# Empirical Rule Calculator (68–95–99.7)

URL: https://speedycalculator.com/calculators/statistics/empirical-rule/
Last updated: 2026-10-30

> The empirical rule calculator applies the 68–95–99.7 rule: from a mean and standard deviation it gives the ranges that hold about 68%, 95% and 99.7% of a bell-shaped data set, and it places any value of yours in the right band with its z-score and percentile.

## Inputs

- Mean (μ)
- Standard deviation (σ)
- Value to locate (optional)

## Results

- About 68% of values lie between
- About 95% of values lie between
- About 99.7% of values lie between
- Bell curve with the 1σ, 2σ and 3σ bands
- Exact shares by number of standard deviations
- z-score of your value
- Percentile of your value
- Where your value falls
- μ − σ
- μ + σ
- μ − 2σ
- μ + 2σ
- μ − 3σ
- μ + 3σ

## Formula

```
68% within μ ± σ · 95% within μ ± 2σ · 99.7% within μ ± 3σ · exact: 68.27%, 95.45%, 99.73%
```

## Explanation

The empirical rule – also called the 68–95–99.7 rule or the three-sigma rule – is a quick way to describe a normal (bell-shaped) distribution without any tables. About 68% of the values lie within one standard deviation of the mean, about 95% within two and about 99.7% within three.

That makes it handy for sanity checks: if a value sits more than three standard deviations from the mean, it happens less than 0.3% of the time and is worth a second look as a possible outlier or error. It also gives a rough idea of the spread when you only know μ and σ, and a test of whether data look normal – if far fewer or far more than 68% of your values fall inside μ ± σ, the data are probably not bell-shaped.

The calculator shows the three ranges, a table with the exact normal percentages (including four standard deviations), and a chart with each band shaded. Enter an optional value to see its z-score, its exact percentile and which band it falls in.

### How to use the Empirical Rule Calculator

- **Mean (μ)**
- **Standard deviation (σ)**
- **Value to locate (optional)**

Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.

### How it works

The ranges are μ ± kσ for k = 1, 2, 3. The exact share inside each range is 2Φ(k) − 1 from the standard normal distribution, and a value x is located with z = (x − μ)/σ and its percentile Φ(z).

### Worked example

IQ scores have μ = 100 and σ = 15. One standard deviation either side gives 85 to 115 (about 68% of people), two give 70 to 130 (95%) and three give 55 to 145 (99.7%). A score of 130 has z = 2: it is at the 97.72nd percentile, so roughly 1 person in 44 scores higher. A score of 145 or more (z ≥ 3) is reached by only 0.13% of people.

### Empirical rule versus Chebyshev

WithinNormal data (empirical rule)Any data (Chebyshev, at least)μ ± 1σ68.27%no guaranteeμ ± 2σ95.45%75%μ ± 3σ99.73%88.9%μ ± 4σ99.994%93.75%

## Worked example

Inputs — Mean (μ): 100; Standard deviation (σ): 15; Value to locate (optional): 130.

Results — **About 68% of values lie between = 85 to 115**; About 95% of values lie between = 70 to 130; About 99.7% of values lie between = 55 to 145; Exact shares by number of standard deviations = Range From To Inside (exact) Outside Outside rate
─────────────────────────────────────────────────────────────
±1σ 85 115 68.2689% 31.7311% 1 in 3
±2σ 70 130 95.45% 4.55% 1 in 22
±3σ 55 145 99.73% 0.27% 1 in 370
±4σ 40 160 99.9937% 0.0063% 1 in 15,787; z-score of your value = 2; Percentile of your value = 97.72 %.

## Key terms

- **Three-sigma rule**: Another name for the empirical rule; 3σ covers 99.7% of a normal distribution.
- **Standard deviation (σ)**: The typical distance of values from the mean.
- **Chebyshev's inequality**: A rule valid for any distribution: at least 1 − 1/k² of values lie within k standard deviations.
- **Outlier**: A value unusually far from the rest, often defined as more than 3σ from the mean.

## FAQ

**What does the 68–95–99.7 rule say?**
For a normal distribution, about 68% of values are within 1 SD of the mean, 95% within 2 SD and 99.7% within 3 SD.

**What are the exact percentages?**
68.27%, 95.45% and 99.73%. The rule rounds them for easy memory.

**What share of values lie above μ + 2σ?**
Half of the 4.55% outside ±2σ, about 2.28%; above μ + 3σ it is only 0.13%.

**For IQ (mean 100, SD 15), what are the ranges?**
85–115 (68%), 70–130 (95%) and 55–145 (99.7%).

**Does the rule work for any data?**
Only for roughly bell-shaped, symmetric data. For any distribution Chebyshev's inequality guarantees at least 75% within 2 SD and 88.9% within 3 SD.

**Can I use it to find outliers?**
Yes – values beyond ±3σ occur about 1 time in 370 under normality, so they are commonly flagged.

**Should I use the sample or population standard deviation?**
Use whichever describes the data; for a sample, the sample standard deviation (dividing by n − 1) is the usual choice.


## Sources

- [NIST/SEMATECH e-Handbook of Statistical Methods](https://www.itl.nist.gov/div898/handbook/)
- [OpenStax – Introductory Statistics 2e](https://openstax.org/details/books/introductory-statistics-2e)
- [Khan Academy – Statistics and probability](https://www.khanacademy.org/math/statistics-probability)