Get reference code

Normal distribution

Plots the CDF and PDF graphs for normal distribution with given mean and variance.
Anton2015-11-28 20:40:11

Normal distribution takes special role in the probability theory. This is most common continues probability distribution, commonly used for random values representation of unknown distribution law.

Probability density function

Normal distribution probability density function is the Gauss function:
f(x) = \tfrac{1}{\sigma\sqrt{2\pi}}\; e^{ -\frac{(x-\mu)^2}{2\sigma^2} }

where μ — mean,
σ — standard deviation,
σ ² — variance,
Median and mode of Normal distribution equals to mean μ.

The calculator below gives probability density function value and cumulative distribution function value for the given x, mean and variance:

Normal distributionCreative Commons Attribution/Share-Alike License 3.0 (Unported)
PDF Graph:
CDF Graph:

Cumulative distribution function

Normal distribution cumulative distribution function has the following formula:
\frac12\left[1 + \operatorname{erf}\left( \frac{x-\mu}{\sigma\sqrt{2}}\right)\right]
where, erf(x) - error function, given as:
\operatorname{erf}(x) = \frac{2}{\sqrt{\pi}}\int\limits_0^x e^{-t^2}\,\mathrm dt

Quantile function

Normal distribution quantile function (inverse CDF) given as inverse error function:

F^{-1}(p) = \mu + \sigma\sqrt2\,\operatorname{erf}^{-1}(2p - 1)
p lays in the range [0,1]

Standard normal distribution quantile function (σ =1, μ=0) looks like this:
 \Phi^{-1}(p)\; =\; \sqrt2\;\operatorname{erf}^{-1}(2p - 1)
This function is called the probit function.

Calculator below gives quantile value by probability for specified by mean and variance normal distribution( set variance=1 and mean=0 for probit function).

Normal Distribution Quantile functionCreative Commons Attribution/Share-Alike License 3.0 (Unported)

Request a calculator
View all calculators
(505 calculators in total. )