3.9 Using the Chi-square Table to Find Areas and Percentiles

I show how to find percentiles and areas for the chi-square distribution using the
chi-square table.

3 thoughts on “3.9 Using the Chi-square Table to Find Areas and Percentiles”

Hi!
First of all, thank you very much for your videos! Even for a non-native english it’s really well explained.

I was wondering, when you have to find the p-value to the right (or left) of something for a given degree of freedom, is there a way to “precisely” calculate it by hand when you have no software to do it? For example for 21.375 with 19 DF, it’s between 18.34 and 27.20…

The short version is no, it is not possible. For some types of tests (e.g. the sign test based on the binomial distribution), then it is possible to calculate the exact p-value by hand. But if you’re asking about tests that involving a continuous distribution that has a table (e.g. normal distribution, t distribution, chi-square distribution, F distribution), then there isn’t a way to do it by hand. To find the area, we need to integrate the curve, and the integral is not of closed form. The functions must be integrated numerically. Conceivably one could come up with a reasonable approximation by hand, but it would take a fair bit of effort.

Some folks recommend using linear interpolation to get close to the true p-value. (e.g. If your chi-square value is 80% of the way between two of the values in the table, take the area to be 80% of the way between the two table areas.) But this is not an exact method. I recommend giving an interval of values from the table, and if we need to be precise, go to software.

Thanks a lot for your quick and detailed answer!
I was asking this because I have an exam tomorrow and I am not allowed to use anything but my small calculator.
So as you said I will give an interval of values and if really needed I will use linear interpolation to get closer to correct result.

Hi!

First of all, thank you very much for your videos! Even for a non-native english it’s really well explained.

I was wondering, when you have to find the p-value to the right (or left) of something for a given degree of freedom, is there a way to “precisely” calculate it by hand when you have no software to do it? For example for 21.375 with 19 DF, it’s between 18.34 and 27.20…

In advance, thanks for your answer!

Thanks for the compliment Ian!

The short version is no, it is not possible. For some types of tests (e.g. the sign test based on the binomial distribution), then it is possible to calculate the exact p-value by hand. But if you’re asking about tests that involving a continuous distribution that has a table (e.g. normal distribution, t distribution, chi-square distribution, F distribution), then there isn’t a way to do it by hand. To find the area, we need to integrate the curve, and the integral is not of closed form. The functions must be integrated numerically. Conceivably one could come up with a reasonable approximation by hand, but it would take a fair bit of effort.

Some folks recommend using linear interpolation to get close to the true p-value. (e.g. If your chi-square value is 80% of the way between two of the values in the table, take the area to be 80% of the way between the two table areas.) But this is not an exact method. I recommend giving an interval of values from the table, and if we need to be precise, go to software.

Cheers.

Jeremy

Thanks a lot for your quick and detailed answer!

I was asking this because I have an exam tomorrow and I am not allowed to use anything but my small calculator.

So as you said I will give an interval of values and if really needed I will use linear interpolation to get closer to correct result.

Thank you again!