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Chapter 3 Variability I. Variability – how scores differ from one another. Which set of scores has greater variability? Set 1: 8,9,5,2,1,3,1,9 Set 2: 3,4,3,5,4,6,2,3 Means are Set 1: 4.75 and Set 2: 3.75. Tells us nothing of variability. Variability is more precisely how different scores are from the mean. II. Computing the Range Subtract the lowest score from the highest (r=h-l) What is the range of these scores? 98,86,77,56,48 Answer: 50 (98-48=50) III. Computing the Standard Deviation The standard deviation (s) is the average amount of variability in a set of scores (average distance from mean). A. Formula: s X X n 1 2 Compute s for the following: 5,8,5,4,6,7,8,8,3,6 So, an s of 1.76 tells us that each score differs from the mean by an average of 1.76 points. *Why n-1? N represents the true population and n-1 represents the sample. Since we are projecting onto the sample, it is better to overestimate the variability (be conservative). The larger the sample size, however, the less of a difference this will make. B. Purpose: to compare scores between different distributions, even when the means and standard deviations are different (e.g., men and women). Larger the s the greater the variability. IV. Computing Variance – simply s2 (really only used to compute other formulas and techniques). Difference: Variance is stated in units that are squared (not original units). SPSS (practice in class p. 43). Chapter 4 Graphing I. Why? Describes data visually, more clearly. II. Frequency Distribution A. Class Interval Column – divides the scores up into categories (0-4, 5-9, etc.). Usually range of 2,5,10, or 25 data points. Main thing: be consistent! B. Frequency Column – number of scores within that range or category. III. Graphs A. Histogram – shows the distribution of scores by class interval. Can compare different distributions on the same histogram. Shows: 1. Variability (p. 60) 2. Skewness (p. 61). If the mean is greater than the median, positive skewness. If median is greater than mean, negative skewness. Relative Frequency Central Tendency and Variability Centre Relative Frequency Central Tendency and Variability Spread Skewness Relative Frequency If the data set is symmetric, the mean equals the median. Median Mean Skewness If the data set is skewed to the right, the mean is greater than the median. Median Mean Skewness If the data set is skewed to the left, the mean is less than the median. Mean Median B. Column Charts – simply tells the quantity of a category according to some scale. SCALE IS IMPORTANT (CSPAN-drug use story). C. Bar Charts – same as Column chart, but reverse the axes. D. Line Chart – Used to show trends (e.g. rise and fall in pres. Popularity – line on website). E. Pie Charts – Great for proportions (percent of MS budget going to each budget category). IV. SPSS and Graphing (southern states and % evangelical-histogram; this class and % gop/dem/other – line/bar)