Chapter 2 Central Tendency & Variability. Measures of Central Tendency The Mean Sum of all the scores divided by the number of scores Mean of 7,8,8,7,3,1,6,9,3,8.
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Measures of Central Tendency The Mean Sum of all the scores divided by the number of scores Mean of 7,8,8,7,3,1,6,9,3,8 –ΣX = 7+8+8+7+3+1+6+9+3+8 = 60 (Σ indicates to ‘sum up’) –N = 10 –Mean = 60/10 = 6 (symbol used in book is M)
The Mode Most common single number in a distribution Mode of 7,8,8,7,3,1,6,9,3,8 = 8 Measure of central tendency for nominal variables (most common category was…)
The Median The middle score when all scores are arranged from lowest to highest Median of 7,8,8,7,3,1,6,9,3,8 –1 3 3 6 7 7 8 8 8 9 median –Median is the average (mean) of the 5 th and 6 th scores, so the median is 7 –Useful when mean is affected too greatly by a few outliers (median is not affected at all).
The Median With large data sets, shortcut… –If odd N, divide by 2, then add ½, gives position of the median score in list. Ex? –If even N, divide by 2, this and score above it need to be averaged for median. ex?
Example of Mean/Median Preference Evolutionary psych example in book (p. 45-46) – competing theories of gender diffs in how many mates we prefer –Buss (evol.) – men should prefer more partners than women (to spread genes) –Miller – men/women should prefer same #
Table 2-1, data from 106 men, 160 women Women’s M=2.8, Median = 1, Mode = 1 Men’s M=64.3, Median = 1, Mode = 1 See Fig 2-8, clearly a few outliers for men (preferring 100-1000 and 1001-10,000 partners), strongly affects M
Measures of Spread The Variance The average of each score’s squared difference from the mean Steps for computing the variance: 1. Subtract the mean from each score 2. Square each of these deviation scores 3. Add up the squared deviation scores 4. Divide the sum of squared deviation scores by the number of scores
Measures of Spread The Variance Formula for the variance: SD=Standard Deviation (when squared = variance) SS- Sum of Squares
What variance tells us Conceptually, it is the average of the squared deviation scores, so… –The more spread out the distribution, the larger the variance What if variance = 0? –Very important for many stat tests –Conceptual difference in unit of variance versus standard deviation? Which is more intuitive?
Measures of Spread The Standard Deviation Most common way of describing the spread of a group of scores Steps for computing the standard deviation: 1. Figure the variance 2. Take the square root Conceptually, it is the average of deviations from the mean. –How much do most scores differ from the mean?
Measures of Spread The Standard Deviation Formula for the standard deviation:
Computational Formula: Easier to use w/large data sets Uses sum of x scores ( X ) and sum of squared x scores ( X 2 ) SD 2 = X 2 – [( X) 2 / N] N Note that your book prefers the definitional formula, not this one Note on p. 56 about when to divide SS by N-1