Quick answer
The empirical rule says about 68% of normal data lie within one standard deviation of the mean, 95% within two and 99.7% within three. With μ = 100 and σ = 15: 85 to 115, 70 to 130 and 55 to 145.
Enter the mean and standard deviation. The calculator returns the three ranges.
Related tools: Binomial Distribution Calculator: Binomial probabilities for n trials. Sampling Distribution of the Mean Calculator: Probability for a sample mean using the central limit theorem. IQ Percentile Calculator: Percentile of an IQ score on a normal scale.
How to use the Empirical Rule Calculator
- Enter mean (μ).
- Enter standard deviation (σ) (above 0).
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Mean (μ) | A number | — |
| Standard deviation (σ) | A number | > 0 |
Formula and method
Formula
μ ± σ (≈68%), μ ± 2σ (≈95%), μ ± 3σ (≈99.7%)
IEEE-754 double precision; shown to at most 6 decimal places.
Approximate coverage for a normal distribution.
Worked examples
This example is verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Mean (μ) = 100; Standard deviation (σ) = 15.
| Result | Value |
|---|---|
| ≈68% (μ ± σ) | 85 to 115 |
| ≈95% (μ ± 2σ) | 70 to 130 |
| ≈99.7% (μ ± 3σ) | 55 to 145 |
The first line is the headline result (≈68% (μ ± σ)). Below it the empirical Rule Calculator lists ≈95% (μ ± 2σ) and ≈99.7% (μ ± 3σ), which describe the same case from other angles. IEEE-754 double precision; shown to at most 6 decimal places.
Example
IQ scores are scaled to mean 100 and SD 15, so about two-thirds of people score 85 to 115 and 95% score 70 to 130. Only about 0.3% fall outside 55 to 145.
Normality needed
The rule assumes an approximately bell-shaped distribution; for skewed data, Chebyshev's inequality gives weaker guarantees (at least 75% within 2σ). Use exact z-scores for other percentages.
How the result changes with mean
Holding every other input at the first worked example, this table changes only mean. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| Mean | ≈68% (μ ± σ) | ≈95% (μ ± 2σ) | ≈99.7% (μ ± 3σ) |
|---|---|---|---|
| 50 | 35 to 65 | 20 to 80 | 5 to 95 |
| 100 | 85 to 115 | 70 to 130 | 55 to 145 |
| 200 | 185 to 215 | 170 to 230 | 155 to 245 |
| 400 | 385 to 415 | 370 to 430 | 355 to 445 |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Standard deviation (σ): entering "-1" shows "Standard deviation (σ) must be greater than 0." beside that field and no result.
Choosing between the Empirical Rule Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Binomial Distribution Calculator](/statistics/binomial-distribution/) | trials, success probability and successes | P(X = 5) |
| [Sampling Distribution of the Mean Calculator](/statistics/normal-probability-sampling-distributions/) | population mean, population SD and sample size | P(x̄ < value) |
| [IQ Percentile Calculator](/statistics/iq-percentile/) | iQ score, mean and standard deviation | Percentile |
| [Poisson Distribution Calculator](/statistics/poisson-distribution-calculator/) | average rate and events | P(X = 3) |
| [Normal Distribution Calculator](/statistics/normal-distribution-calculator/) | mean, standard deviation and probability | P(X < x₁) |
Background: probability and distributions
A probability model only applies when its conditions hold. The binomial model needs a fixed number of independent trials with two outcomes and a constant success probability; its mass function gives the chance of exactly k successes. The normal density is f(x) = 1 ÷ (σ√(2π)) × e^(−(x − μ)² ÷ (2σ²)).
A cumulative probability adds up everything at or below a value, a density or mass function gives the value at a point. For a standard normal variable, z = −1.96 leaves 0.025 in the lower tail, which is why ±1.96 bounds a 95% interval.
Assumptions and limitations
- Assumes an approximately normal distribution
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for standard deviation (σ) to confirm this.
Frequently asked questions
What percentage is within 1.5σ?
About 86.6% for a normal distribution.
Why is it called 68-95-99.7?
For the three percentages.
What if data aren't normal?
Use Chebyshev's inequality.
What does the Empirical Rule Calculator calculate?
68–95–99.7 intervals for a normal distribution. It returns ≈68% (μ ± σ), ≈95% (μ ± 2σ) and ≈99.7% (μ ± 3σ).
What formula does the Empirical Rule Calculator use?
μ ± σ (≈68%), μ ± 2σ (≈95%), μ ± 3σ (≈99.7%). Approximate coverage for a normal distribution.
What do I need to enter in the empirical Rule Calculator?
You enter Mean (μ) and Standard deviation (σ).
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. Assumes an approximately normal distribution.
What is an example of a empirical Rule result?
With mean (μ) 100 and standard deviation (σ) 15, the ≈68% (μ ± σ) is 85 to 115.
Related calculators
- Binomial Distribution Calculator — Binomial probabilities for n trials.
- Sampling Distribution of the Mean Calculator — Probability for a sample mean using the central limit theorem.
- IQ Percentile Calculator — Percentile of an IQ score on a normal scale.
- Poisson Distribution Calculator — Poisson probabilities for a given mean rate.
- Normal Distribution Calculator — Probabilities for a normal distribution.
- Geometric Probability Calculator — Probability the first success occurs on trial k.
Sources
- Standard statistics definition (e.g. NIST/SEMATECH e-Handbook of Statistical Methods)
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.