Find normal-distribution probabilities for a value — cumulative, upper tail, and density. Enter the value, mean, and standard deviation.
Usage Tip
The famous 68-95-99.7 rule: about 68, 95, and 99.7 percent of data lie within 1, 2, and 3 standard deviations of the mean.
Cumulative probability from the standard normal curve
The cumulative probability is the area under the curve up to a value; the density is the curve height there.
P(X less than or equal to x) is the area to the left; P(X greater than x) is to the right.
Probabilities are approximate (error-function based).
The normal distribution is the familiar bell curve, centered on the mean with its spread set by the standard deviation. About 68 percent of values fall within one standard deviation of the mean, about 95 percent within two, and about 99.7 percent within three. The cumulative probability gives the chance of a value at or below x, and the upper tail is the chance of a value above it.
The density is the height of the curve at x, which is not itself a probability. Probabilities for a continuous distribution come from areas under the curve, so the chance of any single exact value is zero. Real data is only approximately normal, so check that the distribution is roughly symmetric and bell shaped before relying on these figures.