Empirical Rule Calculator

What the empirical rule calculator does

The empirical rule, also called the 68-95-99.7 rule, is a quick way to describe how a bell-shaped set of data spreads around its mean. It says that about 68 percent of values fall within one standard deviation of the mean, about 95 percent within two, and about 99.7 percent within three. This calculator turns your mean and standard deviation into those three ranges.

It gives you a fast, no-fuss sense of where almost all of your data lives. Below is how it works and the one condition it depends on.

How to use it

1. Enter the mean of your data.
2. Enter the standard deviation.
3. Press Calculate for the three ranges, or Reset to clear it.

How the ranges are worked out

Each range is the mean give or take a number of standard deviations. The calculator steps out from the mean one, two, and three standard deviations in each direction:

68 percent: mean minus one SD, to mean plus one SD
95 percent: mean minus two SD, to mean plus two SD
99.7 percent: mean minus three SD, to mean plus three SD

So each range is centred on the mean and simply widens by one more standard deviation at a time. The percentages, 68, 95, and 99.7, are rounded figures for a true bell curve, where the exact shares are about 68.3, 95.4, and 99.7 percent.

When the rule applies

The empirical rule holds for data that is roughly normal, the symmetric bell shape. The closer your data is to that shape, the better the three percentages fit. For data that is close to bell-shaped, the rule is a reliable shortcut.

The catch is that it only works for that shape. If your data is strongly skewed, has heavy tails, or two peaks, the 68-95-99.7 splits no longer hold, and reading them off anyway would mislead. For those cases a measure that does not assume a bell curve, like the quartiles and the interquartile range, describes the spread more honestly. So the first question is always whether the data is bell-shaped enough for the rule to apply.

A worked example

IQ scores follow a bell curve with a mean of 100 and a standard deviation of 15. Feeding those in gives the three ranges at a glance.

About 68 percent of people score between 85 and 115, one standard deviation either side. About 95 percent fall between 70 and 130, and about 99.7 percent between 55 and 145. So a score above 145 or below 55 is genuinely rare, landing outside the range that holds all but three people in a thousand.

Entering your values

Enter the mean and the standard deviation of your data. The calculator returns the three ranges directly. Remember that the rule assumes a roughly bell-shaped distribution, so the ranges are most trustworthy when your data fits that shape, and more of a loose guide when it does not.

Questions people ask

What is the empirical rule?

The 68-95-99.7 rule: for bell-shaped data, about 68 percent of values fall within one standard deviation of the mean, about 95 percent within two, and about 99.7 percent within three.

Why 68, 95, and 99.7?

Those are the shares of a normal curve that fall within one, two, and three standard deviations of the mean. They are rounded from the exact figures of about 68.3, 95.4, and 99.7 percent.

When does the rule not work?

When the data is not bell-shaped, such as strongly skewed data or data with two peaks. Then the percentages no longer hold, and quartile-based measures describe the spread better.

What is it good for?

A quick sense of where almost all the data lies, and for spotting rare values. Anything beyond three standard deviations from the mean is uncommon on a bell curve.

References

A quick note on where the methods here come from. The empirical rule and the share of a normal distribution within one, two, and three standard deviations are set out in the NIST/SEMATECH e-Handbook of Statistical Methods, the US government's public statistics reference. OpenStax Introductory Statistics is a free, widely used textbook covering the empirical rule.

1. NIST/SEMATECH e-Handbook of Statistical Methods (the normal distribution). https://www.itl.nist.gov/div898/handbook/
2. OpenStax, Introductory Statistics (the empirical rule and the normal distribution). https://openstax.org/details/books/introductory-statistics-2e