Normal Distribution Calculator

Normal probabilities below, above or between values, the density, and inverse lookups, with the shaded bell curve.

Your result will appear here

Fill in the fields and press Calculate, or pick one of the examples.

About the Normal Distribution Calculator

Computes probabilities for a normal (Gaussian) distribution with any mean and standard deviation: the area below a value, above a value, between two values, or outside them. The matching region is shaded on the bell curve.

The inverse mode goes the other way: given a cumulative probability, it finds the value below which that fraction of the distribution lies.

Limitations

Cumulative probabilities are accurate to about 14 significant digits; inverse values to about 9.

How to use it

Enter the mean and standard deviation (0 and 1 for the standard normal).
Choose the kind of probability and enter the value(s).
Press Calculate.

Example

μ = 100, σ = 15, P(X < 115)

→ 0.8413447461

Formulas used

f(x) = \frac{1}{\sigma\sqrt{2\pi}}\, e^{-\frac{1}{2}\left(\frac{x - \mu}{\sigma}\right)^{2}}

Probability density function.

\Phi(z) = \tfrac{1}{2}\operatorname{erfc}\!\left(-\frac{z}{\sqrt{2}}\right),\quad z = \frac{x - \mu}{\sigma}

Cumulative distribution via the complementary error function.

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About the Normal Distribution Calculator

Limitations

How to use it

Formulas used

Related tools

About the Normal Distribution Calculator

Limitations

How to use it

Formulas used