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Measures of Dispersion

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Here are two sets to look at A = {1,2,3,4,5,6,7} B = {8,9,10,11,12,13,14} Do you expect the sets to have the same means? Median? Mode? What if they did?

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Lets look at three other sets: C = {11,12,13,14,15} D = {5,9,13,17,21} E = {1,7,13,19,25} Find the mean, median and mode C D E

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Yet the set are different. How? Mean Median Mode C = {11,12,13,14,15} 13 13 D = {5,9,13,17,21} 13 13 E = {1,7,13,19,25} 13 13 E x x x x x D x x x x x C x x x x x 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

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E x x x x x D x x x x x C x x x x x 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Set C values are close together Set D values are spread out more from the center Set E values are farthest from the center Conclusion: Mean, median and mode are blind to how the data is spread out. So lets look at the Range: Largest Value – Smallest Value Range c = 4 Range D = 16 Range E = 24

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E x x x x x D x x x x x C x x x x x 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Range: Largest Value – Smallest Value Range c = 4 Range D = 16 Range E = 24 We can see the larger the range the greater the spread between the largest and smallest. Weakness: It is looking at only the extreme values and ignores all the other values. We know nothing about the other data values.

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Here are three set all with same range but very different dispersion of the data values. R x x x x x x x x x S x x x x x x x x x D x x x x x x x x 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 So we should be looking at all the values and their relationship to the center of the data. We will use the mean as the center of measure to do this comparison. We call this looking at the spread of the data. The statistics terms are Variation and Standard Deviation.

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δ Variance is δ 2 or s 2 Formula for Standard Deviation

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C Deviation Deviation 2 value - mean 11 12 13 14 15 Finding Standard Deviation E x x x x x D x x x x x C x x x x x

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D Deviation Deviation 2 value - mean 5 9 13 17 21 Finding Standard Deviation E x x x x x D x x x x x C x x x x x

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E Deviation Deviation 2 value - mean 1 7 13 19 25 Finding Standard Deviation E x x x x x D x x x x x C x x x x x

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Set c standard deviation is Set D standard deviation is Set E standard deviation is We can see the larger the standard deviation is the more spread out the population is E x x x x x D x x x x x C x x x x x

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Population STDSample STD δ Variance is δ 2 or s 2 Formula for Standard Deviation

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Empirical Rule The standard deviation is very useful for estimating probabilities 68 – 95 - 99.7

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IN A NORMAL DISTRIBUTION (Bell shaped curve):68-95-99.7 RULE μ IS THE MEAN OF THE DATA AND σ THE STANDARD DEVIATION 68% OF THE DATA FALLS WITHIN σ OF THE MEAN 95% OF THE DATA FALLS WITHIN 2 σ OF THE MEAN 99.7% OF THE DATA FALLS WITHIN 3 σ OF THE MEAN Empirical Rule μ σ2σ3σ -1σ -2σ-3σ

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Empirical Rule

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Below is the height of 149 females at a local college. Does this data seem to be a symmetrical mound shape? Mean ≈ 66.4 & SD ≈ 2 Applying the empirical rule, what % of females at the college are between 62.4 inches and70.4 inches? Between what two heights are 68% of the females at the college? What percent of the females at the college are above 68.4 inches?

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Amanda39 Amber21 Tim 9 Mike32 Nicole30 Scot45 Erica11 Tiffany12 Glenn39 The following data represents the travel time in minutes to school for nine students enrolled in College. Find the population mean and standard deviation. Find 2 random samples of 3 to estimate the mean and standard deviation for this population.

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Amanda39 Amber21 Tim 9 Mike32 Nicole30 Scot45 Erica11 Tiffany12 Glenn39 Sample 1: mean STD Sample 2: mean STD Sample 3 : mean STD Which samples over or underestimated the population Parameters.?

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END Go to Skewness PowerPoint

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μ

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Empirical Rule The standard deviation is very useful for estimating probabilities 68 – 95 - 99.7

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δ Variance is δ 2 or s 2 Formula for Standard Deviation

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Statistics Review – Part I

Statistics Review – Part I

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