Normal Distribution Calculator
Enter a mean, standard deviation, and a value range to calculate the probability that a normally distributed value falls within that range.
Last updated: September 5, 2026
Frequently asked questions
How do I find the probability a value falls in a range for a normal distribution?
Convert both bounds to z-scores (z = (x − mean) / SD), look up (or calculate) the cumulative probability for each z-score, then subtract: P(lower < X < upper) = CDF(z_upper) − CDF(z_lower). This calculator does that conversion automatically.
What's the probability within 1, 2, or 3 standard deviations of the mean?
This follows the empirical rule (68-95-99.7 rule): about 68% of values fall within 1 standard deviation of the mean, about 95% within 2, and about 99.7% within 3 - try entering mean ± 1, 2, or 3 × standard deviation as your bounds to confirm this.
Is this the same as the Z-Score Calculator?
Related but different - the Z-Score Calculator finds the percentile below a single value. This calculator finds the probability a value falls between two bounds, which is more useful for questions like "what fraction of scores are between 85 and 115?"
What if I set the lower and upper bound to the same value?
The probability comes out to essentially 0% - a continuous normal distribution assigns zero probability to hitting one exact point, only to falling within a range.
How do I find the probability of a value being above or below a single bound instead of between two?
Set the lower bound to a very small number (like mean − 10×SD) to effectively capture everything below your upper bound, or set the upper bound very high to capture everything above your lower bound - this approximates a one-sided probability using the two-bound tool.
Does this work for distributions that aren't normal?
No - this calculator specifically assumes a normal (bell-curve) distribution defined by its mean and standard deviation; skewed or non-normal data won't give accurate probabilities with this method.