Z-Scores are measurements of how far from the center (mean) a data value falls.

Ex: A man who stands 71.5 inches tall is 1 standard deviation ABOVE the mean.(z-score = 1)

Ex: A man who stands 64 inches tall is 2 standard deviations BELOW the mean. (z-score = -2)

Adult Male Heights

Based upon the Empirical Rule, we know the approximate percentage of data that falls between certain standard deviations on a normal distribution curve.

Example:If a class of test scores has a mean of 65 and standard deviation of 9, then what percent of the students would have a grade below 56?

Standardized Scores (aka z-scores)

Z-score represents the exact number of standard deviations a value, x, is

from the mean.

s

xxz

observation (value)

mean

standard deviation

Example: (test score problem) What would be the z-score for a student that received a 70 on the test?

Example: The mean speed of vehicles along a particular section of the highway is 67mph with a standard deviation of 4mph. What is the z-score for a car that is traveling at 72 mph?

What is the z-score for a car that is traveling 60mph?

|----------|----------|----------|----------|----------|----------|(z =) -3 -2 -1 0 1 2 3 55 59 63 67 71 75 79

Mark the z-scores on the number line below

Z-scores are also called standardized scores. To calculate any z-score:

American Adult Males: mean of 69 inches and standard deviation of 2.5 inches.

1) What is the standardized score for a male with a height of 72 inches?

2) What is the standardized score for a male with a height of 62 inches?

To find the proportion or probability that a certain interval is possible, we use the z-score table.

What percentage of the data will have a z-score of less than 1.15? P(z < 1.15)

The z-score table always tells the

proportion to the LEFT of that z-score value.

Other Examples:

1)Find P(z < 0.22)

2) Find P(z > 0.22)

3) Find P(0.22 < z < 1.15)