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Chapter 4 SUMMARIZING SCORES WITH MEASURES OF VARIABILITY

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Going Forward Your goals in this chapter are to learn: What is meant by variability What the range indicates What the standard deviation and variance are and how to interpret them How to compute the standard deviation and variance when describing a sample, when describing the population, and when estimating the population

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

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Three Samples

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Three Variations of the Normal Curve

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Measures of Variability Smaller variability indicates – Scores are consistent – Measures of central tendency describe the distribution more accurately – Less distances between the scores Larger variability indicates – Scores are inconsistent – Measures of central tendency describe the distribution less accurately – Greater distances between the scores

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The Range

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The range indicates the distance between the two most extreme scores in a distribution Range = Highest score – Lowest score

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The Sample Variance and Standard Deviation

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Variance and Standard Deviation The variance and standard deviation indicate how much the scores are spread out around the mean.

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

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

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The Standard Deviation The standard deviation indicates something like 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 S X, the more the scores are spread out around the mean, and the wider the distribution

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Normal Distribution and the Standard Deviation

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Approximately 34% of the scores in any normal distribution are between the mean and the score located one standard deviation from the mean.

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The Population Variance and Standard Deviation

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

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

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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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Estimated Population Variance By dividing by N – 1 instead of N, we have an unbiased estimator of the population variance called the estimated population variance.

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Estimated Population Standard Deviation Taking the square root of the estimated population variance results in the estimated population standard deviation.

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Unbiased Estimators The estimated population variance is an unbiased estimator of the population variance The estimated population standard deviation is an unbiased estimator of the population standard deviation

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Uses of 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 and

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Organizational Chart of Descriptive and Inferential Measures of Variability

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New Symbols The Sum of Squared Xs First square each raw score and then sum the squared Xs The Squared Sum of X First sum the raw scores and then square that sum

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Computing Formulas The computing formula for the sample variance is

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Computing Formulas The computing formula for the sample standard deviation is

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Computing Formulas The computing formula for the estimated population variance is

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Computing Formulas The computing formula for the estimated population standard deviation is

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Example Using the following data set, find The range The sample variance and standard deviation The estimated population variance and standard deviation 14 13151115 131012131413 14151714 15

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

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Example Sample Variance

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Example Sample Standard Deviation

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Example Estimated Population Variance

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Example—Estimated Population Standard Deviation

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