What this calculator does
The range rule of thumb estimates a standard deviation as the range divided by four. Data running from 18 to 82 has a range of 64, giving an estimated standard deviation of 16.
It works because roughly 95% of normally distributed data sits within two standard deviations either side of the mean, a span of four standard deviations in total. If the observed range approximates that span, dividing by four recovers the standard deviation. Everything about that reasoning depends on the data being roughly normal and the sample being a moderate size.
The formula
The range, maximum minus minimum, is divided by four. The approximation holds best for roughly normal data with a sample size around 30. For much smaller samples the range underestimates the true spread, since extremes are unlikely to be observed; for much larger samples it overestimates, because a big sample is more likely to contain genuinely extreme values.
| Term | Meaning |
|---|---|
| Range | Maximum minus minimum. |
| Range rule | Estimated standard deviation as range divided by 4. |
| Empirical rule | The 95% within two standard deviations result that the rule is derived from. |
| Sample size dependence | The divisor that works best changes with n, from about 3 for small samples to 6 for very large ones. |
The inputs explained
| Field | What to enter |
|---|---|
| Minimum value | The smallest value in the data. |
| Maximum value | The largest value in the data. |
When to use it
Sanity-checking a computed standard deviation
If a calculated standard deviation is wildly different from the range over four, one of the two is probably wrong.
Estimating from a published summary
Papers and reports sometimes give only a range, and this recovers an approximate spread for a rough comparison.
Planning a sample size
Sample size formulas need a standard deviation estimate, and the range rule provides one before any data exists.
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 does the rule estimate for different ranges?
Several ranges with the estimated standard deviation.
| Maximum (minimum fixed at 18) | Estimated standard deviation | Range | Note |
|---|---|---|---|
| 38 | 5.000 | 20.000 | a rough approximation, most reliable for roughly normal data with n around 30 |
| 58 | 10.000 | 40.000 | a rough approximation, most reliable for roughly normal data with n around 30 |
| 82 | 16.000 | 64.000 | a rough approximation, most reliable for roughly normal data with n around 30 |
| 118 | 25.000 | 100.000 | a rough approximation, most reliable for roughly normal data with n around 30 |
Questions
How accurate is the range rule of thumb?
Rough. It is generally within about 30% of the true standard deviation for moderate samples of roughly normal data, which is enough for a sanity check or a planning estimate but not for reporting. Use the actual standard deviation whenever the data is available.
Why divide by four?
Because about 95% of normal data lies within two standard deviations of the mean, a total span of four. If the observed range roughly covers that, dividing by four returns the standard deviation. The logic breaks down if the data is not approximately normal.
Does sample size matter?
Considerably. The best divisor grows with n, from around 3 for samples near 10 to 4 near 30 and 6 for samples in the thousands. Dividing by four is the compromise that suits typical sample sizes, and it degrades at both extremes.
When should I not use this?
For skewed data, for data with outliers, and whenever the raw values are available. An outlier inflates the range directly and so inflates the estimate, sometimes by a large factor. It is a fallback for when only a summary exists, not a shortcut to avoid the real calculation.
For the rule this is derived from, see the empirical rule calculator. For a bound that needs no normality assumption, see the Chebyshev theorem calculator.