PLAYGROUND

DISTRIBUTIONS / CONTINUOUS / NORMAL

Normal distribution

What it describes

The Normal distribution is the reference symmetric law for errors and averages. Its importance is not that every dataset is bell-shaped, but that it emerges as the limit of sums of many small, roughly independent effects.

It spans the real line. mu sets the centre and sigma spread; standardization allows comparisons across units through the standard normal.

History and names

De Moivre obtained a normal approximation to Binomial in the eighteenth century, and Gauss used it for astronomy and measurement errors. The central limit theorem secured its broad role.

How it connects to other distributions

It is the limit of many sums and approximates Binomial, Poisson, and t at large sizes under suitable conditions. Its squares give chi-square and its ratios generate t and F.

Where it appears

  • measurement errors, averages, and regression
  • approximations for counts and estimator distributions

Fitting cautions

A bell-shaped histogram does not validate independence, absence of outliers, or normal tails.

The bell is not the real protagonist

The deeper reason for its ubiquity lies in sums. Many different mechanisms, after accumulating small contributions with no single term dominating, produce averages that approach a Normal distribution. The central limit theorem describes the distribution of an average under precise conditions; it does not declare every original observation to be normal.

There is a fruitful geometric curiosity. Two independent standard Normal variables define a random point in the plane. Its distance from the origin is Rayleigh, and the squared distance is Chi Square with two degrees of freedom. Divide a Normal variable by the square root of an independent scaled Chi Square and Student’s t appears. The bell is therefore a hub from which several other families emerge.

Decision guide

A good candidate when: effects are approximately additive, the distribution is symmetric, and tails show no more extremes than expected under a bell curve.

Compare it with: Student’s t for heavy tails, Generalized Normal for different kurtosis, and an asymmetric family for skewness. Inspect a Q–Q plot and dependence, not only a histogram.

References

  • SciPy reference: scipy.stats.norm — definition and parameterization
  • Johnson, N. L., Kotz, S. & Balakrishnan, N. (1994). Continuous Univariate Distributions, 2nd ed., Vol. 1. Wiley.
  • de Moivre, A. (1733). Approximatio ad Summam Terminorum Binomii in Seriem Expansi. Privately printed.
  • Gauss, C. F. (1809). Theoria motus corporum coelestium in sectionibus conicis solem ambientium. Perthes & Besser.

Normal Distribution: equations and calculator

Distribution defintion

X∼Normal(μ,σ)X\sim\mathrm{Normal}\left(\mu,\sigma\right)

Distribution domain

x∈(−∞,∞)x\in\left(-\infty,\infty\right)

Parameters domain and parameters constraints

μ∈R,σ∈R+\mu\in\mathbb{R},\sigma\in\mathbb{R}^{+}

Cumulative distribution function

FX(x)=12[1+erf⁡(x−μσ2)]=Φ(x−μσ)F_{X}\left(x\right)=\frac{1}{2}\left[1+\operatorname{erf}\left(\frac{x-\mu}{\sigma\sqrt{2}}\right)\right]=\Phi\left(\frac{x-\mu}{\sigma}\right)

Probability density function

fX(x)=1σ2πe−12(x−μσ)2=ϕ(x−μσ)f_{X}\left(x\right)=\frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}=\phi\left(\frac{x-\mu}{\sigma}\right)

Percent point function/Sample

FX−1(u)=μ+σ2erf⁡−1(2u−1)=μ+σΦ−1(u)F^{-1}_{X}\left(u\right)=\mu+\sigma\sqrt{2} \operatorname{erf}^{-1}(2u-1)=\mu+\sigma\Phi^{-1}\left(u\right)

Non-central parametric moments

μk′=E[Xk]=∫−∞∞xkfX(x)dx=σk⋅(−i2)kU(−k2,12,−12(μσ)2)\mu'_{k}=E[X^k]=\int_{-\infty}^{\infty}x^{k}f_{X}\left(x\right)dx=\sigma^k\cdot (-i\sqrt 2)^k U\left(-\frac{k}{2},\frac{1}{2},-\frac{1}{2}\left(\frac \mu \sigma\right)^2\right)

Parametric mean

Mean(X)=μ1′=μ\mathrm{Mean}(X)=\mu'_{1}=\mu

Parametric variance

Variance(X)=μ2′−μ1′2=σ2\mathrm{Variance}(X)=\mu'_{2}-\mu'^{2}_{1}=\sigma^{2}

Parametric skewness

Skewness(X)=μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5=0\mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}}=0

Parametric kurtosis

Kurtosis(X)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2=3\mathrm{Kurtosis}(X)=\frac{\mu'_{4}-4\mu'_{1}\mu'_{3}+6\mu'^{2}_{1}\mu'_{2}-3\mu'^{4}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{2}}=3

Parametric median

Median(X)=μ\mathrm{Median}(X)=\mu

Parametric mode

Mode(X)=μ\mathrm{Mode}(X)=\mu

Additional information and definitions

μ:Location parameter\mu:\text{Location parameter}
σ:Scale parameter\sigma:\text{Scale parameter}
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
U(a,b,z):Tricomi’s confluent hypergeometric functionU(a,b,z):\text{Tricomi's confluent hypergeometric function}
Φ(x):CDF normal standard distribution\Phi\left(x\right):\text{CDF normal standard distribution}
Φ−1(x):PPF normal standard distribution\Phi^{-1}\left(x\right):\text{PPF normal standard distribution}
ϕ(x):PDF normal standard distribution\phi\left(x\right):\text{PDF normal standard distribution}
erf(x):Error function\mathrm{erf}(x):\text{Error function}
erf−1(x):Inverse of error function\mathrm{erf}^{-1}(x):\text{Inverse of error function}