Empirical Rule Calculator
Apply the 68–95–99.7 empirical rule to find where most values in a normal distribution fall.
About the Empirical Rule Calculator
Enter the mean and standard deviation. The calculator gives the ranges that hold about 68%, 95% and 99.7% of values, the exact percentages, and a bell curve with the bands shaded.
For data that follow a normal (bell-shaped) distribution, about 68% of values lie within one standard deviation of the mean, 95% within two and 99.7% within three. Anything beyond 3σ happens about 3 times in 1,000.
The formula
P(μ − kσ < X < μ + kσ) = Φ(k) − Φ(−k)
Worked example
IQ scores: mean 100, standard deviation 15
- 68% score 85–115
- 95% score 70–130
- 99.7% score 55–145
Questions people ask
Does it apply to any data?
Only to roughly normal data; for any distribution, Chebyshev's theorem gives weaker bounds.
Is it exactly 95%?
95.45% for 2σ; exactly 95% is 1.96σ.
How do I find the share above a value?
Use the Z-Score or Normal Distribution Calculator for any cut-off.