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)

Σ - The Summation Operator Sum the scores –In general, “Add up all the scores” –Sum all the values specified

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 , or the midpoint 25 Mode

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 (σ 2 ) - the average of the squared deviations of each score from their mean (SS(X)), also known as the Mean Square (MS)

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 4. Divide by N Average

Computing Variance Score (X)μX - μ(X – μ) 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.

Variability Sample Variance (s 2 ) – (sort of) the average of the squared deviations of each score from their mean (SS(X))

Unbiased Estimates M for μ (M is an unbiased estimate of μ) 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 (s 2 ) – (sort of) the average of the squared deviations of each score from their mean; the unbiased estimate of σ 2

Squared the Units? 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 “-2cigarettes- squred” or 4 cigarettes- squared Score (X)μX - μ(X – μ) Σ(X – μ)=0*Σ(X – μ) 2 = SS(X) = 10

Variability Standard Deviation (σ) - the square root of the variance (σ 2 )

Variability in samples Sample Standard Deviation (s) - the square root of the variance (s 2 )