Editing Pearson's chi-squared test (section)

==Problems==
The approximation to the chi-squared distribution breaks down if expected frequencies are too low. It will normally be acceptable so long as no more than 20% of the events have expected frequencies below 5. Where there is only 1 degree of freedom, the approximation is not reliable if expected frequencies are below 10. In this case, a better approximation can be obtained by reducing the absolute value of each difference between observed and expected frequencies by 0.5 before squaring; this is called [[Yates's correction for continuity]].

In cases where the expected value, E, is found to be small (indicating a small underlying population probability, and/or a small number of observations), the normal approximation of the multinomial distribution can fail, and in such cases it is found to be more appropriate to use the [[G-test]], a [[likelihood-ratio test|likelihood ratio]]-based test statistic.  When the total sample size is small, it is necessary to use an appropriate exact test, typically either the [[binomial test]] or, for [[contingency tables]], [[Fisher's exact test]]. This test uses the conditional distribution of the test statistic given the marginal totals, and thus assumes that the margins were determined before the study; alternatives such as [[Boschloo's test]] which do not make this assumption are [[uniformly more powerful]].

It can be shown that the <math>\chi^2</math> test is a low order approximation of the <math>\Psi</math> test.<ref>{{cite book |author-link=Edwin Thompson Jaynes |last=Jaynes |first=E.T. |year=2003 |title=Probability Theory: The Logic of Science |publisher=C. University Press |isbn=978-0-521-59271-0 |page=298 |url=http://www-biba.inrialpes.fr/Jaynes/prob.html}} (''Link is to a fragmentary edition of March 1996''.)</ref> The above reasons for the above issues become apparent when the higher order terms are investigated.