How to interpret a too small chi-square $\chi^2$ value? I use the Chi-square (i.e., $\chi^2$) goodness-of-fit test to measure the distance between chunks of data and a theoretical distribution. 
For most of the data being tested the results make sense.
I got from time to time unusual results, which correspond to very small values (i.e., between 10 and 100). The sample size is not the problem, there is at least 5 elements for each symbol.
After a careful analysis of the observed data, the frequency of each symbol in the data being tested is very close to the theoretical distribution as the $\chi^2$ value suggest.
But it seems to be too perfect for me. It is unlikely according to the chi-square distribution function that a given value would occur (i.e., $z_{0.0001}$).
Should I reject the null hypothesis because the data is too good to be true ?
I did not find clear explanations or I missed something. 
How to interpret unlikely values in the left tail of the chi-square distribution ?
Thanks in advance.
Best regards,
John
 A: You don't.  In a goodness of fit test the left tail part of the null hypothesis condition.  It's a one-sided test.
A corollary:  In a goodness of fit test, you properly use a one sided interval for the null hypothesis.  If your confidence level is 95%, the null hypothesis is $0 \le \chi^2 \le \chi^2_{0.95}$, not$\chi^2_{0.025} \le \chi^2 \le \chi^2_{0.975}$.
A: Usually chi-squared tests of goodness-of-fit are one-sided (because
the squaring involved in computing the test statistic gives both negative and positive differences the effect of increasing the statistic). Thus one rejects
the null hypothesis (data fit the model) if the test statistic is larger
than some critical value. (P-value is small.) The test statistic is never negative.
However, these rules do not apply when vetting a pseudorandom number
generator to be used in probability simulation because a fit that is "too good to be true" (test statistic near 0, P-value near $1)$ indicates the generator is giving nonrandom values as much as does a large
value of the test statistic.
There are also cases, such as the famous one in @Henry's Comment, in which
data fit a model "too closely" and the procedure of data collection or
tabulation comes into doubt. If you ask someone to check whether a die
is fair by rolling it 600 times, and the answer comes back that each of
the six faces showed exactly 100 times, you would be entitled to wonder
whether the 600-roll experiment was done faithfully.
"When the P-value is very small, doubt the null hypothesis; when the P-value
is very near $1$, doubt the model or the data collection."
Note: Interpretation of goodness-of-fit (GOF) tests is often incorrect. In a two-sample
test whether Drug A is better than a placebo, the experimenter may be
wishing for a rejection of $H_0$ (drug has no effect). However, in GOF tests
the experimenter is often hoping not to reject $H_0.$ This means that it is
especially important to assess the power of a GOF test. 
