zScores 8 - Standard scores z-scores Example: Dave gets a...

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Standard scores z-scores
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Example: Dave gets a 50 on his Statistics midterm and an 50 on his Calculus midterm. Did he do equally well on these two exams? Big question: How can we compare a person’s score on different variables?
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GRADE 0 20 40 60 80 100 0 5 10 15 In one case, Dave’s exam score is 10 points above the mean In the other case, Dave’s exam score is 10 points below the mean In an important sense, we must interpret Dave’s grade relative to the average performance of the class Statistics Calculus Mean Statistics = 40 Mean Calculus = 60 Example 1
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Both distributions have the same mean (40), but different standard deviations (10 vs. 20) In one case, Dave is performing better than almost 95% of the class. In the other, he is performing better than approximately 68% of the class. Thus, how we evaluate Dave’s performance depends on how much variability there is in the exam scores GRADE 0 20 40 60 80 100 0 5 10 15 20 25 30 Example 2 Calculus Statistics
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Standard Scores We want to express a person’s score with respect to both (a) the mean of the group and (b) the variability of the scores how far a person is from the mean = X - M variability = SD
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Standard (Z) Scores how far a person is from the mean = X - M variability = SD Standard score or ** How far a person is from the mean, in the metric of standard deviation units ** SD M X Z i i ) ( - =
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GRADE 0 20 40 60 80 100 0 5 10 15 Dave in Statistics: (50 - 40)/10 = 1 (one SD above the mean) Dave in Calculus (50 - 60)/10 = -1 (one SD below the mean) Statistics Calculus Mean Statistics = 40 Mean Calculus = 60 Example 1
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An example where the means are identical, but the two sets of scores have different spreads Dave’s Stats Z-score (50-40)/5 = 2 Dave’s Calc Z-score (50-40)/20 = .5 GRADE 0 20 40 60 80 100 0 5 10 15 20 25 30 Calculus Statistics Example 2
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Thee Properties of Standard Scores 1. The mean of a set of z-scores is always zero
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This note was uploaded on 05/26/2011 for the course PSCH 343 taught by Professor Victoriaharmon during the Spring '11 term at Ill. Chicago.

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zScores 8 - Standard scores z-scores Example: Dave gets a...

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