AP Statistics · Topic 2.11 · Unit 2

AP Stats 2.11: The Normal Distribution

By Jude Wallis · Published July 28, 2026

Topic 2.11 in the Fall 2026 AP Statistics course covers the normal distribution: a continuous, unimodal, bell-shaped, symmetric curve set by a mean and standard deviation. The empirical rule gives about 68, 95, and 99.7 percent within 1, 2, and 3 standard deviations, and areas give probabilities.

AP Statistics: Unit 2 (topics 2.11). In the Fall 2026 AP Statistics course, the normal distribution is Unit 2 Topic 2.11, aligned to skills 3.C, 3.D, and 4.C.

Describing a normal distribution

Topic 2.11 introduces the most-used continuous model in the course. A continuous random variable can take any value within a specified domain, and every interval within that domain has a probability attached to it.

Many continuous random variables are well modeled by a normal distribution, which is a continuous, unimodal, bell-shaped, and symmetric curve. It is identified by two parameters, the mean $\mu$ and the standard deviation $\sigma$ . A smaller $\sigma$ makes the curve taller and more concentrated around its mean, and a larger $\sigma$ makes it shorter and more spread out.

The standard normal and the empirical rule

A standard normal distribution is a normal distribution with mean $\mu = 0$ and standard deviation $\sigma = 1$ . Converting a value to a z-score, its number of standard deviations from the mean, places it on this common scale, as shown in how to find a z-score.

The empirical rule estimates areas under any normal curve. Approximately 68% of observations lie within 1 standard deviation of the mean, approximately 95% within 2 standard deviations, and approximately 99.7% within 3 standard deviations. It is also called the 68-95-99.7 rule.

Areas, intervals, and relative position

If a random variable is approximately normal, the probability that it falls within an interval equals the area under the normal curve over that interval, and the total area under the curve is 1. You find these areas with technology or with z-scores and a standard normal table.

You can also work backward from a given area to the boundary values, assigning inequalities so the lowest, middle, or highest percent of values map to the correct interval. Percentiles and proportions then let you compare relative positions within one normal distribution or between two different normal distributions.

Sketch, shade, and label

Normal-distribution problems go faster and earn more credit when you sketch the curve, mark the mean, and shade the region you want. Labeling the values 1, 2, and 3 standard deviations from the mean turns the empirical rule into a picture you can read directly.

For an area the empirical rule cannot give, convert the boundary to a z-score and look up the area in a standard normal table, or use technology. To work backward from a target area to a boundary value, find the z-score for that percentile first, then undo the z-score formula to recover the original measurement. Keeping direction straight matters: shading the wrong tail is the error that most often turns a correct z-score into a wrong probability.

Resting heart rates on a normal model

Adult resting heart rates are approximately normal with mean 70 beats per minute and standard deviation 8. Give the empirical-rule interval for the middle 68%, and find the percentile of a rate of 82.

Middle 68% lies within 1 standard deviation: $70 - 8 = 62$ to $70 + 8 = 78$ beats per minute.
Convert 82 to a z-score: $z = (82 - 70) / 8 = 12 / 8 = 1.5$ .
A rate of 82 is 1.5 standard deviations above the mean.
From a standard normal table, the area to the left of $z = 1.5$ is $0.9332$ .

About 68% of adults have resting rates between 62 and 78 beats per minute, and a rate of 82 sits at roughly the 93rd percentile.

Frequently asked questions

What two parameters define a normal distribution?

A normal distribution is identified by its mean and its standard deviation. The mean locates the center of the bell-shaped curve, and the standard deviation sets its spread. A smaller standard deviation makes the curve taller and more concentrated; a larger one makes it shorter and wider.

What is the empirical rule?

For a normal distribution, about 68% of observations fall within 1 standard deviation of the mean, about 95% within 2 standard deviations, and about 99.7% within 3 standard deviations. It is also called the 68-95-99.7 rule and gives quick area estimates without a z-table.