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Chi Square distribution

What it represents

The chi-square distribution is the sum of squares of df independent standard normal variables. Its positive skew decreases with degrees of freedom, which makes it useful for variability and for normal approximations.

Historical clue

Karl Pearson introduced it while developing his goodness-of-fit statistic at the beginning of the twentieth century. The chi-square notation is tied to the Greek letter used for the sum of squares.

Relationships that clarify its use

It is a special case of Gamma and, for two independent scaled chi-square variables, leads to the F distribution. In a normal sample, the sum of squared deviations from the mean separates from the sample mean and follows a chi-square law.

Data examples

  • intervals and tests for the variance of a normal population
  • independence and goodness-of-fit tests for contingency-table counts

Modelling warning

Do not treat df as an arbitrary fitted knob when it comes from a design: changing it changes the inferential meaning as well as the curve.

Squared deviations create a new language

Each standard Normal variable contributes a signed deviation. Squaring removes the sign and leaves a positive contribution; adding several gives a Chi Square variable. This construction explains why degrees of freedom count independent dimensions and why the curve becomes less skewed as dimensions are added.

Pearson made this law the foundation of his 1900 goodness-of-fit test. It also appears when estimating the variance of Normal data and as a Gamma case whose shape is half the degrees of freedom. Pearson’s approximation needs adequate expected frequencies and defensibly chosen categories. A statistic computed without those conditions may look precise while being compared with the wrong reference law.

Decision guide

A good candidate when: the quantity is a sum of standardized squares or follows from a theoretical variance-and-degrees-of-freedom construction.

Compare it with: Gamma for positive data without that construction. Do not choose Chi-square merely for skewness: its parameters and scale are imposed by the sum-of-squares mechanism.

References

  • SciPy reference: scipy.stats.chi2 — definition and parameterization
  • Johnson, N. L., Kotz, S. & Balakrishnan, N. (1994). Continuous Univariate Distributions, 2nd ed., Vol. 1. Wiley.
  • Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed. Wiley.
  • Pearson, K. (1900). On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 50(302), 157–175.

Chi Square Distribution: equations and calculator

Distribution defintion

X∼χ2(df)X\sim\mathrm{\chi^{2}}\left(\text{df}\right)

Distribution domain

x∈(0,∞)x\in\left(0,\infty\right)

Parameters domain and parameters constraints

df∈N+\text{df}\in\mathbb{N}^{+}

Cumulative distribution function

FX(x)=γ(df2, x2)Γ(df2)=P(df2, x2)F_{X}\left(x\right)=\frac{\gamma(\frac{\text{df}}{2},\,\frac{x}{2})}{\Gamma(\frac{\text{df}}{2})}=\text{P}\left(\frac{\text{df}}{2},\,\frac{x}{2}\right)

Probability density function

fX(x)=12df/2Γ(df/2) xdf/2−1e−x/2f_{X}\left(x\right)=\frac{1}{2^{\text{df}/2}\Gamma(\text{df}/2)}\,x^{\text{df}/2-1} e^{-x/2}

Percent point function/Sample

FX−1(u)=2P−1(df2,u)F^{-1}_{X}\left(u\right)=2\text{P}^{-1}\left(\frac{\text{df}}{2},u\right)

Non-central parametric moments

μk′=E[Xk]=∫0∞xkfX(x)dx=df(df+2)⋯(df+2k−2)=2kΓ(k+df2)Γ(df2)\mu'_{k}=E[X^k]=\int_{0}^{\infty }x^{k}f_{X}\left(x\right)dx= \text{df}(\text{df}+2)\cdots(\text{df}+2k-2)=2^k\frac{\Gamma\left(k+\frac{\text{df}}{2}\right)}{\Gamma\left(\frac{\text{df}}{2}\right)}

Parametric mean

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

Parametric variance

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

Parametric skewness

Skewness(X)=μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5=8df\mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}}=\sqrt{\frac{8}{\text{df}}}

Parametric kurtosis

Kurtosis(X)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2=3+12df\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+\frac{12}{\text{df}}

Parametric median

Median(X)=2P−1(df2,12)\mathrm{Median}(X)=2\text{P}^{-1}\left(\frac{\text{df}}{2},\frac{1}{2}\right)

Parametric mode

Mode(X)=max(df−2,0)\mathrm{Mode}(X)=\text{max}(\text{df}-2,0)

Additional information and definitions

u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
P(a,x)=γ(a,x)Γ(a):Regularized lower incomplete gamma function\text{P}\left(a,x\right)=\frac{\gamma(a,x)}{\Gamma(a)}:\text{Regularized lower incomplete gamma function}
P−1(a,u):Inverse of regularized lower incomplete gamma function\text{P}^{-1}\left(a,u\right):\text{Inverse of regularized lower incomplete gamma function}
γ(a,x):Lower incomplete gamma function\gamma\left(a,x\right):\text{Lower incomplete gamma function}
Γ(x):Gamma function\Gamma\left(x\right):\text{Gamma function}