Let's say you want to study the association between age (continuous) and exercise frequency (also continuous). If you do a cross-tabulation of these two you will get a cross-table with many, many cells - which will be impossible to interpret. For this reason, you will do a correlation or a regression instead. But let's say you for some (weird?) reason yo choose re-code your two variables into two categorical variables: age (18-30, 31-40, 41-50, and 51 or more) and exercise frequency (0-1 per week, 2-3 per week, and 4 or more per week). Now you can do a cross-tabulation and get a cross-table with 12 cells (4 x 3), and now a chi-square test makes sense if the aim is to find out if your two (new) categorical variables are related or not.

In case where the sample size is too small to affect normal distribution, chi square test is appropriate. Chi squared test is not limited to binary data, continuous data from small sample size is tested by chi square. Recall that the referenced critical value in the chi square table is the T value. As a general rule, large sample produces normal distribution (T test); smaller sample that could not reference by normal distribution is tested by chi square (chi sqr. test) and for two non-normal samples, use F test.

Chi-square test is designed to analyse categorical data. Hence, as per the example given by Chister, as long as such continuous data gets divided into categories, then an analysis using a chi-square test can be conducted. However, it is crucial to note that the performance/power of a chi-square test is largely anchored on the way in which the data is binned or divided.

Dear Arjun Paul

Sorry, but there's one thing missing that makes your question confusing! Are you talking about comparing two samples? If yes, then do you have a small or large sample? If you answer these 2 simple questions, then, of course, there are certain answers available.

But, what about the situation when we have one normal and the other non-normal (or skewed) sample?

And, given the above situation besides having continuous (or categorical) variable, like age, education, household size, income, etc., with 2 small samples, one normal and the other non-normal, can't we apply non-parametric Mann-Whitney U test and/or Kolmogorov-Smirnov test?

Good luck

If you convert the data to classes or normal and abnormal, then you can perform chi square on the new data.

Christer Thrane makes a point. . Can you reasonably recode the two variables to something where a contingency table test is reasonable. then you might be able to do a contingency table Chi square test. Under similar conditions you might also be able to do something with some type of logistic or similar type of regression depending on how you recode. I think the Christer Thrane approach is probably the best suggestion so far. Best, D. Booth However, you may still need to convince a reviewer.

Why?

If you mean can you take your F or t or z and transform it into a chi^2, yes, but why.

If you mean can you compare the likelihoods of two models with continuous data with a Chi^2 distribution, yes, and this is done often.

If you mean is it wise as a general practice to bin data and not take into account either the distance between points or the ordering, no. Is there a law against it, not even here in Las Vegas. But why?

I agree with Paul Louangrath

If you can change the continuous variable into categorical, then you can compare qualitative variables.

Example, you want to determine an association between BMI and level of activity.

Since BMI is given as continuous variable, we can not to Chi-square. We can create categories depending upon the value of BMI (Normal weight, underweight, obese etc.) to determine association between categories of BMI and level of acivity.

Hope this helps!!