2.4 Measures of Variation

Range and Deviation Range - The difference between the maximum and the minimum data entries in the set Range = (Max – Min) Deviation – the difference between the entry and the mean of the data set. Deviation of

Variance and Standard Deviation Population Variance: Population Standard Deviation: Sample Variance: Sample Standard Deviation:

Deviation Squares Heights (in inches) 70 -.3 .09 1.7 2.89 72 .7 .49 71 69 73 68 -.3 1.7 .7 -1.3 2.7 -2.3 .09 2.89 .49 1.69 7.29 5.29 SSx= 2.01 1.418 2.233 1.494 Mean = 70.3

Empirical Rule

About 68% of the data lies within 1 standard deviation of the mean

About 95% of the data lies within 2 standard deviation of the mean

About 99.7 of the data lies within 3 standard deviation of the mean

Examples Heights of Women in the U.S. have a mean of 64 with a standard deviation of 2.75. Use the empirical rule to estimate: The percent of the heights that are between 61.25 and 64 inches. ANS: 34% Between what two values does about 95% of the data lie? ANS: (58.5, 69.5)

Chebychev’s Theorem The portion of any data set lying within k standard deviations (k>1) of the mean is at least Example: k = 2 , 75% of the data is within 2 standard deviations of the mean k = 3; 88.9% of the data lies within 3 standard deviations of the mean.

Sample Standard deviation for grouped data