Empirical rule calculator: 68-95-99.7 intervals

What the empirical rule tells you

The empirical rule describes how values spread out in an approximately normal distribution, the symmetric bell shape. For a distribution that is roughly normal, about 68% of the data falls within 1 standard deviation of the mean, about 95% falls within 2 standard deviations, and about 99.7% falls within 3. Because the curve is symmetric, each interval is centered on the mean, so half of each percentage sits on each side.

The rule is also called the 68-95-99.7 rule after those three percentages. It gives you a fast read on where most values land without any table lookup, as long as the boundary you care about sits exactly 1, 2, or 3 standard deviations from the mean. Almost everything in a normal distribution, about 99.7% of it, lands inside 3 standard deviations, so values past that range are rare.

The three intervals

Each interval runs from the mean minus a whole number of standard deviations to the mean plus the same number:

Here $\mu$ (mu) is the mean, $\sigma$ (sigma) is the standard deviation, and $k$ is 1, 2, or 3. Setting $k = 1$ captures the middle 68%, $k = 2$ captures 95%, and $k = 3$ captures 99.7%.

You can also break the rule into slices to answer tail and between questions. Moving out from the mean, each side holds about 34% between 0 and 1 SD, 13.5% between 1 and 2 SD, 2.35% between 2 and 3 SD, and 0.15% beyond 3 SD. Those four slices add to 50%, one half of the curve.

How to use the calculator

Enter the mean and the standard deviation of your distribution, and the calculator returns all three intervals at once. The default values, a mean of 100 and a standard deviation of 15, match the common scale used for IQ scores, so you can see the rule in action right away. There is no need to work out each interval by hand, because the tool applies the mean plus or minus 1, 2, and 3 standard deviations for you.

With those defaults the tool reports 85 to 115 for the 68% interval, 70 to 130 for the 95% interval, and 55 to 145 for the 99.7% interval. Change either input and every interval updates, since each one is just the mean plus or minus a multiple of the standard deviation. Read the three ranges together as a picture of where the bulk of a normal distribution sits.

When not to use the empirical rule

The empirical rule only gives clean percentages at exactly 1, 2, and 3 standard deviations. If your boundary sits at 1.4 standard deviations, or at a raw value that is not a whole number of standard deviations from the mean, the rule cannot answer the question. Convert the boundary to a z-score and read the area from a z-table, or use the normal distribution calculator for any cutoff.

The rule also assumes the distribution is approximately normal, meaning roughly symmetric and bell-shaped. For strongly skewed data or data with heavy outliers, the 68-95-99.7 percentages can be well off, so check the shape with a histogram first. The guide on the empirical rule walks through the reasoning with more examples.