Enter a mean and standard deviation to find the probability between, below or above any values, or find the cut-off for a given probability.
Default: IQ scores, designed to have mean 100 and SD 15.
z(a) = (85 − 100) ÷ 15 = −1; z(b) = 1 · that is 68.27% of the distribution.
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Pick how many standard deviations to include. The shaded band covers μ ± kσ, and the share of the curve inside it is the same for every normal distribution, whatever the mean and SD.
For your distribution (μ = 100, σ = 15), ±1σ runs from 85 to 115.
68.27% inside
The normal distribution N(μ, σ) is the symmetric bell curve described by Gauss. The mean sets its centre and the standard deviation its width. Every normal distribution becomes the standard normal N(0, 1) once you convert to z-scores, which is how the printed z-table covers them all.
This calculator computes Φ with a rational approximation accurate to about 15 significant digits (W. J. Cody’s algorithm), so tail areas far beyond the printed table are still reliable.
Convert both values to z-scores, find the cumulative area for each, and subtract: P(a < X < b) = Φ(z_b) − Φ(z_a). For IQ scores (mean 100, SD 15), P(85 < X < 115) = Φ(1) − Φ(−1) ≈ 0.6827.
In a normal distribution, about 68.27% of values lie within one standard deviation of the mean, 95.45% within two and 99.73% within three. It is also called the empirical rule or three-sigma rule.
Find the z-score whose cumulative area equals the probability, then convert back: x = μ + zσ. The top 10% of IQ scores start at z = 1.2816, so x = 100 + 1.2816 × 15 ≈ 119.2. Excel’s NORM.INV and the TI-84 invNorm do the same.
Zero. A continuous distribution assigns probability to intervals, not single points. If your values are rounded, use a small interval around x, such as 99.5 to 100.5.
Many measurements that are the sum of many small independent effects are close to normal: heights, measurement error, averages of samples. Counts, times and money are often skewed. Check a dot plot or histogram before relying on normal probabilities.