Empirical Rule Calculator (68-95-99.7)
Find the ranges containing about 68%, 95% and 99.7% of normally distributed values from the mean and standard deviation.
Change the values and press Calculate to work out your own figures.
Enter the mean and standard deviation.
Show the working
- μ ± 1σ, μ ± 2σ, μ ± 3σ
How to use it
Enter the mean and standard deviation.
Probabilities can be typed as decimals, fractions or percentages. Show the working gives the formula with your numbers in it.
Key facts
μ ± σ ≈ 68%; μ ± 2σ ≈ 95%; μ ± 3σ ≈ 99.7%
A rule of thumb
The rule holds for normal distributions; for other shapes Chebyshev's inequality gives looser guarantees (at least 75% within 2 standard deviations).
Questions
What range holds 95% of IQ scores?
70 to 130 (mean 100, SD 15).
Are the percentages exact?
They round 68.27%, 95.45% and 99.73%.
Formulas
μ ± σ ≈ 68%; μ ± 2σ ≈ 95%; μ ± 3σ ≈ 99.7%
Sources
Limitations
- Probabilities can be typed as decimals (0.25), fractions (1/4) or percentages (25%).