Chi-Squared (χ²) Distribution

The distribution of the sum of squared standard normal deviates.

The "Goodness of Fit" Distribution

The Chi-Squared (χ²) distribution is a continuous probability distribution that is widely used in hypothesis testing. It arises as the distribution of a sum of squared independent standard normal random variables.

In finance and econometrics, it is the backbone of the Chi-Squared test, which is used to test the goodness of fit of a model, check for independence between categorical variables, and compare variances. For instance, a risk manager might use it to test if the observed frequency of portfolio losses matches the frequency predicted by their risk model.

The Formula

The probability density function (PDF) is defined by one parameter: the degrees of freedom (

k

f(x; k) = \frac{1}{2^{k/2}\Gamma(k/2)} x^{k/2-1} e^{-x/2}

$x$ is the variable (must be ≥ 0).
$k$ represents the degrees of freedom.
$\Gamma(k/2)$ is the Gamma function.

Interactive χ² Distribution

Adjust the degrees of freedom to see how the shape of the distribution changes. Notice how it becomes more symmetric and bell-shaped as the degrees of freedom increase.

Degrees of Freedom (k): 5