What the empirical rule says
For a normal (bell-shaped) distribution, the empirical rule (or 68-95-99.7 rule) states that about 68% of values fall within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3. With a mean of 100 and SD of 15, 95% of values lie between 70 and 130.
The three percentages are the areas under the standard normal curve between z-scores of ±1, ±2 and ±3, rounded for memorability. To find how many standard deviations a particular value sits from the mean, use the z-score calculator.
How to use this calculator
Enter the mean and standard deviation. The result shows the three ranges. It only applies to roughly normal data — skewed distributions don't follow these percentages. Find SD with the standard deviation calculator.
The standard deviation must be positive; a value of zero would collapse all three ranges to the mean itself. Mean and SD must share the same unit, and the returned bounds carry that same unit. If your SD comes from a sample rather than a whole population, the ranges are estimates and become less reliable with small samples.
Reading the ranges
The middle range is the one most often quoted: mean ± 2 SD covers roughly 95% of values, which is why IQ scores of 70 to 130 are described as the normal range for a test with mean 100 and SD 15. Anything beyond mean ± 3 SD is genuinely rare, occurring about three times in a thousand observations.
Because the rule is symmetric, half of the excluded values fall on each side: outside 2 SD, roughly 2.5% of the data lies above 130 and 2.5% below 70. For a formal interval around a sample mean rather than a rule of thumb, use the confidence interval calculator.
Frequently asked questions
- What is the 68-95-99.7 rule?
- In a normal distribution, about 68% of data is within 1 SD of the mean, 95% within 2 SD, and 99.7% within 3 SD.
- What range holds 95% of the data?
- The mean plus or minus 2 standard deviations. For mean 100 and SD 15, that's 70 to 130.
- Does the empirical rule always apply?
- Only to approximately normal, bell-shaped distributions. Skewed data doesn't follow it.
- What falls outside 3 standard deviations?
- About 0.3% of values — these are rare outliers in a normal distribution.
- When should I use Chebyshev's inequality instead?
- Use Chebyshev's inequality when your data is clearly not bell-shaped. It works for any distribution but gives weaker guarantees — at least 75% within 2 SD rather than about 95% — so the empirical rule is preferable whenever normality is a reasonable assumption.