9.3 – Measures of Dispersion Chapter 1 9.3 – Measures of Dispersion Objective: TSW calculate the range and standard deviation from a set of data. © 2008 Pearson Addison-Wesley. All rights reserved

Measures of Dispersion Sometimes we want to look at a measure of dispersion, or spread, of data. Two of the most common measures of dispersion are the range and the standard deviation. © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Range For any set of data, the range of the set is given by Range = (greatest value in set) – (least value in set). © 2008 Pearson Addison-Wesley. All rights reserved

Example: Find the median and mean for each set of data below. Now, let’s find the range. Set A 1 2 7 12 13 Set B 5 6 8 9 © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Standard Deviation One of the most useful measures of dispersion, the standard deviation, is based on deviations from the mean of the data. To find the deviations of each number: Find the mean of the set of data Subtract the number from the mean © 2008 Pearson Addison-Wesley. All rights reserved

Example: Deviations from the Mean Find the deviations from the mean for all data values of the sample 1, 2, 8, 11, 13. Solution The mean is 7. Subtract to find deviation. Data Value 1 2 8 11 13 Deviation The sum of the deviations for a set is always 0. © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Standard Deviation Variance, to find: Square each deviation add the squares up divide by (n-1) where n = number of items in the set. standard deviation – the square root of the variance. Gives an average of the deviations from the mean. Which is denoted by the letter s. (The standard deviation of a population is denoted the lowercase Greek letter sigma.) © 2008 Pearson Addison-Wesley. All rights reserved

Calculation of Standard Deviation Let a sample of n numbers x1, x2,…xn have mean Then the sample standard deviation, s, of the numbers is given by The smaller the number for standard deviation – the closer your data is together. The higher the number for standard deviation, you data is further apart. © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Two standard deviations, or two sigmas, away from the mean (the red and green areas) account for roughly 95 percent of the data points. Three (3) standard deviations (the red, green and blue areas) account for about 99 percent of the data points. If this curve were flatter and more spread out, the standard deviation would have to be larger in order to account for those 68 percent or so of the points. That's why the standard deviation can tell you how spread out the examples in a set are from the mean. © 2008 Pearson Addison-Wesley. All rights reserved

Calculation of Standard Deviation The individual steps involved in this calculation are as follows Step 1 Calculate the mean of the numbers. Step 2 Find the deviations from the mean. Step 3 Square each deviation. Step 4 Sum the squared deviations. Step 5 Divide the sum in Step 4 by n – 1. Step 6 Take the square root of the quotient in Step 5. © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Example Find the standard deviation of the sample 1, 2, 8, 11, 13. The mean =_____. Data Value 1 2 8 11 13 Deviation (Deviation)2 © 2008 Pearson Addison-Wesley. All rights reserved

Example: Find the Standard Deviation 12, 13, 16, 18, 18, 20 mean = _______ Data Value 12 13 16 18 20 Deviation (Deviation)2 © 2008 Pearson Addison-Wesley. All rights reserved

Example: Find the Standard Deviation 125, 131, 144, 158, 168, 193 Data Value 125 131 144 158 168 193 Deviation (Deviation)2 © 2008 Pearson Addison-Wesley. All rights reserved

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