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Chapter 5: z-Scores - technique using mean M and sd to transform each X…
Chapter 5: z-Scores - technique using mean M and sd to transform each X into a z-score or standard score.
z-Scores for Samples
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NOTE - for Sample, use M and s for mean and SD
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NOTES
value of z-score specifies distance from mean by counting the number of standard deviations between X and μ (# and mean)
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Relationship between z, X, the Mean, and SD
Example 1: In a population with a mean of μ = 65, a score of X = 59 corresponds to z = -2.00. What
is the standard deviation for the population? 65-59=6 points / 2 (-2.0), so one σ = 3 points
Example 2: In a sample with a standard deviation of s = 6, a score of X = 33 corresponds to z = 11.50.
What is the mean for the sample? (1.50)(6) = 9 points, so score located 9 points above the mean. X = 33, so the mean is 33-9 = M = 24.
Example 3: In a population distribution x=54 corresponds to z=+2.00 and X=42 corresponds to z= -1.00. What are the values for the mean and the sd for the population?
Solve for sd: 54-42 = 12 points between; 42 is 1 sd below and 54 is 2 SD above, so 12pts / 3 sd, so σ = 4 points
Solve for Mean: X = 42 is z = -1.00, so 4 points below the mean, so mean is 42+4= μ = 46
Example 4: In a sample distribution, a score of X = 64 corresponds to z = 0.50 and a score of X = 72
has a z-score of z = 1.50. What are the values for the sample mean and sd?
Solve for sd: if M is 60 and 64 is .5 above, 1 above will add an additional 4 points, σ = 8
Solve for Sample Mean: 72-64 = 8 points between; 64 is .50 above and 72 is 1.5 above. 8/2=4 points, so 64-4 is M=60
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