Test Statistic in Chi-Square Test

[ych22]

Can anyone come up with an intuitive explanation or point me to a link that gives a derivation of the test statistic in chi-square test? I am having problems understanding why the particular test statistic is approximated by a Chi-Square random variable under the null hypothesis of the chi-square test. I cannot find any helpful literature online or in my textbooks too.

After all, the chi-square distribution with k degrees of freedom arises from the sum of squares of k standard normal distributions. This implies that for each bin, (observed-expected)^2 /expected ~ chi-square(1). Why?

 

[statdad]

Prepare for the incredibly non-formal discussion:
You have one linear restriction: the sum of the "(observed - expected)" values is zero. You have k statistics, with 1 restriction, hence k - 1 degrees of freedom.