Math

Normal Distribution Calculator: What 'Within One Standard Deviation' Actually Means

By David Brown · January 2026 · 3 min read

The normal distribution — bell curve — describes many natural phenomena: heights, measurement errors, IQ scores, and many physical processes. Knowing how to read it makes statistical claims much easier to evaluate.

The 68-95-99.7 Rule

For any normally distributed data:

68% of values fall within 1 standard deviation of the mean
95% of values fall within 2 standard deviations of the mean
99.7% of values fall within 3 standard deviations of the mean

This is the empirical rule, and it's worth memorizing.

Practical Interpretation

Adult male heights in the US: mean ≈ 70 inches, standard deviation ≈ 3 inches.

Within 1 SD (67–73 inches): 68% of men — roughly 5'7" to 6'1"
Within 2 SD (64–76 inches): 95% of men — roughly 5'4" to 6'4"
Above 3 SD (79+ inches, 6'7"+): 0.15% of men

When someone is described as "3 standard deviations above the mean" on any normally distributed trait, that means they're in roughly the top 0.1% — about 1 in 741 people.

Z-Scores

A Z-score converts any value to "how many standard deviations from the mean is this?"

Z = (value - mean) / standard deviation

A score of 130 on an IQ test (mean 100, SD 15):

Z = (130 - 100) / 15 = 2.0 → 97.7th percentile

When Normal Distribution Doesn't Apply

Income, wealth, city populations, and website traffic follow power law distributions — not normal distributions. Applying normal distribution assumptions to these leads to massive underestimation of extreme values (the "fat tail" problem that famously caused the 2008 financial crisis models to fail).

[Use the normal distribution calculator →](https://doesitaddup.com)

Try it yourself: All free calculators at doesitaddup.com →

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