Empirical Rule Calculator (68-95-99.7)
Turn a mean and standard deviation into the 68%, 95% and 99.7% intervals.
Enter your values
Enter the mean (μ) you are working with.
Enter the standard deviation (σ) you are working with.
Enter the optional value to score (x) you are working with.
Results update instantly as you type — no submit needed. Your values are remembered on this device, and the shareable link reopens the calculator with exactly these numbers.
- 68% of values fall between
- 85 and 115
- 95% of values fall between
- 70 and 130
- 99.7% of values fall between
- 55 and 145
| Interval | Range | Approx. share |
|---|---|---|
| μ ± 1σ | 85 to 115 | 68.27% |
| μ ± 2σ | 70 to 130 | 95.45% |
| μ ± 3σ | 55 to 145 | 99.73% |
Quick answer
The empirical rule says that for a normal distribution about 68% of values fall within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3.
μ ± 1σ ≈ 68%, μ ± 2σ ≈ 95%, μ ± 3σ ≈ 99.7%
At a glance
| What it does | Turn a mean and standard deviation into the 68%, 95% and 99.7% intervals. |
|---|---|
| Category | Statistics |
| Inputs needed | Mean (μ), Standard deviation (σ), Optional value to score (x) |
| Formula | μ ± 1σ ≈ 68%, μ ± 2σ ≈ 95%, μ ± 3σ ≈ 99.7% |
| Cost | Free — no sign-up, no download |
How to use the Empirical Rule Calculator
- 1Enter the mean and standard deviation of your normally distributed data.
- 2Optionally add a single value to get its z-score and percentile.
- 3Read the three empirical-rule intervals.
Inputs explained
- Mean (μ)
- Enter the mean (μ) you are working with.
- Standard deviation (σ)
- Enter the standard deviation (σ) you are working with.
- Optional value to score (x)
- Enter the optional value to score (x) you are working with.
Frequently asked questions
What is the 68-95-99.7 rule?
It is a shortcut for normal distributions: roughly 68% of observations lie within one standard deviation of the mean, 95% within two, and 99.7% within three.
When does the empirical rule not apply?
It assumes an approximately bell-shaped distribution. For skewed or heavy-tailed data use Chebyshev's inequality or the actual distribution instead.
What do I need to enter into the Empirical Rule Calculator?
Just 3 values: mean (μ), standard deviation (σ), optional value to score (x). Every field starts with a realistic example, so you can change one number at a time and watch the result update instantly.
How does the Empirical Rule Calculator work out the answer?
The empirical rule says that for a normal distribution about 68% of values fall within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3. It applies the formula μ ± 1σ ≈ 68%, μ ± 2σ ≈ 95%, μ ± 3σ ≈ 99.7% and shows the working so you can check each step by hand.
Is the Empirical Rule Calculator free, and do I need an account?
It is completely free with no sign-up, no download and no usage limit. Everything is calculated in your browser, so the numbers you type never leave your device.
Results are estimates for general information. For medical, legal, structural or financial decisions, confirm with a qualified professional.