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Measures of Dispersion

Learning Objectives: Explain what is meant by variability Describe, know when to use, interpret and calculate: range, variance, and standard deviation. Measures of Dispersion. More Statistical Notation. indicates the sum of squared Xs. Square ea score (2 2 + 2 2 )

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Measures of Dispersion

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  1. Learning Objectives: Explain what is meant by variability Describe, know when to use, interpret and calculate: range, variance, and standard deviation Measures of Dispersion

  2. More Statistical Notation • indicates the sum of squared Xs. • Square ea score (22+ 22) • Find sum of squared Xs =4+4=8 • indicates the squared sum of X. • (2+2)2

  3. Measures of Variability … describe the extent to which scores in a distribution differ from each other.

  4. A Chart Showing the Distance Between the Locations of Scores in Three Distributions

  5. Variability • Provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together • Figure 4.1

  6. Kurtosis • Kurtosis based on size of a distribution’s tail. • Leptokurtic: thin or skinny dist • Platykurtic: flat • Mesokurtic: same kurtosis (normal distribution)

  7. Three Variations of the Normal Curve

  8. The Range, Semi-Interquartile Range, Variance, and Standard Deviation

  9. The Range • … indicates the distance between the two most extreme scores in a distribution • Crude measurement • Used w/ nominal or ordinal data • Rangedifference btwn upper real limit of max score and lower real limit of min score • Range = highest score – lowest score

  10. The Interquartile Range • Covered by the middle 50% of the distribution • Interquartile range= Q3-Q1 • Semi-Interquartile Range • Half of the interquartile range

  11. Variance and Standard Deviation • Variance & standard deviation communicate how different the scores in a distribution are from each other • We use the mean as our reference point since it is at the center of the distribution and calculate how spread out the scores are around the mean

  12. The Population Variance and the Population Standard Deviation

  13. Population Variance • The population variance is the true or actual variance of the population of scores.

  14. Population Standard Deviation • The population standard deviation is the true or actual standard deviation of the population of scores.

  15. Describing the Sample Variance and the Sample Standard Deviation

  16. Sample Variance • The sample variance is the average of the squared deviations of scores around the sample mean

  17. Sample Variance • Variance is average of squared deviations (usually large) & squared units • Difficult to interpret • Communicates relative variability

  18. Standard Deviation • Measure of Var. that communicates the average deviation • Square root of variance

  19. Sample Standard Deviation • The sample standard deviation is the square root of the average squared deviation of scores around the sample mean.

  20. The Standard Deviation • … indicates • “average deviation” from mean, • consistency in scores, • & how far scores are spread out around mean • larger the value of SD, the more the scores are spread out around mean, and the wider the distribution

  21. Normal Distribution and the Standard Deviation

  22. Normal Distribution and the Standard Deviation Approximately 34% of the scores in a perfect normal distribution are between the mean and the score that is one standard deviation from the mean.

  23. The Estimated Population Variance and the Estimated Population Standard Deviation

  24. Estimating the Population Variance and Standard Deviation • The sample variance is a biased estimator of the population variance. • The sample standard deviation is a biased estimator of the population standard deviation.

  25. Estimated Population Variance • By dividing the numerator of the sample variance by N - 1, we have an unbiased estimator of the population variance.

  26. Estimated Population Standard Deviation • By dividing the numerator of the sample standard deviation by N - 1, we have an unbiased estimator of the population standard deviation.

  27. Unbiased Estimators • is an unbiased estimator of • is an unbiased estimator of • The quantity N - 1 is called the degrees of freedom • Number of scores in a sample that are free to vary so that they reflect variability in pop

  28. Uses of , , and • Use the sample variance and the sample standard deviation to describe the variability of a sample. • Use the estimated population variance and the estimated population standard deviation for inferential purposes when you need to estimate the variability in the population.

  29. Organizational Chart of Descriptive and Inferential Measures of Variability

  30. Always.. • Determine level of measurement • Examine type of distribution • Calculate mean • Calculate variability

  31. American Psychological Association (5th ed) • Mean • M • Standard Deviation • SD

  32. Example • Using the following data set, find • The range, • The semi-interquartile range, • The sample variance and standard deviation, • The estimated population variance standard deviation

  33. Example Range • The range is the largest value minus the smallest value.

  34. ExampleSample Variance

  35. ExampleSample Standard Deviation

  36. ExampleEstimated Population Variance

  37. Example—Estimated Population Standard Deviation

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