Mathematics calculator

Empirical Rule Calculator

Calculate the one-, two-, and three-standard-deviation intervals used by the 68–95–99.7 rule for approximately normal data.

Calculator inputs

Enter your values

Normal distribution

Distribution mean

Positive standard deviation

Calculated output

Result summary

Your result will appear here

Complete the required fields and select the calculate button.

What is Empirical Rule?

The empirical rule summarizes the spread of data that follow an approximately normal, bell-shaped distribution. It places about 68 percent of observations within one standard deviation of the mean, about 95 percent within two, and about 99.7 percent within three. This calculator finds the endpoints of those three intervals.

The percentages are model-based approximations, not counts calculated from a dataset. Before using them, inspect the distribution for strong skew, multiple peaks, major outliers or a natural boundary that conflicts with the proposed normal model. The rule is most helpful as a quick reasonableness check or introduction to normal-distribution thinking.

How to Use This Calculator

Enter the arithmetic mean in the same units as the observations.
Enter a positive standard deviation. Be consistent about whether your source reports a sample standard deviation or a population standard deviation.
Select Calculate. The page subtracts and adds one standard deviation to find the approximate 68 percent interval.
It repeats the calculation with two and three standard deviations for the approximate 95 and 99.7 percent intervals.
Assess whether a normal shape is plausible before using the percentages. If actual observations are available, compare the observed proportions with the rule rather than assuming agreement.

Formula Used

One-standard-deviation interval: μ − σ to μ + σ, containing about 68 percent under a normal model. Two-standard-deviation interval: μ − 2σ to μ + 2σ, containing about 95 percent. Three-standard-deviation interval: μ − 3σ to μ + 3σ, containing about 99.7 percent.

Here μ is the mean and σ is the standard deviation. The calculator finds interval endpoints only. It does not estimate μ or σ from raw observations, test normality, or calculate an exact probability for a non-integer number of standard deviations.

Variables Explained

Input	Meaning and use
Mean (μ)	Center of the distribution, expressed in the original measurement unit.
Standard deviation (σ)	Positive measure of spread in the same unit as the observations.
μ ± σ	Endpoints of the interval expected to contain about 68 percent of normal observations.
μ ± 2σ	Endpoints of the interval expected to contain about 95 percent of normal observations.
μ ± 3σ	Endpoints of the interval expected to contain about 99.7 percent of normal observations.
Normal-model assumption	Approximately symmetric, single-peaked, bell-shaped behavior required for the percentage interpretation.

Worked Example

Consider exam scores modeled with mean 72 and standard deviation 8. Within one standard deviation, the endpoints are 72 − 8 = 64 and 72 + 8 = 80. The two-standard-deviation interval is 72 − 16 = 56 through 72 + 16 = 88. The three-standard-deviation interval is 48 through 96.

If the score distribution is reasonably normal, about 68 percent would be expected between 64 and 80, about 95 percent between 56 and 88, and about 99.7 percent between 48 and 96. Real exam scores are bounded and may be skewed, so the fit should be checked rather than assumed.

Why Use This Calculator?

Each wider interval contains the narrower one. For a normal model, roughly 32 percent lies outside one standard deviation, 5 percent outside two, and 0.3 percent outside three, split symmetrically between the two tails. Those rounded shares are useful for orientation but are not exact observed frequencies.

Values outside two or three standard deviations are unusual under a normal model, but unusual does not automatically mean erroneous. Investigate data collection, context and the planned analysis before excluding an observation. For skewed or heavy-tailed distributions, percentiles or distribution-specific methods often describe spread more honestly. A histogram and normal probability plot can reveal departures that the mean and standard deviation alone cannot show. With small samples, apparent shape is uncertain, so combine graphics with subject knowledge and a documented analysis plan.

Common Mistakes

Applying 68–95–99.7 to any dataset without checking its shape.
Entering variance instead of standard deviation. Standard deviation is the square root of variance.
Using a negative or zero standard deviation, which cannot describe a non-degenerate spread.
Claiming exactly 68, 95 and 99.7 percent of a finite sample must fall in the intervals.
Calling every point beyond three standard deviations a data error.
Mixing a mean in one unit with a standard deviation in another.

Frequently Asked Questions

Does the empirical rule work for every dataset?

No. Its percentages rely on an approximately normal shape. Strong skew, several peaks, truncation or heavy tails can make the coverage quite different.

Is standard deviation the same as variance?

No. Variance is in squared units; standard deviation is its square root and returns to the original unit. Enter standard deviation here.

Are the percentages exact?

They are rounded normal-distribution probabilities. A finite sample will rarely contain exactly those percentages, even when it comes from a normal population.

What if my lower endpoint is negative?

The arithmetic can be correct while signaling that a normal model is questionable for a variable that cannot be negative. Examine the distribution and consider a better model.

Does a value beyond three standard deviations have to be removed?

No. It is rare under a normal model, but it may be valid. Check collection and context, then use a preplanned, defensible rule for any exclusion.

What is the difference between sample and population standard deviation?

They use different denominators when estimated from data. Use the value appropriate to your problem and state which one was used; both must be positive.

Can I find an exact probability with this page?

No. It reports only the three conventional intervals. Exact probabilities for other cutoffs require a normal cumulative-distribution calculation or another suitable model.