Module 6 Measures of Dispersion /Variation S.D =

MODULE 6 MEASURES OF DISPERSION /VARIATION A measure of dispersion is a statistic signifying the extent of the scatteredness of items around a measure of central tendency. The degree to which numerical data tend to spread about an average value is called the variation or dispersion of the data. Measures of the basic variability in data are of interest for two main reasons: 1. It serves as a measure of the reliability or confidence to be placed on an average. 2. We measure variation just to know exactly how much variation exists in the data. If the gap between the rich and the poor can be regarded as a measure of variation in income, then to close the gap, the government must know just how much gap there is. A zero measure means there is no variation in the data and therefore all the observations are the same. When the variation is large, the values are widely scattered; when it is small, they are tightly clustered. Possible measures of dispersion are range, interpercentile range, interdecile range, interquartile range, semi-interquartile range, quartile deviation, mean deviation and standard deviation. 1. Range is the highest value – the lowest value 2. Interpercentile range = 푃90 - 푃10 3. Interdecile range = 퐷9 − 퐷1 4. Interquartile range = 푄3 − 푄1 푄 −푄 5. Semi-Interquartile range = 3 1 2 ∑ |푑| 6. Mean deviation = for ungrouped data 푁 and ∑ 푓|푑| Mean deviation = for grouped data 푁 where d = x – mean Standard Deviation A standard deviation measures the amount of variability or spread in a data set. It used to describe where where most of the data should fall, in relation to the mean. If the standard deviation is relatively large, it means the data is quite spread out away from the mean and vice-versa. For ungrouped data, ∑(푥−푥̅)2 S.D = 휎 = √ for ungrouped data and 푁−1 ∑ 푓(푥−푥̅)2 휎 = √ for grouped data and ∑ 푓−1 Variance ∑ 푓(푥−푥̅)2 This is the square of the standard deviation i.e. 휎2 = for grouped data. ∑ 푓−1 Co-efficient of variation The coefficient of variation shows the extent of variability of data in a sample in relation to the mean of the population. 휎 Coefficient of variation = x 100% 푥̅ Example Weight (lb) Frequency 118 – 126 3 127 – 135 5 136 – 144 9 145 – 153 12 154 – 162 5 163 – 171 4 172 - 180 2 Find the standard deviation. Solution ∑ 푓(푥−푥̅)2 Recall, 휎 = √ ∑ 푓−1 Weight (lb) f x fx x - 푥̅ (x − 푥̅)2 f(x − 푥̅)2 118 – 126 3 122 336 -24.225 586.8506 1760.5518 127 – 135 5 131 655 -15.225 231.8006 1159.003 136 – 144 9 140 1260 -6.225 38.7506 348.7554 145 – 153 12 149 1788 2.775 7.7006 92.4072 154 – 162 5 158 790 11.775 138.6506 693.253 163 – 171 4 167 668 20.775 431.6006 1726.4024 172 - 180 2 176 352 29.775 886.5506 1773.1012 ∑ 푓= 40 5849 7553.474 ∑ 푓푥 5849 푥̅ = = = 146.225 ∑ 푓 40 ∑ 푓(푥−푥̅)2 7553.474 휎 = √ = 휎 = √ = √193.6788 = 13.92 ∑ 푓−1 40−1 .

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