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3.3 Day II: Measures of Variance By the end of class you will be able to make estimates about standard deviations by using the “rule of thumb”, Chebyshev’s Theorem, and the Empirical Rule Heights of Students (inches) Enter class data in a list Calculate the mean Calculate the standard deviation Is there an easier way to describe how spread out the data is? Is there a way to predict the shortest and tallest student? Estimating Standard Deviation It takes some time to find the standard deviation. We will estimate it using the “Rule of Thumb”: range s 4 Estimate: Real St. Deviation: s = Estimating Data (“rule of thumb”) If the standard deviation and mean are known we can estimate the lowest and highest data value Mean = _____, st. deviation = _____ Predicted minimum value= X 2s Predicted maxiumum value = X 2s How many heights are 1 standard deviation away from the mean? How many are two standard deviations away from the mean? Counting = tedious! Empirical Rule (for symmetric/bell shaped distributions) Chebyshev’s Theorem (for any distribution) Empirical Rule About 68% of data are within 1 standard deviation of the mean. About 95% of the data are within 2 standard deviations of the mean. About 99.7% of the data are within 3 standard deviations of the mean. Empirical Rule: SAT Scores Chebyshev’s Theorem The proportion of data values from a data set that will fall within k standard deviations of the mean will be at least 1 1 2 k This means that if k=2, at least 75% of all data values are within 2 standard deviations of the mean. This means that if k = 3…. Chebychev’s Theorem Estimate the standard deviation of wait times at the bank: 6.5, 6.6, 6.7,6.8, 7.1, 7.3, 7.4, 7.7, 7.7, 7.7 The mean of heights of men is 69in with a standard deviation of 2.8in. Use the range of thumb rule to find the minimum and maximum heights. Minimum: 63.4in Maximum: 74.6in The mean of heights of men is 69in with a standard deviation of 2.8in. What percentage of men have heights between 60.6 in and 77.4in? (assume bell shape) 99.7% using the empirical rule The mean of heights of men is 69in with a standard deviation of 2.8in. What percentage of men have heights between 60.6 in and 77.4in? (assume no bell shape) 89% using Chebyshev’s Theorem How many heights would be in this range of 50 men were sampled? 44.5 Remember: You must always add and subtract standard deviations from the mean to determine how many standard deviations you are away from the mean! Summarizer 3 Facts about data variation 2 Facts about estimating variation 1 Question/Concern you still have