# $t$ test vs $\chi^{2}$ once sample is split into two groups?

I would appreciate some guidance regarding the use of a t test vs $\chi^{2}$.

I am looking at some small set of demographic data (n only 13). I have divided up the sample into two sets, those with Body Mass Index (BMI) >= 25 (n = 5) and those with BMI < 25 (n = 8) and computed the means and standard deviations for the various attributes/data points, these are all numeric values (i.e, weight, height, etc).

If I want to determine if the respective resulting means for both groups are statistically different from each other I would have used a t test since these are quantitative values. However, a related similar paper I'm reading about this uses a $\chi^{2}$ test to assess this in table 1 (http://link.springer.com/article/10.1186/1471-2393-13-115/fulltext.html#Tab1). Is this a correct thing to do? Is this because by splitting the sample into two, you now have two categories, hence the application of the $\chi^{2}$? This seems incorrect. I am trying to summarize our data using R, for now I'm concerned about the numerical data (I'll have to summarize the categorical answers too).

My understanding is that I would use the t test for numerical data, and $\chi^{2}$ for categorical. It's been a while since I've had stats, but I'm reviewing, so any guidance would be appreciated.

(Also, it seems to be a consensus from reading messages here that a test of normality for the data would not make much sense with such small n, comments?)

I.e., the question is about the appropriateness of the test as used here, not the study/design itself.

• Don't dichotomize body mass index unless you've good reason to think that people with a BMI of 25.1 are going to have the same distributions of other variables as people with a BMI of 40; & different ones from those with BMIs of 15 to 24.9. See here. – Scortchi - Reinstate Monica Jul 30 '14 at 16:37
• @Scortchi Thanks for the link. The reason I'm splitting the sample is because I'm interested in the difference between the two groups of people in terms of physical activity level, nutrition etc. and hope to see if there are differences based on their BMI classification (>= 25 overweight/obese vs < 25 "normal"/underweight). I have collected this information and computed their means - I now want to know if those differences are statistically significant between these two groups. I would have expected to use a t-test for this, rather than a chi-squared test as the similar paper does. – Levon Jul 30 '14 at 16:55
• If you think biological differences are better modelled by people's BMI classification rather than by their BMI, then go ahead. And do read Dr Harrell's page - he's a medical statistician. – Scortchi - Reinstate Monica Jul 30 '14 at 17:57
• The table in the article you cite reports an N of 582 and you have an N of 13. With your small sample size, you cannot afford any loss of statistical power. The need to maximize power with a very small N argues for going with the t test approach. If you have many various attributes, you might worry about (or at least consider) experiment-wise alpha. – Joel W. Jul 31 '14 at 0:03
• Are you interested in whether there is a relationship between BMI and each of the many attributes? If so, have you considered calculating a correlation coefficient between BMI and each? Also, can you gather more data? – Joel W. Jul 31 '14 at 11:37

$$\sqrt{\frac{(n_1-1)s^2_1+(n_2-1)s^2_2}{n_1+n_2-2}\cdot\left(\frac{1}{n_1}+\frac{1}{n_2}\right)}$$
$$=\sqrt{\frac{(196-1)5.1^2+(386-1)5.2^2}{196+386-2}\cdot\left(\frac{1}{196}+\frac{1}{386}\right)} \ \text{years}$$
$$= 0.4532 \ \text{years}$$