What this calculator does
For roughly normal data, about 68% of values lie within one standard deviation of the mean, 95% within two and 99.7% within three. For a mean of 100 and a standard deviation of 15, those intervals are 85 to 115, 70 to 130 and 55 to 145.
The rule is a description of the normal distribution, not a property of data in general. Apply it to skewed data such as income and it will be wrong, often badly. The value of knowing it is that it makes a standard deviation concrete: it turns an abstract measure of spread into a statement about where values actually fall.
The formula
The intervals are the mean plus and minus one, two and three standard deviations. The percentages come from the normal distribution itself, where the exact figures are 68.27%, 95.45% and 99.73%. The familiar 95% figure for two standard deviations is a rounding; the more precise multiplier for exactly 95% is 1.96, which is where that number in confidence intervals comes from.
| Term | Meaning |
|---|---|
| Empirical rule | The 68-95-99.7 approximation for normal data. |
| Standard deviation | The unit the intervals are measured in. |
| Three sigma | The ±3 SD interval, covering 99.7% and widely used as a control limit. |
| 1.96 | The multiplier giving exactly 95%, as against the rounded 2. |
The inputs explained
| Field | What to enter |
|---|---|
| Mean | The mean of the distribution. |
| Standard deviation | The standard deviation. The intervals are multiples of this either side of the mean. |
When to use it
Interpreting a test score
Many standardised tests are built to a mean of 100 and a standard deviation of 15, so the rule places any score immediately.
Setting control limits
Quality control commonly uses three standard deviations as the action limit, because only 0.3% of output should fall outside it by chance.
Judging whether a value is unusual
A value beyond two standard deviations happens about 5% of the time, which is the conventional threshold for surprising.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
What intervals does the rule give?
The same mean with a range of standard deviations.
| Standard deviation | 68% of values fall within | 95% of values fall within | 99.7% of values fall within |
|---|---|---|---|
| 5 | 95.000 to 105.000 | 90.000 to 110.000 | 85.000 to 115.000 |
| 10 | 90.000 to 110.000 | 80.000 to 120.000 | 70.000 to 130.000 |
| 15 | 85.000 to 115.000 | 70.000 to 130.000 | 55.000 to 145.000 |
| 20 | 80.000 to 120.000 | 60.000 to 140.000 | 40.000 to 160.000 |
Questions
Does the empirical rule apply to all data?
No, only to approximately normal data. Skewed distributions such as income, waiting times or house prices do not follow it, and applying it there gives misleading intervals. For data of unknown shape, Chebyshev theorem gives a weaker but universally valid bound.
Why is it 95% and not 95.45%?
Because 68-95-99.7 is a memorable rounding of 68.27, 95.45 and 99.73. The exact multiplier for 95% is 1.96 standard deviations rather than 2, which is why 1.96 rather than 2 appears in confidence interval formulas where precision matters.
What does three sigma mean?
Three standard deviations from the mean, an interval containing 99.7% of normal data. It is a common quality control limit: output beyond it is unlikely enough by chance that it suggests something has genuinely changed in the process rather than ordinary variation.
How do I know if my data is normal enough?
Check skewness and kurtosis for obvious departures, and look at a histogram or a normal probability plot. Mild departures leave the rule approximately usable; strong skew or heavy tails do not. If it matters, use a formal normality test rather than judging by eye.
For a bound valid for any distribution, see the Chebyshev theorem calculator. For standardising an individual value, see the z-score calculator.