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Descriptive Statistics: Measures of Central Tendency Donnelly, 2 nd edition Chapter 3
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Descriptive Statistics: Central Tendency Central Tendency: How the distribution of data clusters around the middle 1. Mean/Average = arithmetic average of all the data values ( most common measure) -- Population Mean = -- ∑ is the Greek letter Sigma and stands for "add together“ -- x i is the value of each variables -- i=1 is where to start (in this case the first observation in set x) -- Sample Mean = -- n number of data values in data set EX: x = {2, 5, 7, 8, 9} i = 1 and n = 4 => / 5 = 6.2 Population or Sample? Population measurement is parameter (greek letter) Sample measurement is statistic (english letter) Hope that sample statistic is similar to population parameter (use sample statistic to estimate population parameter) If not, then sample is over/under represented ** Typically deal with sample, but will present both population and sample measurements. 2
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Descriptive Statistics: Central Tendency, cont. 2. Median = the value in the middle when data values are arranged in ascending order (highest to lowest) -- value for which half of the values are greater and half are less -- if n is odd: median is always the middle value -- if n is even: median is the average of the two middle values Ex: 1) 95, 92, 90, 90, 88 = 90 (mean = 91) 2) 95, 92, 90, 90, 88, 60 = 90 (mean = 85.8) 3.Mode = the value that occurs most frequently -- if all values occur with the same frequency there will not be a mode -- possible to have more than one mode – bimodal distribution Ex: 1) 95, 90, 92, 88, 90 = 90 2) 95, 90, 92, 88, 90, 60 = 90 3
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Revisiting Frequency Distributions Shapes of Frequency Distributions: Mean and median will influence shape of distribution Symmetrical Distribution: mean = median (Graph A)A. Left-skewed Distribution: mean < median (Graph B) Higher data values dominate data set Ex: grade inflation Right-skewed Distribution: mean > median (Graph C) B Lower data values dominate data set C Ex: annual income Ex: 1) 95, 90, 92, 88, 90 mean = 91, median = 92, mode = 90 fairly symmetric; slightly left skewed 2) 95, 90, 92, 88, 90, 60 mean = 85.8, median = 90, mode = 90 left skewed Mean, Median, or Mode? Mean => simplest and most common; influenced by outliers Median => with outliers; more computational burdensome Mode => more than 1 mode or no mode 4
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Using Excel 1.Mean = Average(cell range) 2.Median = Median(cell range) 3. Mode(cell range) => excel will only report the smallest mode / if there is no mode => excel reports “N/A”
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