z - Statistic


By the end of this lesson, you should be able to:

1. Determine the z-score given a mean and standard deviation, and vice versa. 

z - Statistic

The z statistic has a normal distribution with a mean of zero and a standard deviation of one. Therefore, if we know the mean and standard deviation for a population, we can identify a z-score given the raw score, or we can identify the raw score given the z-score.

Example 1:
A parts manufacturer produces a framis. The framis has a mean weight of 8 pounds, with a standard deviation of 2 ounces. You measure one particular framis and find it weighs 8 pounds, three ounces. What is its z-score?
Answer: The framis you measured weighed the mean, plus 1.5 standard deviations. Therefore, the z score is 1.5.

Example 2:
The boss then asks you to reject all but the middle 90% of the framises. Where do you establish the minimum and maximum acceptable weights? Use a table that shows the area under the normal curve, such as the one at:
Answer: You consult a table of areas under a normal curve, and find that 90% of the curve is contained within about 1.65 standard deviations around the mean. This is because half of 90% is 45%, and the value corresponding to .450 seems near to 1.65. Since each standard deviation is 2 ounces, the mean minus 1.65 standard deviations is 7 pounds, 12.7 ounces, and the mean plus 1.65 standard deviations is 8 pounds, 3.3 ounces.

Test your knowledge:
3. Using the table under the normal curve, calculate the percentile score of a student who scores 490 on a test that has a mean of 460 and a standard deviation of 40.
Highlight the answer => This corresponds to the mean plus .75 standard deviations. On the table, .75 standard deviations corresponds to 0.273. When added to the 50th percentile, this results in the seventy-seventh percentile.

4. A T-test (that's a capital T) has a mean of 50 and a standard deviation of 10.

A. Convert the following z-score to a T-score: -1.4
Highlight the answer => 36

B. Convert the following T-score to a z-score: 74
Highlight the answer => 2.4

All information is subject to change without notification.
© Jim Flowers
Department of Technology, Ball State University