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Honors Math 3 Unit 3: Inferential Statistics Name___________________ Date____________________ 3.07: Three measures of spread Here are three ways of measuring how spread out a data set is. The inputs for all three measures are the deviations (how far each data value is from the mean), but each measure combines these numbers in a different way. For any of the three measures, a larger value indicates that a data set is more spread out. How to calculate the mean absolute deviation Step 1. Given a data set, calculate the mean (notation: x ) Step 2. Find the absolute deviation for each data value x by calculating | x – x |. Step 3. Find the mean of the Step 2 numbers. The result is called the mean absolute deviation. How to calculate the variance (a.k.a. mean squared deviation) Step 1. Given a data set, calculate the mean. Step 2. Find each data value’s deviation. Step 3. Calculate the square of each number found in Step 2. Step 4. Now calculate the mean of the numbers from Step 3. The result is called the variance. How to calculate the standard deviation (a.k.a. root mean squared deviation) Steps 1–4. Same as for finding the variance. Step 5. Now take the square root of the variance. The result is called the standard deviation. Example Data set: {3, 3, 4, 6, 6, 6, 6, 10, 14, 15, 15, 15, 15, 16, 16}. First step is finding the mean: x = 10. x 3 3 4 6 6 6 6 10 14 15 15 15 15 16 16 |x– x| 7 7 6 4 4 4 4 0 4 5 5 5 5 6 6 sum of data values = 150 mean = 150/15 = 10 sum of absolute deviations = 72 mean absolute deviation = 72/15 = 4.8 ( x – x )2 49 49 36 16 16 16 16 0 16 25 25 25 25 36 36 sum of squares of deviations = 386 variance = 386/15 ≈ 25.733 standard deviation ≈ 25.733 ≈ 5.073 You try it 1. For the data set {3, 4, 5, 6, 9, 10, 10, 10, 10, 10, 11, 15, 15, 16, 16}, answer the following questions using the table below. Follow the model of the page 1 example. a. Find the mean. b. Find the mean absolute deviation. c. Find the variance. d. Find the standard deviation. e. A good way to check your answers is to find the standard deviation on your calculator. Here’s how: type the data set into your calculator under STAT Edit L1, then do STAT Calc 1-Var Stats f. Compare the answers to b, c, and d with the values for the example from page 1. According to each of the measures of spread, which data set is more spread out? x 3 4 5 6 9 10 10 10 10 10 11 15 15 16 16 |x– x| ( x – x )2 If you have time, try the following. We will discuss this more next class. 2. Generalize using 𝚺 notation: Now suppose the data set is {x1, x2, x3, …, xn}. Write expressions using 𝚺 notation for each of the following. mean absolute deviation = variance = standard deviation =