Assumption of normality in a sample size of 3 I was wondering whether it is appropriate to take a subjective impression of a data set of 3 observations, say (31, 36 and 42) and say that, as they are fairly symmetrical, there is no evidence of non-normality?
And thus, might a two-sample t-test be appropriate to analyse this data over a Mann-Whitney test?
Also, if the standard deviation of this dataset is somewhere between 5-7, would it be fair to say that this is relatively small?
Many thanks in advance, I look forward to your insights.
 A: Honestly, with a sample size of three, there's really not much you can say at all. Those numbers are more likely to come from a normal distribution than, say, $1$, $2$, and $1000$ are, but there are plenty of non-normal distributions that can produce those numbers. 
With regards to a standard deviation being relatively small, that entirely depends on the underlying data. A standard deviation of 5-7 for the number of drinks consumed in a night is big, a standard deviation of 5-7 for total dollar earnings in a year is tiny. 
A: You don't say what null hypothesis you're testing.
If you want to know if your observations indicate that the population mean or median is above $20,$ then a one-sample, one-sided Wilcoxon signed-rank test
cannot give a P-value below $1/8 = 0.125.$
wilcox.test(x, mu=20, alt="g")$p.val
[1] 0.125

By contrast, if you have prior experience with the kind of mechanism
or population that produced these must be normally distributed and with reasonably small variance, then
a one-sided, one-sample t-test of $H_0:  \mu=20$ vs. $H_a: \mu > 20$
will give a highly significant result with a P-value of about $0.018 < 0.05.$
x = c(36, 31, 42)
t.test(x, mu=20, alt="greater")

        One Sample t-test

data:  x
t = 5.1366, df = 2, p-value = 0.01794
alternative hypothesis: true mean is greater than 20
95 percent confidence interval:
 27.04837      Inf
sample estimates:
mean of x 
 36.33333 

However, depending on the plausibility of your assumptions of
normality and small or moderate variance, you may have difficulty
persuading anyone else that the population mean truly exceeds $20.$
If it's an important issue for you and you are the one doing the
experimental work, then you may feel further experimentation is
worthwhile---even if others are sceptical.
A: Samples of size 3 cannot be easily used to visually assess whether they might be consistent with having come from a normal population.

*

*Tiny samples actually from a normal distribution may be highly asymmetric.


*Symmetry doesn't imply normality. In some cases population symmetry would be very helpful, in others it's far from the most important consideration. For example, in a two-sample test, where you're taking differences, if the two sample means have the same distribution then the  difference in means is symmetric, so skewness of the original variable may not be at all consequential for the skewness of the numerator. Further, symmetry of the parent distribution would be insufficient for the properties of the denominator and for the dependence between the two.


*Very small samples from  quite non-normal distributions can look perfectly consistent with a normal. You will have little power.
I contend that with three observations your assessment of skewness is based on the relative length of the longer and shorter intervals between adjacent observations (in that almost any reasonable measure will be a monotonic function of that ratio).
If the two intervals between adjacent values ($x_{(3)}-x_{(2)}$ and $x_{(2)}-x_{(1)}$) are similar in size you'll "see" symmetry and if they're very different in size you'd interpret that as skewness.
As a measure of symmetry, then, we could take the ratio of the shorter interval to the longer, which will range from 0 - very different sized intervals, from which you'd see asymmetry - to 1 - identical intervals, from which you'd see symmetry.
I could have taken the reciprocal of that (the ratio of longer interval to shorter - which would range from 1 upward) as a measure of skewness, but this has a very highly skewed distribution and isn't so convenient to work with. It's easier to see what's going on with its inverse which has support on the unit interval.
For the purpose of illustration, let's look at three parent distributions; one is skewed (its density is monotonic decreasing, so it's unambiguously skewed), one is symmetric but heavy-tailed (has infinite kurtosis), and one is normal.
I drew 100,000 samples of size 3 from each distribution, and calculated the ratio of the shorter interval to the longer for each sample, then plotted histograms of these 10000 symmetry measures (I have not identified which is from which distribution, merely numbering them):

As we see, the distributions of this measure is extremely hard to distinguish for those chosen distributions. Imagine you had a sample of size three where the longer interval was 5 times as long as the shorter interval (S/L = 0.2) -- pretty skew, right? Which distribution did it come from? I certainly couldn't do much better than guess at random, and two of the parent distributions these symmetry measures came from are symmetric!
Now let's imagine that the longer interval was about 11% longer than the shorter one (S/L = 0.9) -- very nearly symmetric right? But which of those three distributions did it come from? If I hadn't made the plots I could not hope to know with any degree of confidence which was from the skewed parent.
With extremely skew or extremely heavy-tailed distributions it's somewhat easier to tell the distributions of S/L apart from that for a normal, but it's still not much help, because the distributions aren't typically different enough to reliably tell from a single sample of size 3. If you see a long interval ten or twenty times as long as the short one, you still don't have a reliable indicator of whether it came from a normal distribution or not -- there's still a good chance of seeing a very skewed sample with a normal and of seeing a nearly-symmetric sample with something that very much isn't normal.
I'll leave aside my usual advice about not using the sample to choose your specific hypothesis and end with a simple warning:
Do not use a Mann-Whitney test nor signed rank test with n=3
Because of this difficulty of assessing normality, you will no doubt see people recommending a (Wilcoxon-)Mann-Whitney test or signed rank test as fits the circumstances; such advice is very common but simply disastrous with sample sizes of 3. Besides the obvious lack of power you'd experience with any location test, you lack available significance levels. With two samples of size 3, the smallest possible two sided p-value you can observe is $0.1$. That is, if you're trying to reject p values $\leq 0.05$, you'll never see one, no matter how different the samples are).  Unless you're prepared to do a test at the 10% level, you're wasting your time.
The situation is even worse with a signed rank test.

Spoiler - no doubt some people will wish to know the details of the distributions involved. To that end:

 One of the distributions is a folded normal($\mu=1,\sigma=1$) parameterized the same way as the Wikipedia page; this one has a monotonic decreasing pdf. One is standard normal. One is a $t$ distribution with 3 degrees of freedom. You can always simulate them to see, but even when you know which is which, regarded as members of location-scale families in each case (so only shape information matters for distinguishing them), they're very hard to tell apart with just three observations.

A: With classic statistics, I would still do the usual test and arrive at large confidence intervals, for example, which still may be informative in postulating say a prior distribution for any subsequent low data exercise. [EDIT] The rationale is also that it offers a parametric opinion to contrast to the nonparametric/bootstrap path discussed below. 
One could also assume that each of the three numbers are equally likely to recur in the future (as to whether this is rationale depends on the nature of the data).
If so, generate all 27 sets of data (3x3x3). Compute statistics of interest across all sets. Tabulate the statistics into an empirical distribution which may again have some probabilistic value for decision making. [EDIT] What I just described is called a nonparametric bootstrap technique, which usually has a problem in that even small data sets can produce an extremely large number of data sets to examine, but not for the current case.
A: As Norvia points out, there you cannot make this assumption with a sample size of 3. I would not assume normal distribution. The mann-whitney U test is the better choice. 
