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Which statistics would be ideal for comparing two groups with three unpaired data points in each group? With just three data points, I suppose a t-test would not be ideal.

Tim
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Arindam Ghosh
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    Three points is very little data. You can formally run a t-test, but I would find a plot of the six data points in two groups of three more enlightening than a formal test. – Stephan Kolassa Oct 21 '22 at 12:05

2 Answers2

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You can use the IOTT. That's the interocular trauma test. It hits you between the eyes.

As Stephen and Tim point out, there's no minimum sample size for a t-test (or some others) to be valid. But will it be useful? Let's compare professional basketball players to the general population. According to lines.com, the average NBA player is 6'6". I couldn't find the SD, but I guessed it at 4 inches. The average American man is about 5'10" and I used the same SD.

set.seed(1234) #Set a seed

BB <- rnorm(3, 78, 4)
GenPop <- rnorm(3, 70, 4)


t.test(BB, GenPop)

And the difference is not statistically significant (t = 2.19, df = 4, p = 0.10). Now, maybe the SD is actually smaller, but still, if you can't tell basketball players from the average adult man ... You can't tell much.

And the nonparametric tests generally have less power.

Peter Flom
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As discussed in the Can you do statistics with 4 data points? thread, you can do statistics with such small samples, but you would usually end up with high-uncertainty estimates. For hypothesis tests, you would need to observe very strong differences to achieve the significance. This means that the hypothesis tests would mostly confirm the obvious differences that would be already visible to the naked eye from the raw data.

As Stephen Kolassa suggested, you can just plot the data instead. Below you can see an example of such a plot. What it shows is that there might be some difference between the groups, though it is not clear and it could be just by chance.

Plot showing two groups: blonde and dark-haired people, scattered around the x-axis. Blonde people are concentrated around 3.5-5.5 and dark-haired people around 4.5-6.5 but there is an overlap between the three groups.

If you run the $t$-test for the same data, it'll give you a similar answer.

        Two Sample t-test

data:  x and y
t = 1.0842, df = 4, p-value = 0.3393
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -1.200519  2.738769
sample estimates:
mean of x mean of y 
 5.389796  4.620671

(In the example, the groups do come from different distributions, but the sample size is just too small to tell.)

See also Is there a minimum sample size required for the t-test to be valid? and How to choose between t-test or non-parametric test e.g. Wilcoxon in small samples if you'd insist on using hypothesis test.

Tim
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