Copyright © Texas Education Agency, All rights reserved. Z-Score Statistics & Risk Management 1 Copyright © Texas Education Agency, All rights reserved.

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Copyright © Texas Education Agency, All rights reserved. Standardization A “Standard” Normal Distribution has a Mean of ZERO and a Standard Deviation of 1.0. A “Standard” Normal Distribution has a Mean of ZERO and a Standard Deviation of

Copyright © Texas Education Agency, All rights reserved. Visualize the Standard Normal Distribution 4

Copyright © Texas Education Agency, All rights reserved. The Ζ-Score Is based upon Standard Deviation divisions. Is based upon Standard Deviation divisions. Gives you a standardized way to measure the distance left(-) or right(+) of the Mean value. Gives you a standardized way to measure the distance left(-) or right(+) of the Mean value. Works regardless of the value of the mean, because we make the mean ZERO and the Standard Deviation 1.0 Works regardless of the value of the mean, because we make the mean ZERO and the Standard Deviation 1.0 Related PERCENTILES Related PERCENTILES Related T-SCORES Related T-SCORES 5

Copyright © Texas Education Agency, All rights reserved. Formula for a Ζ-Score Mean of the PopulationSample Score Population Standard Deviation z-Score 6

Copyright © Texas Education Agency, All rights reserved. Application We have a 100 test scores. The Mean is 79 and the standard deviation is 4. We have a 100 test scores. The Mean is 79 and the standard deviation is 4. If you earned a score of 85, where do you sit within the distribution? If you earned a score of 85, where do you sit within the distribution? z = ( ) / 4 = 6 / 4 = 1.5 You are sitting around the 94th percentile…very good. You are sitting around the 94th percentile…very good. 7

Copyright © Texas Education Agency, All rights reserved. Formula for a Z-Score Mean of the Population Sample Score Population Standard Deviation z-Score 8

Copyright © Texas Education Agency, All rights reserved. Application We have a 100 test scores. The Mean is 79 and the standard deviation is 4. We have a 100 test scores. The Mean is 79 and the standard deviation is 4. What score do you need to earn a 90% to get that “A”? What score do you need to earn a 90% to get that “A”? X = (1.2 x 4) + 79 = 83.8 You need to earn a score of 83.8 or above for the “A”. You need to earn a score of 83.8 or above for the “A”. 9