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Chapter 3

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Chapter 3

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  1. Chapter 3 Central Tendency and Variability

  2. 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

  3. 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)

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

  5. 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”

  6. 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

  7. 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

  8. Variability • Range - the highest score minus the lowest score

  9. Variability • Variance (σ2) - the average of the squared deviations of each score from their mean (SS(X)), also known as the Mean Square (MS)

  10. 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

  11. 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.

  12. Variability • Sample Variance (s2) – (sort of) the average of the squared deviations of each score from their mean (SS(X))

  13. 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

  14. 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

  15. Therefore: • Sample Variance (s2) – (sort of) the average of the squared deviations of each score from their mean; the unbiased estimate of σ2

  16. 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

  17. Variability • Standard Deviation (σ) - the square root of the variance (σ2)

  18. Variability in samples • Sample Standard Deviation (s) - the square root of the variance (s2)