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Why is the study of variability important? • Allows us to distinguish between usual & unusual values • In some situations, want more/less variability – scores on standardized tests – time bombs – medicine Measures of Variability • range (max-min) • interquartile range (Q3-Q1) • deviations x x Lower case Greek letter 2 sigma • variance • standard deviation Suppose that we have these data values: 24 16 34 28 26 21 30 35 37 29 Find the mean. Find the deviations. x x What is the sum of the deviations from the mean? 24 16 34 28 26 21 30 35 37 29 x x 2 Square the deviations: Find the average of the squared deviations: x x 2 n The average of the deviations squared is called the variance. Population parameter 2 Sample s 2 statistic Calculation of variance of a sample xn x s n 1 2 2 df A standard deviation is a measure of the average deviation from the mean. Calculation of standard deviation xn x s 2 n 1 Degrees of Freedom (df) • n deviations contain (n - 1) independent pieces of information about variability Use calculator Which measure(s) of variability is/are resistant? Linear transformation rule • When multiplying or adding a constant to a random variable, the mean changes by both. • When multiplying or adding a constant to a random variable, the standard deviation changes only by multiplication. • Formulas: ax b ax b ax b a x An appliance repair shop charges a $30 service call to go to a home for a repair. It also charges $25 per hour for labor. From past history, the average length of repairs is 1 hour 15 minutes (1.25 hours) with standard deviation of 20 minutes (1/3 hour). Including the charge for the service call, what is the mean and standard deviation for the charges for labor? 30 25(1.25) $61.25 1 25 $8.33 3 Rules for Combining two variables • To find the mean for the sum (or difference), add (or subtract) the two means • To find the standard deviation of the sum (or differences), ALWAYS add the variances, then take the square root. • Formulas: a b a b a b a b 2 a a b 2 b If variables are independent Bicycles arrive at a bike shop in boxes. Before they can be sold, they must be unpacked, assembled, and tuned (lubricated, adjusted, etc.). Based on past experience, the times for each setup phase are independent with the following means & standard deviations (in minutes). What are the mean and standard deviation for the total bicycle setup times? Phase Mean SD Unpacking Assembly Tuning 3.5 21.8 12.3 0.7 2.4 2.7 T 3.5 21.8 12.3 37.6 minutes T 0.7 2 2.42 2.7 2 3.680 minutes