Z scores convert data (the x values) to standard deviations, so all data can be expressed as a number of standard deviations from the mean, which has a z-score of zero. To convert a number x to a z-score, find the distance from the mean in standard deviations. The formula which does this is:

The z-score is the standardized value of x.

Hardness measures of a certain steel part are known to be distributed approximately normally, with a mean of 54 and a standard deviation of 0.75, Rockwell scale. What is the standardized value of a part with a hardness of 55.5?

This piece is 2 standard deviations harder than the average.

Note that z has no units; it is a ratio.

#24.5 When x is below the mean, what sign will z have?

A student scores 95 on an English test with a class average of 84 and a standard deviation of 6.4 points, while scoring 80 on a math test with a class average of 70 and a standard deviation of 5.1 points. Is the student better in English or in math? We can use z scores to compare the student's performance on the two tests.

A student's English test score: x=95 The class average and standard deviation for the English test:=84 s = 6.4 z = = 1.72 for English

A student's math test score: x=80 The class average and standard deviation for the math test:=70 s = 5.1 z = = 1.96 for math

The student did better on the math test, in comparison to the rest of the class.

The normal distribution curves below show the student did very well in relation to the rest of the class. About 96% and 98% of the area under each curve is to the left (below) the students standardized score. This portion is shown shaded.

#25 What percent of the class scored above this student in English?

#26 What percent of the class scored above this student in math?

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