Why does Chi-square formula have two different ones?

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SUMMARY

The discussion clarifies the existence of two distinct chi-square formulas: one defined as the sum of squared standard-normal random variables divided by variance, and the other as the sum of squared differences between observed and expected frequencies divided by the expectation value. The first formula is a theoretical definition of the chi-squared distribution, while the second is specifically applied for goodness-of-fit tests. For hypothesis testing, the second formula is recommended to determine whether to reject or accept the null hypothesis.

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  • Understanding of chi-squared distribution
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  • Knowledge of observed versus expected frequencies
  • Basic statistics, particularly regarding variance and standard-normal random variables
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EigenCake
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One is:
http://en.wikipedia.org/wiki/Chi-squared_distribution

where at the bottom of the page, chi-square = sum of something devided by variance.

However, here:
http://www.napce.org/documents/research-design-yount/23_chisq_4th.pdf

where the chi-square formula is that the sum of the same thing devided by expectation value.

Which formula is right then?
 
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The first reference is the definition of Chi-squared :- the distribution of a RV formed from the sum of squared standard-normal random variables.

The second reference is an application of the above distribution to the case of expect versus observed frequencies, where (O-E)/sqrt(E) is asymptotically standard-normal and hence (O-E)^2/E is asymptotically Chi-squared.
 
Thanks!

It seems that for goodness of fit I should use the second formula to either reject or accept the null hypothesis, rather than use the first formula.
 

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