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## Chapter 3

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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; Mor 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”**Mode**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**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)**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***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 (s2) – (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 (s2) – (sort of) the average of the squared deviations of each score from their mean; the unbiased estimate of σ2**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**Variability**• Standard Deviation (σ) - the square root of the variance (σ2)**Variability in samples**• Sample Standard Deviation (s) - the square root of the variance (s2)