Empirical rule (68-95-99.7) and normal distribution

What the empirical rule says

The empirical rule, also called the 68-95-99.7 rule, describes how values spread out in a normal distribution. For data that is approximately normal:

• About 68% of values fall within 1 standard deviation of the mean.
• About 95% of values fall within 2 standard deviations of the mean.
• About 99.7% of values fall within 3 standard deviations of the mean.

The mean is written $\mu$ (the center of the distribution) and the standard deviation is written $\sigma$ (a measure of how spread out the values are). So the three intervals are $\mu \pm \sigma$, $\mu \pm 2\sigma$, and $\mu \pm 3\sigma$.

The word 'about' matters. These are rounded from the exact normal-curve percentages of 68.27%, 95.45%, and 99.73%, and real data only approximates them.

The normal distribution and its bell shape

A normal distribution is a symmetric, bell-shaped curve. Picture a smooth mound that is highest at the center and tails off evenly on both sides. The peak sits at the mean $\mu$, and the left half is a mirror image of the right half. The College Board describes the normal curve as continuous, unimodal, bell-shaped, and symmetric.

Because the curve is symmetric, half of the values (50%) sit below the mean and half sit above it. Standard deviations mark off equal steps as you move away from the center. If you stand at the mean and walk out one $\sigma$ in each direction, you capture the middle 68% of the area under the curve. Walk out to $2\sigma$ and you capture 95%; walk out to $3\sigma$ and you capture 99.7%.

The size of $\sigma$ controls the width, not the shape. A small standard deviation makes the curve tall and narrow with values clustered near the mean, while a large standard deviation makes it short and wide. Either way the 68-95-99.7 percentages stay the same, because they are measured in standard deviations rather than in raw units.

Almost all of a normal distribution (99.7%) sits within 3 standard deviations of the mean, so the tails beyond that point are very thin.

How to sketch the curve and label it

Drawing the curve makes empirical-rule problems much easier. Here is the sketch in words.

Start with a horizontal axis. Put a tall tick in the middle and label it with the mean $\mu$. Draw six evenly spaced ticks around it, three on each side, and label them $\mu - 3\sigma$, $\mu - 2\sigma$, $\mu - \sigma$, then $\mu + \sigma$, $\mu + 2\sigma$, $\mu + 3\sigma$. For a distribution with $\mu = 150$ and $\sigma = 20$, those ticks read 90, 110, 130, 150, 170, 190, 210.

Now draw a smooth bell over the axis: highest above the mean, dropping symmetrically, and nearly touching the axis by the $3\sigma$ ticks. Write 34% in each band next to the mean, 13.5% in each band between 1 SD and 2 SD, and 2.35% in each band between 2 SD and 3 SD. To find the area for any question, shade the matching region and add the labeled percentages.

The normal distribution percentages in each region

You can split the rule into smaller pieces. Each standard-deviation band holds a known share of the area:

• From the mean out to 1 SD, on one side: 34% (half of 68%).
• Between 1 SD and 2 SD, on one side: 13.5%.
• Between 2 SD and 3 SD, on one side: 2.35%.
• Beyond 3 SD, on one side: 0.15%.

These come from the symmetry of the curve. For example, 95% lies within 2 SD and 68% lies within 1 SD, so the two bands between 1 SD and 2 SD together hold $95\% - 68\% = 27\%$, which is 13.5% on each side.

This breakdown answers 'what percent is above' or 'what percent is below' a whole-SD cutoff. The percent above the point 1 SD above the mean is $13.5\% + 2.35\% + 0.15\% = 16\%$, which also equals $(100\% - 68\%) / 2$.

When not to use the empirical rule

The empirical rule only works when the distribution is approximately normal, meaning roughly symmetric and single-peaked. If the data is skewed or has strong outliers, the 68-95-99.7 percentages do not apply.

Skewed data is the most common trap. Incomes and house prices often have a long right tail, so the interval $\mu \pm \sigma$ can stretch past values that are impossible, and the true percentage within 1 SD will not be 68%. Applying the rule to skewed data gives wrong answers.

Before using the rule, check the shape. A dotplot, histogram, or boxplot that looks roughly symmetric and single-peaked is a good sign. If you see two peaks, a long tail, or extreme outliers, use the actual distribution instead of the rule.

For values that are not a whole number of SDs, use z-scores

The empirical rule gives clean answers only at 1, 2, and 3 standard deviations. For any other cutoff you need a z-score, which counts how many standard deviations a value $x$ sits from the mean:

A z-score of 0.75 means a value sits 0.75 standard deviations above the mean, which the empirical rule cannot handle directly. Once you have the z-score, a z-table or the normal distribution calculator gives the area, and that area is the percentage of values below your cutoff.

See how to find a z-score for the full method. The second worked example below shows the empirical rule and the z-score approach side by side.

Where this fits in AP Statistics

The normal distribution is Unit 2, topic 2.11 in the Fall 2026 AP Statistics course (Probability, Random Variables, and Probability Distributions). You use the 68-95-99.7 rule for quick estimates at whole standard deviations and z-scores with the provided table for precise probabilities.

On the exam, a graphing calculator with statistical capabilities is expected, and formulas and tables are provided for both sections. For the official course framework, see AP Central.