Performing a t-test with discrete (currency) data

Question

I want to perform a 2 sample t-test assuming unequal variances, however my variable is currency. Currency is discrete, however when checking the assumptions of the t-test, I see that the data should be continuous.

Technically, they aren't continuous, but I guess it's closer to a ratio scale (maybe interval). Is this assumption violated? What else should I check?

What do they look like (within the groups)? Are they vaguely normal looking? — gung - Reinstate Monica, Jun 12 '16 at 15:54
What do you precisely mean by your "variable is currency." As in, your variable is measured in dollars and there cannot be fractions of a cent (discrete units are $1/100$ of a dollar)? And you're trying to compare the mean of two samples? — Matthew Gunn, Jun 12 '16 at 17:32
Yes, dollars and cents only. No fractional values less than 1 cent. — Ryan Wildry, Jun 12 '16 at 17:34
Unless the prices are 5 cents, 2 cents, etc..., that's getting a bit pedantic. Do what @gung suggests to check normality, and if it looks good, you're fine. If it's somewhat off with right skew, another thing to check might be log normality: that comes up a lot in economic situations. — Matthew Gunn, Jun 12 '16 at 17:58

gung - Reinstate Monica · Accepted Answer · 2018-05-24T16:39:00.647

Discrete isn't continuous, so technically the assumption of the t-test is not met, and that's that. However, the t-test is fairly robust and having $N=48$ with equal groups is a decent sample, so it might be OK. After all, in practice all data are discrete at some level because we don't record data to infinite decimal places.

I would look (i.e., plot; cf. R - QQPlot: how to see whether data are normally distributed & How to interpret a QQ plot) at your data to see if they appear reasonably normal and meet the other assumptions (e.g., homoscedasticity). I would not run formal tests for normality (cf. Is normality testing 'essentially useless'?). If they look 'close enough', you're probably OK.

On the other hand, it is generally considered sub-optimal to follow the procedure 'check assumptions, if OK, run test'. If you think this might be a problem, you can just go straight to a more robust option. (Cf. How to choose between t-test or non-parametric test e.g. Wilcoxon in small samples.)

Worth pointing out too that with more data, he could lean on asymptotic arguments. Under certain regularity conditions, the sample mean $\frac{1}{n}\sum_i x_i$ would converge to a normally distributed random variable even if $x$ were discrete. — Matthew Gunn, Jun 12 '16 at 17:44

Performing a t-test with discrete (currency) data

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