I have a basic question about the chi-squared statistic. Perhaps I am misunderstanding, or perhaps I am looking for a different test. I am mathematical to a degree but am (becoming) self-educated in statistics.
I understand how to calculate the statistic and do hypothesis testing, according to the textbooks. However, the statistic is very sensitive to the total frequency, in that, if X2(O,E) is the statistic, then increasing all the frequencies by some factor N increases the statistic by the same factor, i.e.,
X2(N*O,N*E) = N * X2(O,E)
where N*O is multiplying a vector by a scalar. Therefore, the statistic is much more sensitive to the total frequency than to the degrees of freedom. When testing goodness-of-fit for a distribution, it seems like the total frequency should NOT be the most significant factor.
This seems odd and I am wondering what I am missing, if anything.