Math 141-copyright Joe Kahlig, 010B Page 1Section 8.3: Variance and Standard DeviationExample: Find the mean and median of these data sets.A) 40, 40, 40, 100, 100, 100B) 60, 65, 65, 70, 75, 75, 80Definition: The standard deviation of a data set is the measure of how the data is spread aboutits mean.The variance of a data set is the average of the square of the distance from the data value to the mean.Variance =(x1− µ)2+ (x2− µ)2+ ... + (xn− µ)2nNotation:Sample PopulationSxst. deviation σx mean µSample Variance formula =(x1−x)2+ (x2− x)2+ ... + (xn− x)2n − 1Example: Compute the standard deviation and the variance.data 7 9 12 15freq. 9 9 7 10Example: Compute the standard deviation and the variance. Let X = the number of Dr. Peppersdrank during a semester.X10 23 43 26 250freq. 50 30 49 73 1Example: Compute the standard deviation.data1 3 8 10prob. 0.3 0.2 0.4 0.1Math 141-copyright Joe Kahlig, 010B Page 2Example: Let X be a discrete random variable with integer values such that 0 ≤ X ≤ 90. The randomvariable has an expected value of 15.3 and standard deviation of 3.5636.A) What values of X are within 1.25 standard deviations of the mean?B) What values of X are upto 2 standard deviations above the mean?Chebyshev’s Inequality: Let X be a random variable with mean µ and standard deviation σ. Thenthe probability that X will be within k standard deviations of the mean isP (µ − kσ ≤ X ≤ µ + kσ) ≥ 1 −1k2Example: The random variable X has a mean of 50 and a standard deviation of 14. EstimateP (30 ≤ X ≤ 70).Example: The expected lifetime of a product is 2 years with a standard deviation of 3.5 months.For a shipment of 5000 items, estimate the number of item that will last between 17.7 months and30.3
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