Definition:Chi-Squared Test
Definition
Goodness of Fit
The chi-squared test for goodness of fit is a test of goodness of fit of observations to some theoretical probability distribution.
Let $n \in \Z_{>0}$.
Let a value $x_i$ for $i \in \set {1, 2, \ldots, n}$ be expected to occur $E_i$ times.
Let $x_i$ actually occur $O_i$ times.
Then the statistic:
- $\ds \chi^2 = \sum_i \dfrac {\paren {O_i - E_i}^2} {E_i}$
has a $\chi$-squared distribution with $n - p$ degrees of freedom where $p$ is the number of distribution parameters estimated from the data and used to compute the $E_i$.
Significantly high values of $\chi^2$ lead to the rejection of the hypothesised distribution.
Lack of Association
Let $C$ be a contingency table with $r$ rows and $c$ columns.
The expected number in an arbitrary cells can be calculated from the fixed marginal totals.
A statistic in the form $\chi^2$ as defined in $\chi$-squared test for goodness of fit can be calculated by taking all the observed and expected numbers in each cell and summing over all cells.
The number of degrees of freedom is $\paren {r - 1} \paren {c - 1}$
Large values of $\chi^2$ indicate rejection of the hypothesis that the numbers in the cells are independent.
Examples
Also see
- Results about the $\chi$-squared test can be found here.
Sources
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