Empirical rule calculator
Enter a mean and standard deviation: where 68%, 95% and 99.7% of a bell-shaped data set lie, and what Chebyshev guarantees for any shape.
What is the empirical rule (68–95–99.7 rule)?
For data that follow a bell-shaped (normal) distribution, about 68% of the values lie within 1 standard deviation of the mean, about 95% within 2 standard deviations and about 99.7% within 3. If the mean is 50 and the standard deviation is 6, those intervals are 44 to 56, 38 to 62 and 32 to 68 (OpenStax Introductory Statistics 2e, section 6.1).
What are the exact percentages behind the rule?
For a normal distribution the share within k standard deviations is 2Φ(k) − 1: 68.27% for k = 1, 95.45% for k = 2 and 99.73% for k = 3 (0.682689, 0.954500 and 0.997300). Further out, 4 standard deviations cover 99.9937%, 5 cover 99.99994% and 6 cover 99.9999998%. So about 1 value in 370 falls outside 3 standard deviations. NIST’s e-Handbook lists 68.26% and 95.44%, truncating instead of rounding.
How do I find the share between any two values?
Convert both ends to z-scores, z = (x − mean) ÷ SD, and subtract the normal probabilities: share = Φ(z₂) − Φ(z₁). With a mean of 50 and a standard deviation of 6, the range 40 to 65 has z-scores of −1.667 and 2.5, so about 94.6% of a normal distribution falls inside it. The Range mode does this for you and also shows what lies below and above.
What is Chebyshev’s inequality?
For any distribution, at least 1 − 1/k² of the values lie within k standard deviations of the mean, for k above 1: at least 75% within 2 standard deviations, 88.89% within 3 and 93.75% within 4 (NIST/SEMATECH e-Handbook, section 7.2.6.1). OpenStax rounds these to 75%, 89% and, at 4.5 standard deviations, 95% (section 2.7). It is a guarantee, so it is much weaker than the empirical rule: a normal distribution has 95.45% within just 2.
How many standard deviations do I need to cover a given share?
Chebyshev gives k = 1 ÷ √(1 − share) for any data, and a normal distribution needs k = Φ⁻¹((1 + share) ÷ 2). For 90% that is 3.162 versus 1.645 standard deviations; for 95%, 4.472 versus 1.960; for 99%, 10 versus 2.576. Use the Chebyshev mode and choose “share of the data” to see your own numbers.
When can I use the empirical rule?
Only when the data are roughly bell-shaped and symmetric. For skewed data or an unknown shape, use Chebyshev’s inequality, which holds for every distribution (OpenStax, section 2.7). To find the mean and standard deviation of your own numbers, use the standard deviation calculator; for a single value’s position, the z-score calculator.
Sources: OpenStax, Introductory Statistics 2e, section 6.1 “The Standard Normal Distribution” and section 2.7 “Measures of the Spread of the Data”; NIST/SEMATECH e-Handbook of Statistical Methods, section 7.2.6.1 “Approximate intervals that contain most of the population values”; Wikipedia, “68–95–99.7 rule” (exact shares). Related: normal distribution calculator, z-score calculator and standard deviation calculator.