# Normal distribution

Plots the CDF and PDF graphs for normal distribution with given mean and variance.

Normal distribution takes a unique role in the probability theory. This is the most common continuous 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 equal to mean μ.

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

#### Normal distribution

Digits after the decimal point: 5
Probability density function value

Cumulative distribution function value

PDF Graph
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CDF Graph
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#### 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) equates like this:
$\Phi^{-1}(p)\; =\; \sqrt2\;\operatorname{erf}^{-1}(2p - 1)$
This function is called the probit function.

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

#### Normal Distribution Quantile function

Digits after the decimal point: 2
Quantile

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PLANETCALC, Normal distribution