Empirical Rule Calculator
Enter a mean and standard deviation — or paste your data — and get the intervals that hold about 68%, 95% and 99.7% of values, drawn on a shaded bell curve and set beside the exact normal shares and Chebyshev's guaranteed minimums. Paste data and it also counts how many values really fall inside. Free, no sign-up.
Gives the expected count inside each interval.
Adds a row for mean ± k SD, with the exact normal share and Chebyshev's bound.
Enter a mean and a standard deviation above zero — or switch to a list of data — to see the 68–95–99.7 intervals.
How to Use the Empirical Rule Calculator
- 1. Choose your starting point. Enter a mean and a standard deviation, or switch to “A list of data” and paste your numbers — the calculator then finds the mean and sample standard deviation for you.
- 2. Add the optional extras. A number of values gives the expected count inside each interval; your own number of SDs adds a row for any distance you like, such as 2.5.
- 3. Read the intervals. The three cards give the ranges holding about 68%, 95% and 99.7% of values, and the shaded curve shows them nested.
- 4. Compare. The table sets the rule beside the exact normal shares and Chebyshev's minimum, which holds for any distribution. With pasted data, it also shows how many of your values actually fall in each interval.
The 68–95–99.7 Rule
For data that follow a normal distribution, about 68% of values lie within one standard deviation of the mean, about 95% within two, and about 99.7% within three. In symbols, with mean μ and standard deviation σ, the intervals are μ ± σ, μ ± 2σ and μ ± 3σ. It is also called the three-sigma rule.
| Interval | Rule says | Exact normal share | Each tail |
|---|---|---|---|
| μ ± 1σ | 68% | 68.27% | 15.87% |
| μ ± 2σ | 95% | 95.45% | 2.28% |
| μ ± 3σ | 99.7% | 99.73% | 0.13% |
The exact shares are 2Φ(k) − 1, where Φ is the standard normal cumulative distribution. “Each tail” is half of what is left over.
The rule is a rounded summary, so use the exact figures when precision matters — a z of 2 leaves 2.28% above it, not the 2.5% the rule implies. Because the curve is symmetric, the rule also splits in half: about 34% of values lie between the mean and one standard deviation above it, about 47.5% between the mean and two, and about 49.85% between the mean and three.
When the Data Aren't Normal: Chebyshev's Inequality
The empirical rule needs a bell-shaped, roughly symmetric distribution. Chebyshev's inequality needs nothing but a finite variance: at least 1 − 1 ÷ k² of the values lie within k standard deviations of the mean, for any distribution. That is only informative beyond one standard deviation — it guarantees at least 75% within 2σ and at least 88.89% within 3σ. The price of generality is that it is a floor, not an estimate; for normal data the true shares (95.45% and 99.73%) are much higher.
Worked Examples
Click “Try an example”: exam marks with a mean of 72 and a standard deviation of 8, from 1,000 students. About 68% of marks lie between 72 − 8 = 64 and 72 + 8 = 80; about 95% between 56 and 88; and about 99.7% between 48 and 96. By the exact normal shares, you would expect 682.7, 954.5 and 997.3 of the 1,000 inside those intervals. With the optional k of 2.5, the interval is 52 to 92, which holds 98.76% of a normal distribution and at least 84% of any distribution.
Now “Try a data example”, 25 marks from 52 to 95. Their mean is 72.08 and their sample standard deviation 9.183, so the one-SD interval is 62.897 to 81.263. Of the 25 marks, 19 (76%) lie inside, 23 (92%) lie within two standard deviations and all 25 within three. That is a little above 68% at one SD and a little below 95% at two — the kind of gap a sample this small produces by chance, so there is no sign of a real departure from normality.
Checking Whether the Rule Applies
Look at the histogram first: a single hump, roughly symmetric, no long tail. Then compare the shares in your own data with the table, as above. A big excess inside one SD with too many beyond three signals heavy tails; a lopsided pattern signals skew. For a formal check, use the Shapiro-Wilk calculator or look at skewness and kurtosis.
Frequently Asked Questions
- What is the empirical rule in statistics?
- It says that in a normal distribution about 68% of values fall within one standard deviation of the mean, about 95% within two and about 99.7% within three. It is a quick way to judge how unusual a value is without a z-table, and it is also called the 68–95–99.7 rule or the three-sigma rule.
- How do you use the empirical rule?
- Find the mean and standard deviation, then add and subtract one, two and three standard deviations from the mean. For a mean of 72 and an SD of 8, about 68% of values lie between 64 and 80 and about 95% between 56 and 88. To find a percentage between two values, convert them to distances from the mean in SDs and add the matching pieces.
- What percentage lies between the mean and one standard deviation?
- About 34%, half of the 68% within one standard deviation, because the normal curve is symmetric. The exact figure is 34.13%. Between one and two standard deviations above the mean lies about 13.5% (13.59% exactly), and between two and three about 2.14%.
- What is the difference between the empirical rule and Chebyshev's theorem?
- The empirical rule gives approximate percentages but only for bell-shaped, symmetric data. Chebyshev's theorem gives a guaranteed minimum for any distribution, 1 − 1 ÷ k², which is much lower: 75% within two standard deviations instead of about 95%. Use the rule when data look normal and Chebyshev when you know nothing about the shape.
- Does the empirical rule work for non-normal data?
- Only roughly, and not for skewed or heavy-tailed data. In a skewed distribution the two tails are unequal, so the rule's symmetric splits fail, and heavy tails put more than 0.3% of values beyond three standard deviations. Check the shape first, or fall back on Chebyshev's guarantee.
- Why is it 99.7% and not 100%?
- The normal curve never touches zero, so some values always lie beyond three standard deviations — about 0.27% in total, or 0.13% in each tail. That is why the three-sigma rule treats a point beyond three SDs as unusual, though in a large dataset a few are expected.