How it works
The normal distribution is the symmetric bell curve described by its mean and standard deviation. Many measurements, such as heights, test scores and manufacturing tolerances, are approximately normal.
The calculator converts your values to z-scores and finds the area under the curve with the standard normal cumulative distribution function. About 68% of values fall within one standard deviation of the mean, 95% within two and 99.7% within three.
For a continuous distribution, the probability of any single exact value is zero, so “below” and “at or below” give the same answer.
Formula
z = (x − μ) ÷ σ · P(X < x) = Φ(z) · P(a < X < b) = Φ(z_b) − Φ(z_a)
Questions people ask
What is the 68-95-99.7 rule?
In a normal distribution, about 68.27% of values lie within 1 standard deviation of the mean, 95.45% within 2 and 99.73% within 3.
What share of IQ scores are above 130?
With mean 100 and SD 15, 130 is z = 2, so about 2.28% of people score above it.
What if my data are not normal?
The results can be misleading for skewed or heavy-tailed data. Check a histogram first, or use the actual percentiles of your data instead.