Empirical Rule Calculator

For a normal (bell-shaped) distribution, about 68% of values lie within 1 standard deviation of the mean, 95% within 2 and 99.7% within 3. Enter a mean and standard deviation to see those ranges, the exact percentages and the tail areas.

Empirical rule ranges
RangeFromToRuleExact (normal)
μ ± 1σ8511568%68.27%
μ ± 2σ7013095%95.45%
μ ± 3σ5514599.7%99.73%
Inside ±2σ
95%
Below 70 (exact)
2.28%
Above 130 (exact)
2.28%
557085100115130145

Step-by-step working

  1. Within 1 standard deviation: μ ± σ100 ± 15 → 85 to 115 (about 68%)
  2. Within 2 standard deviations: μ ± 2σ100 ± 30 → 70 to 130 (about 95%)
  3. Within 3 standard deviations: μ ± 3σ100 ± 45 → 55 to 145 (about 99.7%)
  4. Because the curve is symmetric, what is left over splits equally between the two tailsoutside ±2σ: 4.55% total, 2.28% in each tail

Formula μ ± 1σ ≈ 68% μ ± 2σ ≈ 95% μ ± 3σ ≈ 99.7%

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The bell curve in standard deviations

Every normal curve is the same shape once you measure in standard deviations. That’s what a z-score does: it counts how many SDs a value sits from the mean.

−1+1

68.27% of values lie within ±1 SD of the mean

That is the 68–95–99.7 rule (the empirical rule). Try a single z below: the shaded area to its left is the proportion of values below it, which is its percentile.

1.20

Using the 68-95-99.7 rule

IQ scores are scaled to a mean of 100 and SD of 15. So about 68% of people score 85–115, about 95% score 70–130 and about 99.7% score 55–145. Because the curve is symmetric, the 5% outside 70–130 splits into 2.5% below 70 and 2.5% above 130.

The rounded figures are approximations: the exact normal areas are 68.27%, 95.45% and 99.73%. For any other cut-off, use the normal distribution calculator or the z-score calculator.

When the rule does not apply

The empirical rule only holds for data that are roughly normal. For skewed data, use Chebyshev’s theorem instead: for any distribution at least 1 − 1/k² of values lie within k standard deviations (at least 75% within 2σ, at least 88.9% within 3σ).

Frequently asked questions

What percentage is within 2 standard deviations?

About 95% (exactly 95.45%) for a normal distribution. The interval that holds exactly 95% is μ ± 1.96σ.

What percentage is above 1 standard deviation?

About 16%: 100% − 68% leaves 32% outside ±1σ, split equally between the two tails.

Is the empirical rule the same as Chebyshev’s theorem?

No. The empirical rule is for normal data and gives approximate percentages. Chebyshev gives guaranteed minimums for any distribution, which are much lower.