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Normal Distribution Calculator

Probability Between Values and Value at a Percentile on Any Bell Curve

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How the Normal Distribution Calculator works

This normal distribution calculator finds the area under any bell curve. Give it the mean and standard deviation, then ask for the chance of a value falling below, above, between or outside your bounds, or turn it round and find the value at a chosen percentile. The working is shown.

Heights, exam marks, measurement errors and many other quantities pile up around an average and thin out symmetrically on each side. That shape is the normal distribution, and two numbers fix it completely: the mean μ, which sets the centre, and the standard deviation σ, which sets the spread.

How the normal distribution calculator works

The formula converts each bound to a z-score, z = (x − μ) ÷ σ, then reads the cumulative area Φ(z) from the standard normal curve. The area between two bounds is Φ(zb) − Φ(za); the area above a bound is 1 − Φ(z). For a percentile the calculator runs that in reverse, x = μ + z × σ.

First example. IQ scores have μ = 100 and σ = 15. The share between 85 and 115 is Φ(1) − Φ(−1) ≈ 68.27%.

Second example. Bolts are cut to 50 mm with σ = 0.2 mm. The 99th percentile length is 50 + 2.326 × 0.2 ≈ 50.47 mm.

How many values fall within two standard deviations? About 95.45%, which is where the familiar 68–95–99.7 rule comes from.

Share of a normal distribution within and beyond k standard deviations of the mean

k (SDs) Within ±k Beyond ±k About 1 in
0.25 19.74% 80.26% 1.25
0.5 38.29% 61.71% 1.62
0.674 49.97% 50.03% 2
0.75 54.67% 45.33% 2.21
1 68.27% 31.73% 3.15
1.15 74.99% 25.01% 4
1.25 78.87% 21.13% 4.73
1.282 80.02% 19.98% 5
1.5 86.64% 13.36% 7.48
1.645 90% 9.997% 10
1.75 91.99% 8.012% 12.5
1.96 95% 5% 20
2 95.45% 4.55% 22
2.25 97.56% 2.445% 40.9
2.326 98% 2.002% 50
2.5 98.76% 1.242% 80.5
2.576 99% 0.9995% 100
2.75 99.4% 0.596% 168
3 99.73% 0.27% 370
3.29 99.9% 0.1002% 998
3.5 99.95% 0.04653% 2,150
3.891 99.99% 0.009983% 10,000
4 99.99% 0.006334% 15,800
5 100% 0.000057% 1,740,000

Tips

FAQ

What is the 68–95–99.7 rule?

In a normal distribution about 68% of values lie within one standard deviation of the mean, about 95% within two and about 99.7% within three. It is a quick sanity check: if your data has far more values beyond three standard deviations, it is probably not normal.

How do I find the probability between two values?

Choose P(a < X < b), enter the mean, the standard deviation and both bounds. The calculator turns each bound into a z-score, looks up the cumulative area for each and subtracts the smaller from the larger, showing every step of that working.

What is an inverse normal calculation?

It answers the opposite question: instead of the probability for a value, it gives the value for a probability. Choose Value at a percentile, enter 95, and you get the cut-off that 95% of the distribution falls below, which is how pass marks and size ranges are often set.

Does the normal distribution work for small samples?

The normal curve describes a population, not a sample. If you are estimating a mean from a small sample with an unknown standard deviation, use the t distribution instead, which the confidence interval calculator applies automatically for you.

See also: all Statistics tools, Arithmetic, Number Theory, Algebra and Geometry, Trigonometry and Calculus.

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