for sample) Tariq Mahmood Bajwa UVAS, LHR. 16 Variance and Standard Deviation Co-efficient of Variation. The standard deviation is an absolute measure of dispersion its relative measure of dispersion is called co-efficient of variation (CV) and is defined by : S .D. s CV 100 100 Mean X Tariq Mahmood Bajwa UVAS, LHR. 17 Variance and Standard Deviation Example: Find Variance, S.D and Co-efficient of Variation. X 2 3 6 8 11 30 (X-6)2 16 9 0 4 25 54 2 (x - x )2 54 S = = = 10.8 n 5 (x - x )2 S= = 10.5 = 3.286 n S 3.286 C.V = x 100 = x 100 = 54.76 % x 6 Tariq Mahmood Bajwa UVAS, LHR. 18 Variance and Standard Deviation Example:- Find Variance, S.D and Co-efficient of Variation. Class f X (X-X) (X-X)2 f(X-X)2 20---24 1 22 -17 289 289 25---29 4 27 -12 144 576 30---34 8 32 -7 49 392 35---39 11 37 -2 4 44 40---44 15 42 3 9 135 45---49 9 47 8 64 576 50---54 2 52 13 169 338 TOTAL 50 2350 Tariq Mahmood Bajwa UVAS, LHR. 19 Variance and Standard Deviation Example:- X 39 f(x - x )2 2350 S2 47 n 50 f(x - x )2 S = = 47 = 6.85 n 6.85 C.V = x 100 = 17.56 39 Tariq Mahmood Bajwa UVAS, LHR. 20

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