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Chapter 3 Central Tendency and Variability Characterizing Distributions Central Tendency Most people know these as “averages” scores near the center of the distribution - the score towards which the distribution “tends” – Mean – Median – Mode Arithmetic Mean (Mean) Mean (μ; M or X ) - the numerical average; the sum of the scores (Σ) divided by the number of scores (N or n) X å m= N X å M= n Σ - The Summation Operator Sum the scores – In general, “Add up all the scores” – Sum all the values specified n åX i=1 i = X1 + X2 + + Xn Central Tendency (CT) Median (Md) - the score which divides the distribution in half; the score at which 50% of the scores are below it; the 50%tile Order the scores, and count “from the outside, in” Central Tendency (CT) Mode (Mo) - the most frequent score To find the mode from a freq. dist., look for the highest frequency For this distribution, the mode is the interval 24 - 26, or the midpoint 25 Mode Interval 15 - 17 18 - 20 21 - 23 24 – 26 27 – 29 30 – 32 33 – 35 36 – 38 39 – 41 42 – 44 45 – 47 48 – 50 Total f 1 2 3 6 2 2 2 0 1 0 1 1 21 cf 1 3 6 12 14 16 18 18 19 19 20 21 Characterizing Distributions Variability Variability is a measure of the extent to which measurements in a distribution differ from one another Three measures: – Range – Variance – Standard Deviation Variability Range - the highest score minus the lowest score Variability Variance (σ2) - the average of the squared deviations of each score from their mean (SS(X)), also known as the Mean Square (MS) n 1 2 s = å (Xi - m X ) N i=1 1 2 s x = SS(X) N 2 x Variance the average of the squared deviations of each score from their mean 1. Deviation of a score from the mean 2. Squared 3. All added up Average 4. Divide by N Computing Variance Score (X) μ X-μ (X – μ)2 2 4 -2 4 3 4 -1 1 4 4 0 0 4 4 0 0 5 4 1 1 6 4 2 2 Σ(X – μ)=0* Σ(X – μ)2= SS(X) = 10 *When computing the sum of the deviations of a set of scores from their mean, you will always get 0. This is one of the special mathematical properties of the mean. s 2 (X - m ) å = N 10 s = 6 2 s = 1.67 2 2 Variability Sample Variance (s2) – (sort of) the average of the squared deviations of each score from their mean (SS(X)) n 1 2 s = (Xi - M X ) å n - 1 i=1 1 2 sx = SS(X) n -1 2 x Unbiased Estimates M for μ (M is an unbiased estimate of μ) X å m= N X å M= n The average M (of all the Ms) from all random samples of size n is guaranteed to equal μ Samples systematically underestimate the variability in the population If we were to use the formula for population variance to compute sample variance We would systematically underestimate population variance by a factor of 1 in the denominator Therefore: Sample Variance (s2) – (sort of) the average of the squared deviations of each score from their mean; the unbiased estimate of σ2 n 1 2 s = (Xi - M X ) å n - 1 i=1 1 2 sx = SS(X) n -1 2 x Squared the Units? Score (X) μ X-μ (X – μ)2 2 4 -2 4 3 4 -1 1 4 4 0 0 4 4 0 0 5 4 1 1 6 4 2 2 Σ(X – μ)=0* Σ(X – μ)2= SS(X) = 10 Let’s say that these scores represent cigarettes smoked per day In the first column, for example, “2” represents the quantity “2 cigarettes” The third column represents 2 fewer cigarettes than the mean The fourth column represents “-2cigarettessqured” or 4 cigarettessquared Variability Standard Deviation (σ) - the square root of the variance (σ2) sX = 1 SS(X) N sX = s 2 X Variability in samples Sample Standard Deviation (s) - the square root of the variance (s2) 1 sX = SS(X) n -1 sX = s 2 X