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AP Statistics
z-scores and transforming data
Notes 2.1b
One way to measure location is to calculate the standardized score or z-score.
z
x  mean
xx

standard deviation
sx
Example
The following are test scores on Mr. Pryor’s first Statistics Test.
79, 81, 80, 77, 73, 83, 74, 93, 78, 80, 75, 97, 73, 77, 63, 86, 90, 79, 85, 83, 89, 84, 82, 97, 92
A. Calculate the mean, standard deviation and 5-number summary for the test grades.
B. Calculate and interpret the z-score for the score 93?
C. Calculate and interpret the z-score for the score73?
Transforming Data
Effect of adding (or subtracting) a constant
Adding the same positive number a to (subtracting a from) each observation
 adds a to (subtracts a from) measures of center and location (mean, median, quartiles, percentiles), but
 does not change the shape of the distribution or measures of spread (range, IQR, standard deviation)
Effect of multiplying (or dividing) by a constant
Multiplying (or dividing) each observation by the same positive number b
 multiplies (divides) measures of center and location (mean, median, quartiles, percentiles) by b,
 multiplies (divides) measures of spread (range, IQR, standard deviation) by b, but
 does not change the shape of the distribution
Example
D. If Mr. Pryor adds 5 to each student’s score, calculate the mean, standard deviation and 5-number summary for
the transformed test grades.
E. If Mr. Pryor multiplied each student’s score by 1.1, calculate the mean, standard deviation and 5-number
summary for the test grades.
F. If Mr. Pryor multiplied each student’s score by .9 then added 10, calculate the mean, standard deviation and 5number summary for the test grades.
G. Suppose Mr. Pryor standardized the student’s scores, what would be the shape, center, and spread of the new
distribution?