Empirical Rule Calculator (68-95-99.7)

The 68%, 95% and 99.7% ranges for a normal distribution, plus the exact share between any two values.

About 68% of a normal distribution lies within μ ± 1σ, 95% within μ ± 2σ and 99.7% within μ ± 3σ. For IQ scores (mean 100, SD 15) that is 85–115, 70–130 and 55–145.

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How to use it

Enter the mean and standard deviation.
Read the 68%, 95% and 99.7% ranges.
For any other range, enter its limits for the exact percentage.

What you enter

Mean
Standard deviation

What you get

1σ, 2σ and 3σ ranges
Bell curve
Exact share between two values

Formula

μ ± 1σ ≈ 68.27% | μ ± 2σ ≈ 95.45% | μ ± 3σ ≈ 99.73%

Worked example: IQ scores

Input: Mean 100, SD 15

Result: 68%: 85–115; 95%: 70–130; 99.7%: 55–145

An IQ of 130 or above is in the top 2.5%.

Questions

What is the empirical rule?

For data that follow a normal (bell-shaped) distribution, about 68% of values fall within one standard deviation of the mean, 95% within two and 99.7% within three.

What are the exact percentages?

68.27%, 95.45% and 99.73%. The exact 95% range is μ ± 1.96σ.

How do I find the percentage above or below a value with the empirical rule?

Use symmetry: 50% lies above the mean, so above μ + 1σ is (100 − 68) ÷ 2 = 16%, and above μ + 2σ is 2.5%.

When can I not use the empirical rule?

When the data are skewed or have heavy tails. Chebyshev's theorem gives guaranteed minimums for any distribution: at least 75% within 2σ.

Limitations

Only for approximately normal data.

More Statistics

All free tools · By Dr. Zubair Khalid, DVM, MS, PhD.