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Social Statistics: Difference Review Social statistics Descriptive statistics Inferential statistics Mean, median and mode Z519--Spring 2015 Outline Range Variance Standard deviation Using Excel and SPSS to calculate them The whole story Descriptive statistics Centrality tendency (average) Measurement of variability (variability) Average+Variability = describe the characteristics of a set of data Measures of variability Variability How scores differ from one another Three sets of data 7, 6, 3, 3, 1 3, 4, 4, 5, 4 4, 4, 4, 4, 4 Variability = the difference from the mean Measures of variability Three ways Range Variance Standard deviation Range The most general measure of variability How far apart scores are from one another Range = highest score – lowest score What is the range for the follow numbers? 98, 86, 77, 56, 48 Variance Variance The average of the squared deviations from the mean s 2 ( x x) n 1 2 Standard deviation Standard deviation (SD) Average deviation from the mean (average distance from the mean) Represents the average amount of variability s ( x x) n 1 2 Variance and SD 𝑥 − 𝑥 : the deviation from the mean 2 (𝑥 − 𝑥) : the sum of the squared deviations from the mean 𝑛 : the number of scores (𝑥−𝑥)2 𝑛−1 : the variance (𝑥−𝑥)2 𝑛−1 : the standard deviation Exercise The right figure shows 2013 GDP per Capita for select countries-current US$ Please calculate the variance and standard deviation by hand or by using excel (VAR.S() & STDEV.S()) Country Afghanistan Brazil Canada Chile Greece Japan Lebanon Norway Panama Saudi Arabia Singapore Tunisia United States Vietnam Zimbabwe GDP per Capita 678 11208 51958 15732 21910 38492 9928 100819 11037 25852 55182 4329 53143 1911 905 Data collected from http://data.worldbank.org/ STDEV.S and STDEV.P s s 2 ( x x ) n 1 2 ( x x ) n STDEV.S 𝑠 2 VAR.S STDEV.P 𝑠 2 VAR.P STDEV.S and STDEV.P STDEV.S is standard deviation for sample (biased SD) STDEV.P is standard deviation for population (unbiased SD) If your dataset is the whole population, use STDEV.P to calculate standard deviation If you dataset is the sample of something, use STDEV.S to calculate standard deviation Why n or n-1? To be conservative STDEV This is the standard deviation for sample Take n-1 in order to make STDEV a bit larger than it would be. If we have error, we compensate by overestimating the STDEV Why n or n-1? Sample size Numerator in standard deviation formula Denominator Population standard deviation STDEV.P (dividing by n) Denominator Sample standard deviation STDEV.S (dividing by n-1) Difference between STDEVP and STDEV 10 500 7.07 7.45 0.38 100 500 2.24 2.25 0.01 1000 500 0.7071 0.7075 0.0004 SD vs.Variance Often appears in the “Results” sections of journals They are quite different Variance is squared SD SD vs.Variance 9 8 7 mean 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 Average distance to mean=(2+2+2+1+1+1+2+3)/10=1.4 SD = 1.76 Variance = 3.1 What to remember Standard Deviation (SD) = the average distance from the mean The larger SD, the more different data are from one another Since mean is sensitive to extreme scores, so do SD If SD=0, this means that there is no variability in the set of scores (they are all identical in value) – this happens very rarely. Exercise 1 Average life expectancy for females in 2010-2011 is reported for 10 countries. Please calculate the range, variance and SD for European counties, NonEuropean countries, and all countries. Exercise 2 You and your friends have just measured the heights of your dogs (in millimeters): The heights (in mm) (at the shoulders) are: 600, 470, 170, 430 and 300. Find out the Mean, the Variance, and the Standard Deviation. Exercise 3 Western Airlines Flight Report Morning Flights Number of passengers Evening Flights Number of passengers Thursday Friday To Kansas To Kansas 258 251 Thursday Friday To Kansas To Kansas 312 Thursday To Philadelphia 303 Thursday To Philadelphia 331 321 Friday Thursday Friday To To To Philadelphia Providence Providence 312 166 176 Friday Thursday Friday To To To Philadelphia Providence Providence 331 210 274 Exercise 3 Write a half page summary report to your boss Form a group to discuss it