StatReason

Normal Distribution Calculator

Given a mean and standard deviation, find the probability (area under the curve) between, above, or below specific values.

What this answers

This calculator answers "if a quantity follows a normal distribution with this mean and standard deviation, what fraction of values fall in this range?" It converts your bounds to z-scores (standardized distances from the mean) and returns the corresponding area under the normal curve, the probability of landing in that range.

When to use it, and when not to

Use this when you have a reasonable basis for believing a quantity is approximately normally distributed (test scores, measurement error, and many biological measurements often are) and you know or can assume its mean and standard deviation. This tool cannot tell you whether the normal model actually fits your specific data; for that, use the Data Normality Checker.

How it is calculated

Each bound is converted to a z-score, the number of standard deviations it sits from the mean, then the standard normal cumulative distribution function gives the area to the left of each z-score. Supplying both a lower and upper bound returns the area between them; supplying only one returns the area beyond that single bound in the corresponding direction.

Worked example

With mean 100 and standard deviation 15 (a classic IQ-scale example), the range 85 to 115 is exactly one standard deviation on either side of the mean. That area works out to about 68.27%, the well-known "68 percent within one SD" rule for any normal distribution, regardless of its specific mean and SD.

Assumption audit

Calculated from your data: the z-scores corresponding to your entered bounds, and whether your bounds are ordered correctly.
Evidence to review: whether your actual data looks bell-shaped and symmetric. Use the Histogram Maker to check before trusting a normal-model probability.
You must verify: that the mean and standard deviation you supplied are themselves accurate, and that a normal model is actually appropriate for your quantity, since this calculator takes both as given rather than testing them.

What this result does not mean

This probability is entirely conditional on the normal model and the mean and SD you provided. It is not a real-world forecast guaranteed to be calibrated. If the true distribution is skewed, heavy-tailed, or has a different mean or SD than assumed, the real proportion of values in your range can differ substantially from this result.

Limitations

Extreme tail probabilities (very large or very small z-scores) become vanishingly small but are always computed with a numerically stable method here, never rounded to exactly zero, which matters when comparing two very rare events to each other.