Lecture+4+su10+_z-scores+and+standardized+distribution_-1

# Lecture+4+su10+_z-scores+and+standardized+distribution_-1 -...

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02/23/11 Soc 210 Summer 2010 1 Sociology 210 Lecture 4: Variance, Standard Deviation, and Z Scores

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02/23/11 Soc 210 Summer 2010 2 The Variance • The sum of squares considers how far the values X i deviate from the mean The mean of the sum of squares is the variance The variance is also called the mean square (short for mean squared deviation) 1 ) ( variance Sample ) ( variance Population 2 2 2 2 - - = = - = = n X X s N X i i μ σ
02/23/11 Soc 210 Summer 2010 3 A mean is computed by finding a sum and dividing by the number of scores: mean = sum of scores/number of scores The variance for the population can be expressed as To calculate sample variance , we find the sum of squared deviations and divide by the number of scores that are free to vary N SS or scores of number deviations squared of sum 2 = σ df SS 1 - n SS vary to free scores of number deviations squared of sum s 2 = = = The Variance is the Mean Squared Deviation

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02/23/11 Soc 210 Summer 2010 4 Standard Deviation When we square deviation scores to get rid of the plus and minus signs, we also convert the units to squared distances – thus, the variance can no longer be interpreted on the same scale as the variable The variance (mean squared distance) is not the best descriptive measure for variation What we’d like to have is a measure of the standard distance from the mean To correct for having squared the deviations, we can take the square root of the variance, which is called the standard deviation (aka root mean square deviation or root mean square ): It is the typical distance of scores from the mean (the typical (X - μ ) variance deviation standard =
02/23/11 Soc 210 Summer 2010 5 Visual Example of Variance and Standard Deviation 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 Sample 1 Sample 2 n = 8 Mean = 4.5 Variance = 6 Standard Deviation = 2.45 n = 8 Mean = 4.5 Variance = 2.57 Standard Deviation = 1.60

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02/23/11 Soc 210 Summer 2010 6 Bad Statistics Joke of the Day
02/23/11 Soc 210 Summer 2010 7 Standardizing a Distribution The purpose of a standardized score (e.g., z -score) is to identify and describe the exact location of a given score in a distribution Example of test score: if you get a 76 on an exam, you want to locate your score in the distribution to see how you did To understand how well you did, you need to know both the mean and the standard deviation of the distribution Here is an example of how the standard deviation can affect the location of a score within a distribution (holding constant the mean)

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02/23/11 Soc 210 Summer 2010 8 Standardized Variable Z Let X be a variable with mean μ and standard deviation σ > 0. Then the standardized variable Z is defined by
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## This note was uploaded on 02/21/2011 for the course SOCIOLOGY 210 taught by Professor Ybarra during the Summer '10 term at University of Michigan.

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Lecture+4+su10+_z-scores+and+standardized+distribution_-1 -...

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