AP STATISTICS · THE NORMAL DISTRIBUTION

Normal Distributions & z-Scores

Standardize any value to see how unusual it is.

15 min read 4 practice questions

A normal distribution is the familiar symmetric bell curve, described entirely by its mean

\mu

and standard deviation

\sigma

. A z-score tells you how many standard deviations a value is from the mean.

z = \frac{x - \mu}{\sigma}

The empirical rule: about

68\%

of values fall within

1\sigma

of the mean,

95\%

within

2\sigma

, and

99.7\%

within

3\sigma

Normal area explorerInteractive

Set a mean and standard deviation, then drag the z-bounds to shade an area under the curve.

\mu

100

\sigma

Lower z-1.00Upper z1.00

Values85.0 to 115.0

Shaded area68.27%

Worked example · Comparing scores

SAT math scores are

N(530, 110)

. A student scores

750

. What's their z-score?

1
$z = \frac{750 - 530}{110} = \frac{220}{110} = 2$ .
2
The score is $2$ standard deviations above the mean — about the 97.5th percentile.

CHECK YOUR UNDERSTANDING

Practice

Answer each question, then check.

Heights are normal with

\mu = 64

in and

\sigma = 3

in. What is the z-score of

70

in?

About what percent of a normal distribution lies between

z = -1

and

z = 1

Using the empirical rule, what percent of values lie above

z = 2

? (Enter a number like 2.5)

A test has

\mu = 80

and

\sigma = 5

. What score has

z = -1.4

Numeric answers accept decimals or fractions like 3/4.