# Why do the critical values of the chi-square distribution increase with degrees of freedom?

This makes no sense to me since we usually want more degrees of freedom and the t-and F-distributions reflects this by demanding smaller critical values for significance. The tables I have found only display 1-30 degrees of freedom for the Chi-square which seems odd given that t- and F-distribution tables usually show at least 100+ as that would be the standard if you have a decently large dataset to subtract number of variables and constant from the number of observations to get your degrees of freedom.

Can anyone make it clear to me how I should read such a table (just click the first Google search picture and you will see the same). Or tell me if I have completely lost it. I get that it looks different in a graph but the values makes no sense.

• The chi-squared distribution arises as the sum of squares of DF independent standard Normal variables. Since squares are nonnegative, this makes it obvious that both the average and the variance of chi-squared distributions increase with DF. Although that doesn't prove that critical values must also increase with DF, it strongly suggests it! – whuber May 19 '19 at 13:20

The $$t$$ distribution is set up so that the critical value converges to 1.96 as $$n \rightarrow \infty$$. It is for a one d.f. hypothesis. The $$F$$ distribution is similar to a $$\chi^2$$ distribution divided by its d.f. This always makes things confusing, because e.g. you can add a variable to the regression model and have $$F$$ decrease even when the variable is important. $$\chi^2$$ is such that adding information (e.g., a predictor) increases the statistic. The d.f. is the number of opportunities that the statistic had to be large. More opportunities come from adding variables or adding categories in a frequency (contingency) table. The more opportunities (d.f.) the higher the critical value you must achieve to have evidence above the noise level.