1. Descriptive Statistics: Measures of Central Tendency Donnelly, 2 nd edition Chapter 3

2. 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

3. 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

4. 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

5. 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”