Normal Distribution Calculator

The normal distribution is the familiar bell-shaped curve where most values cluster near the average and fewer fall far from it. This normal distribution calculator finds the probability that a value lands below, above, or between the points you choose. It works by converting your x value into a z-score, then reading the matching area under the standard normal curve.

How to use this calculator

1. Enter the mean of your distribution (the center of the bell curve).
2. Enter the standard deviation (how spread out the values are).
3. Enter the x value you want to test — and the upper bound too, if you are using the between mode.
4. Pick a mode: P(X < x) for the area to the left, P(X > x) for the area to the right, or P(x < X < upper) for the area between two points.
5. Read off the probability and the z-score(s) the calculator returns.

Worked example

Suppose test scores follow a normal distribution with a mean of 100 and a standard deviation of 15, and you want the probability of scoring below 130. Set the mean to 100, the standard deviation to 15, x to 130, and the mode to P(X < x).

First, convert x to a z-score:

z = (x - mean) / standard deviation
z = (130 - 100) / 15 = 2

A z-score of 2 means 130 sits two standard deviations above the mean. The calculator then reads the area to the left of that point:

P(X < 130) = area to the left of x under the bell curve = 0.9772

So about 97.72% of scores fall below 130. If you switched the mode to P(X > x), the calculator would return the area to the right instead: 1 - 0.9772 = 0.0228, or about 2.28%. For the exact formula and reference tables, see the NIST/SEMATECH e-Handbook on the normal distribution.

Frequently asked questions

What is the normal distribution?

The normal distribution, also called the gaussian distribution, is a symmetric, bell-shaped curve defined by its mean and standard deviation. The mean marks the center of the bell curve, and the standard deviation sets how wide it spreads.

What does the probability mean?

The probability is the area under the curve for the region you selected — the proportion of values expected to fall there. Because the total probability under the curve always equals 1, the answer is a number between 0 and 1.

Why does the calculator show a z-score?

The z-score restates your x value as the number of standard deviations from the mean. Converting to a z-score lets the tool look up the probability under the curve from a single standard normal table, no matter what mean or spread your data has.

Is the bell curve always symmetric?

Yes — every normal distribution is perfectly symmetric around its mean, so the left and right halves of the bell curve mirror each other. That symmetry is why P(X < x) and P(X > x) for the same x always add up to 1.