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Measures of Variation • Range • Standard Deviation • Variance The Range the difference between the largest and smallest values of a distribution Find the range: 10, 13, 17, 17, 18 The range = largest minus smallest = 18 minus 10 = 8 The standard deviation a measure of the average variation of the data entries from the mean Standard deviation of a sample s (x x) n 1 n = sample size 2 mean of the sample • • • • • • To calculate standard deviation of a sample Calculate the mean of the sample. Find the difference between each entry (x) and the mean. These differences will add up to zero. Square the deviations from the mean. Sum the squares of the deviations from the mean. Divide the sum by (n 1) to get the variance. Take the square root of the variance to get the standard deviation. The Variance the square of the standard deviation Variance of a Sample ( x x ) 2 s n 1 2 Find the standard deviation and variance x 30 26 22 78 x x 4 0 4 mean= 26 (x x ) Sum = 0 16 0 16 ___ 32 2 The variance s 2 ( x x) n 1 2 = 32 2 =16 The standard deviation s= 16 4 Find the mean, the standard deviation and variance mean = 5 x xx (x - x) 4 1 1 5 0 0 5 0 0 7 2 4 4 1 1 25 6 2 The mean, the standard deviation and variance Mean =5 S tan dard deviation 1.5 1.22 Variance 6 1 .5 4 Computation formula for sample standard deviation: s SS x n 1 x 2 where SS x x 2 n To find x 2 Square the x values, then add. To find ( x ) 2 Sum the x values, then square. Use the computing formulas to find s and s2 x x2 n=5 4 16 (Sx) 2 = 25 2 = 625 5 25 Sx2 = 131 5 25 SSx = 131 – 625/5 = 6 7 49 s2 = 6/(5 –1) = 1.5 4 25 16 131 s = 1.22 Population Mean and Standard Deviation x population mean N population standard deviation 2 x x N where N number of data values in the population COEFFICIENT OF VARIATION: a measurement of the relative variability (or consistency) of data s CV 100 or 100 x CV is used to compare variability or consistency A sample of newborn infants had a mean weight of 6.2 pounds with a standard deviation of 1 pound. A sample of three-month-old children had a mean weight of 10.5 pounds with a standard deviation of 1.5 pounds. Which (newborns or 3-month-olds) are more variable in weight? To compare variability, compare Coefficient of Variation For newborns: For 3month-olds: CV = 16% Higher CV: more variable CV = 14% Lower CV: more consistent Use Coefficient of Variation To compare two groups of data, to answer: Which is more consistent? Which is more variable? CHEBYSHEV'S THEOREM For any set of data and for any number k, greater than one, the proportion of the data that lies within k standard deviations of the mean is at least: 1 1 k 2 CHEBYSHEV'S THEOREM for k = 2 According to Chebyshev’s Theorem, at least what fraction of the data falls within “k” (k = 2) standard deviations of the mean? 1 3 At least 1 2 2 4 75 % of the data falls within 2 standard deviations of the mean. CHEBYSHEV'S THEOREM for k = 3 According to Chebyshev’s Theorem, at least what fraction of the data falls within “k” (k = 3) standard deviations of the mean? 1 8 At least 1 3 2 9 88 . 9 % of the data falls within 3 standard deviations of the mean. CHEBYSHEV'S THEOREM for k =4 According to Chebyshev’s Theorem, at least what fraction of the data falls within “k” (k = 4) standard deviations of the mean? 1 15 At least 1 4 2 16 93 . 8 % of the data falls within 4 standard deviations of the mean. Using Chebyshev’s Theorem A mathematics class completes an examination and it is found that the class mean is 77 and the standard deviation is 6. According to Chebyshev's Theorem, between what two values would at least 75% of the grades be? Mean = 77 Standard deviation = 6 At least 75% of the grades would be in the interval: x 2 s to x 2 s 77 – 2(6) to 77 + 2(6) 77 – 12 to 77 + 12 65 to 89