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

2. New Statistical Notation
indicates the sum of squared Xs.
indicates the squared sum of X.
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3. Measures of Variability
Measures of variability describe the extent to which scores in a distribution differ from each other.
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4. A Chart Showing the Distance Between the Locations of Scores in Three Distributions
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5. Three Variations of the Normal Curve
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6. The Range, Variance, and Standard Deviation
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7. Range = highest score – lowest score
The Range
The range indicates the distance between the two most extreme scores in a distribution
Range = highest score – lowest score
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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
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9. The Sample Variance and the Sample Standard Deviation
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10. Sample Variance
The sample variance is the average of the squared deviations of scores around the sample mean
Definitional formula
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11. Computational formula
Sample Variance
Computational formula
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12. Sample Standard Deviation
The sample standard deviation is the square root of the sample variance
Definitional formula
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13. Sample Standard Deviation
Computational formula
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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 SX, the more the scores are spread out around the mean, and the wider the distribution
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15. Normal Distribution and the Standard Deviation
[Insert new Figure 5.2 here.]
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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.
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17. Standard Deviation and Range
For any roughly normal distribution, the standard deviation should equal about one-sixth of the range.
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18. The Population Variance and the Population Standard Deviation
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19. Population Variance
The population variance is the true or actual variance of the population of scores.
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20. Population Standard Deviation
The population standard deviation is the true or actual standard deviation of the population of scores.
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21. The Estimated Population Variance and the Estimated Population Standard Deviation
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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.
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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
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24. Estimated Population Variance
Computational formula
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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
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26. Estimated Population Standard Deviation
Computational formula
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27. is an unbiased estimator of
Unbiased Estimators
is an unbiased estimator of
The quantity N - 1 is called the degrees of freedom
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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.
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29. Organizational Chart of Descriptive and Inferential Measures of Variability
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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
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31. Example Using the following data set, find The range,
The sample variance and standard deviation,
The estimated population variance and standard deviation 14 13 15 11 10 12 17
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32. The range is the largest value minus the smallest value.
Example Range
The range is the largest value minus the smallest value.
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33. Example Sample Variance
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34. Example Sample Standard Deviation
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35. Example Estimated Population Variance
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36. Example—Estimated Population Standard Deviation
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