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Normal distribution

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

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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 distribution

Digits after the decimal point: 5
Probability density function value

Cumulative distribution function value

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 function

Digits after the decimal point: 2
Quantile

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PLANETCALC, Normal distribution