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# Variability

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### Variability

1. 1. Chapter Five Measures of Variability: Range, Variance, and Standard Deviation
2. 2. New Statistical Notation <ul><li> indicates the sum of squared X s. </li></ul><ul><li> indicates the squared sum of X . </li></ul>
3. 3. Measures of Variability <ul><li>Measures of variability describe the extent to which scores in a distribution differ from each other. </li></ul>
4. 4. A Chart Showing the Distance Between the Locations of Scores in Three Distributions
5. 5. Three Variations of the Normal Curve
6. 6. The Range, Variance, and Standard Deviation
7. 7. The Range <ul><li>The range indicates the distance between the two most extreme scores in a distribution </li></ul>Range = highest score – lowest score
8. 8. Variance and Standard Deviation <ul><li>The variance and standard deviation are two measures of variability that indicate how much the scores are spread out around the mean </li></ul><ul><li>We use the mean as our reference point since it is at the center of the distribution </li></ul>
9. 9. The Sample Variance and the Sample Standard Deviation
10. 10. Sample Variance <ul><li>The sample variance is the average of the squared deviations of scores around the sample mean </li></ul><ul><li>Definitional formula </li></ul>
11. 11. Sample Variance <ul><li>Computational formula </li></ul>
12. 12. Sample Standard Deviation <ul><li>The sample standard deviation is the square root of the sample variance </li></ul><ul><li>Definitional formula </li></ul>
13. 13. Sample Standard Deviation <ul><li>Computational formula </li></ul>
14. 14. The Standard Deviation <ul><li>The standard deviation indicates the “average deviation” from the mean, the consistency in the scores, and how far scores are spread out around the mean </li></ul><ul><li>The larger the value of  X , the more the scores are spread out around the mean, and the wider the distribution </li></ul>
15. 15. Normal Distribution and the Standard Deviation
16. 16. Normal Distribution and the Standard Deviation <ul><li>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. </li></ul>
17. 17. Standard Deviation and Range <ul><li>For any roughly normal distribution, the standard deviation should equal about one-sixth of the range. </li></ul>
18. 18. The Population Variance and the Population Standard Deviation
19. 19. Population Variance <ul><li>The population variance is the true or actual variance of the population of scores. </li></ul>
20. 20. Population Standard Deviation <ul><li>The population standard deviation is the true or actual standard deviation of the population of scores. </li></ul>
21. 21. The Estimated Population Variance and the Estimated Population Standard Deviation
22. 22. Estimating the Population Variance and Standard Deviation <ul><li>The sample variance is a biased estimator of the population variance. </li></ul><ul><li>The sample standard deviation is a biased estimator of the population standard deviation. </li></ul>
23. 23. Estimated Population Variance <ul><li>By dividing the numerator of the sample variance by N - 1, we have an unbiased estimator of the population variance. </li></ul><ul><li>Definitional formula </li></ul>
24. 24. Estimated Population Variance <ul><li>Computational formula </li></ul>
25. 25. Estimated Population Standard Deviation <ul><li>By dividing the numerator of the sample standard deviation by N - 1, we have an unbiased estimator of the population standard deviation. </li></ul><ul><li>Definitional formula </li></ul>
26. 26. Estimated Population Standard Deviation <ul><li>Computational formula </li></ul>
27. 27. Unbiased Estimators <ul><li>The quantity N - 1 is called the degrees of freedom </li></ul><ul><li>is an unbiased estimator of </li></ul><ul><li>is an unbiased estimator of </li></ul>
28. 28. Uses of , , , and <ul><li>Use the sample variance and the sample standard deviation to describe the variability of a sample. </li></ul><ul><li>Use the estimated population variance and the estimated population standard deviation for inferential purposes when you need to estimate the variability in the population. </li></ul>
29. 29. Organizational Chart of Descriptive and Inferential Measures of Variability
30. 30. Proportion of Variance Accounted For <ul><li>The proportion of variance accounted for is the proportion of error in our predictions when we use the overall mean to predict scores that is eliminated when we use the relationship with another variable to predict scores </li></ul>
31. 31. Example <ul><li>Using the following data set, find </li></ul><ul><ul><li>The range, </li></ul></ul><ul><ul><li>The sample variance and standard deviation, </li></ul></ul><ul><ul><li>The estimated population variance and standard deviation </li></ul></ul>15 14 14 17 15 14 13 14 13 12 10 13 15 11 15 13 14 14
32. 32. Example Range <ul><li>The range is the largest value minus the smallest value. </li></ul>
33. 33. Example Sample Variance
34. 34. Example Sample Standard Deviation
35. 35. Example Estimated Population Variance
36. 36. Example—Estimated Population Standard Deviation