About the Normal Distribution Calculator
The normal distribution, or bell curve, describes many measurements such as heights, test scores and errors. This calculator takes the mean, the standard deviation and a value, and gives the z-score, the probability of a result below the value, above it and, with a second value, between the two. The height of the curve at that point is shown as well. Type the mean, the standard deviation, a value and, optionally, a second value and the normal probability appears at once, with a few related figures underneath, and you can copy the result with one tap. The sums run in your browser, so nothing you enter is uploaded.
Updated October 2026
How to use the Normal Distribution Calculator
- 1Enter the mean, the standard deviation, a value and, optionally, a second value.
- 2Pick the units or options from the menus if the normal probability tool offers them.
- 3Read the normal probability in the large result box.
- 4Look at the extra rows for related figures.
- 5Press Copy result to keep the answer.
When you need the normal probability
Teachers use it to grade on a curve, quality engineers to find the share of parts outside tolerance, and researchers to turn test statistics into probabilities. Anyone working with measurements that cluster around an average can use it.
About 68 percent of values lie within one standard deviation of the mean, 95 percent within two and 99.7 percent within three. The tool calculates exact areas, so you are not limited to these round numbers.
How the normal probability is worked out
z = (x − mean) ÷ standard deviation. The probability below x is the area under the standard normal curve to the left of z. IQ scores have a mean of 100 and a standard deviation of 15. A score of 130 has a z-score of 2, and 97.7 percent of people score below it, 2.3 percent above. The share of people between 85 and 115 is 68.3 percent, the familiar one-standard-deviation rule.
Every answer on this page comes from that one rule, so you can check any result by hand with a calculator. The figures shown underneath, such as other units, are conversions of the same answer.
- Below, above and between
- z-score and curve height
- Exact areas
Mistakes to avoid with normal probability
The calculation assumes the data really follow a normal distribution, which many do not. Skewed data, such as incomes, give misleading results. The standard deviation must be above zero. Probabilities are accurate to about ten decimal places.
On phone and computer
On a phone, type into the boxes one after another and read the normal probability underneath; Copy result puts it on the clipboard for a message or a note. On a computer the inputs sit on the left and the answer on the right, so you can change one figure and watch the normal probability move. The normal distribution calculator works the same on both. A bell curve needs a mean and a spread. Use the Standard Deviation Calculator to find the spread, the Z-Score Calculator to place a single value, and the Poisson Distribution Calculator for counts of rare events.
Features
- Check the data are bell-shaped
- SD above zero
- 68-95-99.7 rule
- Clear messages when the normal probability inputs make no sense
- Copy the result with one tap
- Private: nothing is uploaded
Frequently asked questions
How do I calculate normal probability?
z = (x − mean) ÷ standard deviation. The probability below x is the area under the standard normal curve to the left of z. IQ scores have a mean of 100 and a standard deviation of 15. A score of 130 has a z-score of 2, and 97.7 percent of people score below it, 2.3 percent above. The share of people between 85 and 115 is 68.3 percent, the familiar one-standard-deviation rule.
What is the formula for normal probability?
z = (x − mean) ÷ standard deviation. The probability below x is the area under the standard normal curve to the left of z. The normal distribution calculator applies it exactly and rounds only the displayed answer to six significant figures.
What is a z-score?
The number of standard deviations a value lies from the mean.
What is the probability of exactly one value?
For a continuous distribution it is zero; only ranges have a probability.
How do I find the percentile of a score?
Calculate P(X < x) and multiply by 100.
Is this normal distribution calculator free and private?
Yes. It is free, needs no account and runs in your browser, so the numbers you type are not uploaded or stored.