Z-Score Notes
Normal Distribution
The distribution of heights of adult American
men is approximately Normal with a mean of 69
inches and standard deviation of 2.5 inches.
a. What percent of men are taller than 74
inches?
b. What percent of men are no taller than
71.5 inches?
c. What percent of men are between 64
and 66.5 inches?
d. What percent of men are over 67 inches
tall?
Using z-score to find probability
What happens when the value is not exactly 1, 2, or 3 standard deviations of the mean?
The z-score allows us to identify how many standard deviations from the mean a value is and, using the z-score chart,
find the probability under the curve to the left (less than) that value.
𝑥−µ
𝑧 = σ
µ = 𝑚𝑒𝑎𝑛
σ = 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛
𝑥 = 𝑣𝑎𝑙𝑢𝑒
Example 1:
The Welcher Adult Intelligence Test Scale is composed of a number of subtests. On one subtest, the raw scores have
a mean of 35 and a standard deviation of 6. Find the z-scores for the following:
a. A test score of 30
b. A test score of 52
c. A test score of 41
Example 2:
Three students take equivalent stress tests. Find the following z-scores.
a. Student A: A score of 144 on a test with a mean of 128 and a standard deviation of 34.
b. Student B: A score of 90 on a test with a mean of 86 and a standard deviation of 18.
c. Student C: A score of 18 on a test with a mean of 15 and a standard deviation of 5.
d. Who scored the best on their stress test? Why?
Using a z-score table, find the percent of individuals scoring below the following z-scores:
a. 𝑧 = − 0. 47
b. 𝑧 = 2. 24
c. 𝑧 = 1. 17
d. 𝑧 = − 1. 37
Example 3:
A patient recently diagnosed with Alzheimer’s disease takes a cognitive abilities test and scores a 45. The mean on
this test is 52 and the standard deviation is 5. What percent of the patients have his score and below?
Example 4:
A fifth grader takes a standardized achievement test (mean = 125, std. dev = 15) and scores a 148. What percent of
children made higher than a 148?
Example 5:
Pat and Chris both took a spatial abilities test (mean = 80, st. dev = 8). Pat scores a 76 and Chris scored a 94. What
percent of individuals would score between Pat and Chris?
