Statistics - Chi-squared Distribution


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The chi-squared distribution (chi-square or ${X^2}$ - distribution) with degrees of freedom, k is the distribution of a sum of the squares of k independent standard normal random variables. It is one of the most widely used probability distributions in statistics. It is a special case of the gamma distribution.

Chi-squared Distribution

Chi-squared distribution is widely used by statisticians to compute the following:

  • Estimation of Confidence interval for a population standard deviation of a normal distribution using a sample standard deviation.

  • To check independence of two criteria of classification of multiple qualitative variables.

  • To check the relationships between categorical variables.

  • To study the sample variance where the underlying distribution is normal.

  • To test deviations of differences between expected and observed frequencies.

  • To conduct a The chi-square test (a goodness of fit test).

Probability density function

Probability density function of Chi-Square distribution is given as:

Formula

${ f(x; k ) = } $ $ \begin {cases} \frac{x^{ \frac{k}{2} - 1} e^{-\frac{x}{2}}}{2^{\frac{k}{2}}\Gamma(\frac{k}{2})}, & \text{if $x \gt 0 $} \\[7pt] 0, & \text{if $x \le 0 $} \end{cases} $

Where −

  • ${\Gamma(\frac{k}{2})}$ = Gamma function having closed form values for integer parameter k.

  • ${x}$ = random variable.

  • ${k}$ = integer parameter.

Cumulative distribution function

Cumulative distribution function of Chi-Square distribution is given as:

Formula

${ F(x; k) = \frac{\gamma(\frac{x}{2}, \frac{k}{2})}{\Gamma(\frac{k}{2})}\\[7pt] = P (\frac{x}{2}, \frac{k}{2}) }$

Where −

  • ${\gamma(s,t)}$ = lower incomplete gamma function.

  • ${P(s,t)}$ = regularized gamma function.

  • ${x}$ = random variable.

  • ${k}$ = integer parameter.



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