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Summarizing Symmetric Distributions

# Summarizing Symmetric Distributions - 2.3 Describing...

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1 2.3 Describing Distributions Numerically – cont. Describing Symmetric Data

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2 Symmetric Data Body temp. of 93 adults
3 Recall: 2 characteristics of a data set to measure center measures where the “middle” of the data is located variability measures how “spread out” the data is

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4 Measure of Center When Data Approx. Symmetric mean (arithmetic mean) notation x i x x x x n x x x x x i n i i n n : th measurement in a set of observations number of measurements in data set; sample size 1 2 3 1 1 2 3 , , , , : = = + + + +
5 N x n x n x x x x x x N i i n i i n = = = = + + + + = 1 1 3 2 1 size population = N known) not typically (value mean Population mean Sample μ μ

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Connection Between Mean and Histogram A histogram balances when supported at the mean. Mean x = 140.6 Histogram 0 10 20 30 40 50 60 70 118.5 125.5 132.5 139.5 146.5 153.5 160.5 More Absences from Work Frequency Frequency
7 Mean: balance point Median: 50% area each half right histo: mean 55.26 yrs, median 57.7yrs

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8 Properties of Mean, Median 1.The mean and median are unique; that is, a data set has only 1 mean and 1 median (the mean and median are not necessarily equal). 2.The mean uses the value of every number in the data set; the median does not. 1 4 20 4 6 Ex. 2, 4, 6, 8. 5; 5 4 2 21 4 6 Ex. 2, 4, 6, 9. 5 ; 5 4 2 x m x m + = = = = + = = = =
9 Example: class pulse rates 53 64 67 67 70 76 77 77 78 83 84 85 85 89 90 90 90 90 91 96 98 103 140 23 1 23 84.48; 23 :location: 12th obs. 85 i i n x x m m = = = = =

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10 2002, 2006 baseball salaries 2002 n = 805 μ = \$2,384,779 median = \$900,000 max = \$25,000,000 2006 n = 821 μ = \$2,831,674 median = \$950,000 max = \$21,680,727
11 Disadvantage of the mean Can be greatly influenced by just a few observations that are much greater or much smaller than the rest of the data

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