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I am stuck with understanding of simple chi-squared test on proportion. We have

$$\sum \frac{(Observed_i - Expected_i)^2}{Expected_i}$$

and say that it follows chi squared distribution (sum of squared standard normals). Observed ~ $\mathcal N$ (by central limit theorem). Expected is a true parameter so it does not follow any distribution. Why this normalisation (division by Expected) leads to standardisation? After understanding this it will be clear what are the sources of the error (why this test is approximate), I would finally like to understand this.

UPDATE: I have found this question and answer by @Glen_b. However, it still sounds to me like "$F=ma$ because it is". I believe that it is not a standard normal, but why it is negatively dependent and why it converges to chi square?

UPDATE II: another nice answer by @Glen_b tells, finally, why.

Steffen Moritz
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German Demidov
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    This statistic does not approximate a $\chi^2$ distribution unless one has been careful about how the "Expected" values are computed. See http://stats.stackexchange.com/questions/16921/how-to-understand-degrees-of-freedom/17148#17148 for some remarks about this issue. – whuber Feb 05 '16 at 13:51

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