Empirical Rule Calculator
Calculate 1, 2, and 3 standard deviation ranges from mean and standard deviation using the 68-95-99.7 empirical rule.
What Is the Empirical Rule?
The empirical rule (68-95-99.7 rule) describes how data in a normal distribution clusters around the mean. About 68% of values fall within one standard deviation, 95% within two, and 99.7% within three. Enter mean and standard deviation to get all three intervals instantly. For broader bounds that do not assume normality, try the Chebyshev's Theorem Calculator or the Standard Deviation Calculator.
How the 68-95-99.7 Rule Works
For a normal distribution with mean $\mu$ and standard deviation $\sigma$, the intervals are:
$$\mu \pm \sigma \quad (68\%), \quad \mu \pm 2\sigma \quad (95\%), \quad \mu \pm 3\sigma \quad (99.7\%)$$
IQ scores with mean 100 and standard deviation 15 illustrate the rule: 68% of people score between 85 and 115, 95% between 70 and 130, and 99.7% between 55 and 145.
When to Use the Empirical Rule
Use it when data is approximately bell-shaped: test scores, heights, measurement errors, and many natural variables. It helps spot outliers and estimate how much data lies in a given range without testing every observation.
Frequently Asked Questions
Does data need to be perfectly normal?
The rule is most accurate for symmetric, bell-shaped data. Skewed or multimodal distributions may not match the 68-95-99.7 percentages closely.
What if I only know the variance?
Take the square root of variance to get standard deviation, then enter that value here.
How is this different from Chebyshev's theorem?
Chebyshev's theorem applies to any distribution but gives minimum bounds. The empirical rule is tighter but requires approximate normality.
Can standard deviation be zero?
No. A zero standard deviation means every value equals the mean, so the intervals would collapse to a single point.
What counts as an outlier?
Values beyond three standard deviations from the mean are unusual under normality and may warrant further investigation.
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