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

Question

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.

Answer 1

Noting the major blunder in the paper you reference of discarding much information about BMI from the outset by dichotomizing its calculated values, a prudent reader should be alert to the possibility of minor blunders. The labelling of the "Difference" column in Table 1 as "test for significance chi squared" appears to be one such. Elsewhere it is stated that "Differences between groups were assessed using independent sample t tests or Mann Whitney U tests for continuous variables, and chi squared tests for categorical variables"^†. The pooled-variance estimate of the standard error for the difference in mean ages of the two groups is

$$\sqrt{\frac{(n_1-1)s^2_1+(n_2-1)s^2_2}{n_1+n_2-2}\cdot\left(\frac{1}{n_1}+\frac{1}{n_2}\right)}$$

$$=\sqrt{\frac{(196-1)5.1^2+(386-1)5.2^2}{196+386-2}\cdot\left(\frac{1}{196}+\frac{1}{386}\right)} \ \text{years}$$

$$= 0.4532 \ \text{years}$$

so the cited p-value implies a difference in sample means of 0.1096 years if a pooled-variance, two-sided t-test were used, in agreement with the tabulated ages of 30.0 & 29.9 years for each group. Where medians rather than means are reported, in Table 3, the use of the Mann–Whitney U test is clearly indicated in the footnotes.

† The chi-square test the authors refer to is Pearson's chi-square test for association between categorical variables.