Normal Distribution Calculator

What the normal distribution calculator does

The normal distribution is the bell curve that so much data follows, from heights to measurement errors to test scores. This calculator answers questions about it for any mean and standard deviation: the probability of a value falling in some range, the z-score behind a given probability, or the quantiles that mark off a share of the data.

It does the work a printed z-table used to do, but for any bell curve rather than just the standard one. Below is how the curve works and what the calculator can find.

How to use it

1. Choose what to find: a probability from a score, a z-score from a probability, or quantiles.
2. Enter the mean and standard deviation of your distribution, and then the raw score or probability the mode asks for.
3. Press Calculate for the result, or Reset to clear it.

The normal curve and its two numbers

A normal distribution is completely described by just two numbers: the mean, which sets where the peak of the bell sits, and the standard deviation, which sets how wide and how flat it is. A small standard deviation gives a tall, narrow bell, and a large one gives a short, wide one, but the symmetric bell shape is always the same.

The key idea is that area under the curve is probability. The whole area is 1, and the area over any stretch of values is the probability of landing in that stretch. So every question the calculator answers is really a question about measuring an area under the bell, and the total always balances to a probability of 1.

The three things it can find

Probability from a score takes a raw value and measures the areas around it: the chance of being above it, below it, between it and its mirror on the other side of the mean, or outside that range. This is how you turn a value into a probability.

Z-score from a probability goes the other way, finding the z-score, and the matching raw score, that cuts off a given probability. This is how you get critical values for a significance level. Quantiles finds the value below which a given share of the data falls, along with the central range that holds the rest, which is how you mark off, say, the bottom 5 percent or the middle 90.

A worked example

IQ scores are set up to follow a normal distribution with a mean of 100 and a standard deviation of 15. Suppose you want the probability of an IQ above 130.

A score of 130 is 30 above the mean, which is 30 ÷ 15 = 2 standard deviations, a z-score of 2. The area beyond a z-score of 2 in the upper tail is about 0.0228, so the probability of an IQ above 130 is roughly 2.3 percent. The same setup would tell you that about 97.7 percent of people score at or below 130.

The standard normal, and z-scores

Leave the mean at 0 and the standard deviation at 1, and the calculator becomes a standard normal calculator, the modern version of the z-table in the back of a statistics book. In that setting the raw score you enter is a z-score, and the probability it returns is the one you would have looked up in the table.

This is why z-scores matter so much. Every normal distribution can be shifted and rescaled onto that one standard curve, so once you have a z-score, a single table, or this calculator, handles the probabilities for all of them at once.

Entering your values, and the precision

Enter the mean and standard deviation for your distribution, then the raw score or probability, with any probability strictly between 0 and 1. The precision box sets how many digits the answer shows. The underlying areas are computed with a well-established approximation that is accurate to around seven or eight significant figures, so digits beyond that, if you raise the precision high, are not meaningful.

Questions people ask

What is the normal distribution?

The symmetric bell-shaped distribution described by a mean and a standard deviation. Area under its curve gives probability, with the whole area equal to 1.

Why does the calculator talk about area?

Because on the normal curve, the probability of a value falling in a range is the area under the curve over that range. Every probability it reports is a measured area.

What is the standard normal distribution?

The normal distribution with a mean of 0 and a standard deviation of 1. Setting those values turns the tool into a z-table calculator, where the score you enter is a z-score.

What is a quantile here?

The value below which a given share of the distribution falls. The 5 percent quantile, for instance, is the value that only 5 percent of the data sits below.

References

A quick note on where the methods here come from. The normal distribution, its cumulative area, and its inverse are set out in the NIST/SEMATECH e-Handbook of Statistical Methods, the US government's public statistics reference. OpenStax Introductory Statistics is a free, widely used textbook covering the normal distribution and z-scores.

1. NIST/SEMATECH e-Handbook of Statistical Methods (the normal distribution). https://www.itl.nist.gov/div898/handbook/
2. OpenStax, Introductory Statistics (the normal distribution and the standard normal distribution). https://openstax.org/details/books/introductory-statistics-2e