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University of Toronto St. George
Political Science
Kenichi Ariga

amyc pol322 feb.03.2014 2.Center of Data • Mode • Median • Mean • Median: ◦ definition: order smallest value to largest value. Middle is median ◦ if even number of observations, then divide the middle numbers by 2 ◦ if odd number of observations, then compute via MS Excel • Mean: ◦ avg.value ◦ y=1/N << N i=1 y1 ◦ N= # observations ◦ y=value of variable of observations ◦ i=index ◦ yi=value of observation I ◦ <<=sum of ◦ example: 1/11 (48+58+52...) ~ 52.09 (approx. ◦ - ◦ the mean is sensitive: mean may differ if you include/exclude outliers ◦ outliers are extreme values. Outliers always pull the mean to the right ◦ example: if you exclude outliers on the left, mean is larger ◦ example: if you exclude outliers on the right, mean is smaller 3.Variability of Data • Why? ◦ The spread of distribution is greater in b), but it has the same mean as a) • Variance and Standard Deviation ◦ S^2=1/N-1 Sum Of N, i=1 (y1,-y)^2 ◦ definition: sum of squared diffs b/w each observation's value and the mean, divided by number of observations minus one ◦ yi-y=diff b/w each observation and mean ◦ (yi-y)^2=squared of the diff ◦ Sum Of N, i=1(yi-y)^2=sum of squared diffs for all observations ◦ 1/N-1 Sum Of N, i=1 (y1,-y)^2=divided by number of observations minus one • Standard Deviation ◦ square root of variance ◦ S=Root S ^2 ◦ standard deviation ~ avg. distance b/w each observation value (yi) and the mean (y) ◦ we want to show the variability of values of a variable across its mean ◦ avg. distance from each observation nd mean is nice, but if we compute it, it would equal to 0! ◦ thus, take the square of the distance to change everything into +ve values, then take the square root of their avg to express it in the original unit of the variables ◦ we use N-1 instead of N because N-1 is more appropriate for statistical inference ◦ so, going back to a) and b), we can now describe the EXTENT of the 2 spreads: • Percentile, Quartile, IQR ◦ percentile: the pth percentile is a value such that ◦ median is 50 percentile ◦ ex: the 10 percentile here is 32.3 th ◦ lower quartile: 25 percentile. ¼ of data fall below the lower quartile ◦ ex: figure 2 • Box Plot ◦ a type of graph: • Box Plot: Outliers ◦
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