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Normal Distribution

NIST/SEMATECH Section 1.3.6.6.1 Normal Distribution

PDF

Normal(0, 1) PDF -4 -3 -2 -1 0 1 2 3 4 x 0 0.1 0.2 0.3 0.4 f(x)

CDF

Normal(0, 1) CDF -4 -3 -2 -1 0 1 2 3 4 x 0 0.2 0.4 0.6 0.8 1 F(x)

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PDF

CDF

Formulas

Probability Density Function

f(x)=1σ2πexp⁡ ⁣(−(x−μ)22σ2)f(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp\!\left(-\frac{(x - \mu)^2}{2\sigma^2}\right)

Cumulative Distribution Function

F(x)=12[1+erf⁡ ⁣(x−μσ2)]F(x) = \frac{1}{2}\left[1 + \operatorname{erf}\!\left(\frac{x - \mu}{\sigma\sqrt{2}}\right)\right]

Properties

Mean

μ\mu

Variance

σ2\sigma^2

Overview

The normal (Gaussian) distribution is the most important continuous probability distribution, characterized by its symmetric bell-shaped curve. It is fully defined by its mean μ\mu and standard deviation σ\sigma and arises naturally via the central limit theorem.

Related Distributions

Student's t-Distribution Chi-Square Distribution Lognormal Distribution