This document discusses measures of variability used to describe how spread out data values are from the mean or average. It defines and provides formulas for calculating range, variance, standard deviation, sample variance, sample standard deviation, population variance, population standard deviation, estimated population variance, and estimated population standard deviation. These measures are important in statistical analysis to understand the distribution of data values.

1. Chapter Five Measures of Variability: Range, Variance, and Standard Deviation

2. New Statistical Notation indicates the sum of squared X s. indicates the squared sum of X .

3. Measures of Variability 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. Three Variations of the Normal Curve

6. The Range, Variance, and Standard Deviation

7. The Range The range indicates the distance between the two most extreme scores in a distribution Range = highest score – lowest score

8. Variance and Standard Deviation The variance and standard deviation are two measures of variability that indicate how much the scores are spread out around the mean We use the mean as our reference point since it is at the center of the distribution

9. The Sample Variance and the Sample Standard Deviation

10. Sample Variance The sample variance is the average of the squared deviations of scores around the sample mean Definitional formula

11. Sample Variance Computational formula

12. Sample Standard Deviation The sample standard deviation is the square root of the sample variance Definitional formula

13. Sample Standard Deviation Computational formula

14. The Standard Deviation 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 The larger the value of  X , the more the scores are spread out around the mean, and the wider the distribution

15. Normal Distribution and the Standard Deviation

16. 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.

17. Standard Deviation and Range For any roughly normal distribution, the standard deviation should equal about one-sixth of the range.

18. The Population Variance and the Population Standard Deviation

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

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

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

22. 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.

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

24. Estimated Population Variance Computational formula

25. 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. Definitional formula

26. Estimated Population Standard Deviation Computational formula

27. Unbiased Estimators The quantity N - 1 is called the degrees of freedom is an unbiased estimator of is an unbiased estimator of

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. Proportion of Variance Accounted For 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

31. Example Using the following data set, find The range, The sample variance and standard deviation, The estimated population variance and standard deviation 15 14 14 17 15 14 13 14 13 12 10 13 15 11 15 13 14 14

32. Example Range The range is the largest value minus the smallest value.

33. Example Sample Variance

34. Example Sample Standard Deviation

35. Example Estimated Population Variance

36. Example—Estimated Population Standard Deviation