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EXAMPLE

The Chi-square Statistic is a sum of differences between observed and Expected Outcome frequencies, each squared and divided by the expectation:

: \chi^2 = \sum { rac{(O - E)}{E}^2}
where:
O

E


The resulting value can be compared to the Chi-square Distribution to determine the goodness of fit.

In order to determine the degrees of Freedom of the Chi-Squared distribution, one takes the total number of observed frequencies and subtracts one. I.E. If there are 8 different frequencies, one would compare to a Chi-Squared with Seven degrees of Freedom.

There is also a reduced chi-squared statistic, which is weighted based on measurement error.
: \chi^2 = \sum { rac{(O - E)^2}{\sigma^2}}
where \sigma^2 is the Variance of the observation. http://www.google.com/url?sa=t&ct=res&cd=5&url=http%3A%2F%2Fwww.sns.gov%2Fworkshops%2Fsns_hfir_users%2Fposters%2FLaub_Chi-Square_Data_Fitting.pdf&ei=cPxBRqCHKJa4iwH739SyDg&usg=AFrqEzeYr2RT39isRAal15gfXFXuWb032g&sig2=fUdIIzi-kdrfQzEe7vxfvQ


BINOMIAL CASE


A binomial experiment is a sequence of independent trials in which the trials can result in one of two outcomes, success or failure. There are n trials each with probability of success, denoted by p.
Provided that npi \geq 5 for every i, where i=1,2,...,k).

where,

\chi^2 = \sum_{i=1}^{k} { rac{(N_i - np_i)^2}{np_i}} = \sum_{all\,cells}^{} { rac{(Observed - Expected)^2}{Expected}}.

This has approximately a chi-squared distribution with k-1 df. The fact that df= k - 1 is a consequence of the restriction \sum N_i=n. We know there are k observed cell counts, however, once any k - 1 are known, the remaining one is uniquely determined. Basically, one can say, there are only k - 1 freely determined cell counts, thus df= k - 1.




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