# If a sample is normally distributed, is its population always normally distributed?

I know this is correct: "If a population is normally distributed, a sample randomly chosen from it must be normally distributed." For instance, If the sizes of all apples of my farm are normally distributed, 50 apples chose from it should be normally distributed unless randomness was violated.

But, is the following as well correct? "If a sample randomly chosen from a certain population is normally distributed, the population must be normally distributed." That is, can I believe that its population is normally distributed, if I find a certain sample is normally distributed?

• Just think about it: what can it mean to say that a sample is normally distributed? All you have to work with is a finite set of numbers. Aug 1, 2022 at 0:16

I'll be talking somewhat loosely here, but hopefully this can give a way of thinking about it.

Once you observe it, a sample is a collection of numbers, not a collection of random variables (with corresponding distribution(s)).

To the extent that you can say a sample 'has a distribution', its empirical distribution function is discrete, it's a step function. The random variables that the sample values are realizations of, of course could have some continuous distribution and those may well be seen as being independent and identically distributed. But in the limit as the sample size of a simple random sample goes to $$\infty$$, the ecdf does approach the population distribution you were sampling from.

"If a population is normally distributed, a sample randomly chosen from it must be normally distributed."

The $$n$$ variables making up a simple random sample of some population distribution (/data generating process) each have that population distribution; $$X_1\sim F$$, $$X_2\sim F$$,..., $$X_n\sim F$$. Once you realize these random variables (actually observe their values), they're each just a number.

"If a sample randomly chosen from a certain population is normally distributed, the population must be normally distributed."

If your simple random sample is the collection of variates $$X_1, X_2, ..., X_n$$, then each of them has the same distribution as the population; so if you know the distribution of the variable $$X_k$$, you know the population distribution; they're the same object.

Once you realize that set of random variables, and literally have the numbers $$x_1, x_2, ..., x_n$$, even if you make the assumption that they're an ideal case of a simple random sample of some population, you simply cannot determine what distribution they came from.

You can be quite sure that they didn't come from some distributions. e.g. if you're observing necessarily positive quantities (like weights of apples), you can be absolutely certain -- even before you see any of the numbers -- that they're not a sample from an actual normal distribution.

Indeed the chance that real data actually represent a simple random sample from any simple-form distribution is fantastically unlikely (that very specific mathematical shape... exactly?).

Not that this is important in practice; distributions are abstractions/idealizations -- literally 'models' of the thing they're modelling, not the thing itself. The question is generally not 'is this model literally the truth' but more 'is this model useful for our present purpose'.

So it may make perfect sense to model apple weights by a normal distribution, but that's not the real thing; the normal distribution is a convenient abstraction, a tool.

That is, can I believe that its population is normally distributed, if I find a certain sample is normally distributed?

You literally can't "find a certain sample is normally distributed".

I presume you're hinting at having tested normality, but if so, you're misunderstanding what that can tell you.

Note first that the actual null hypothesis in that case is that the population is normally distributed. Not about the sample; you're seeing if the sample is inconsistent with that hypothesis about the population.

If you test normality, you may well fail to reject normality, but you would also fail to reject an infinite number of non-normal distributions at the same time, and at least some of those would have a better chance to produce your sample* than whatever distribution you decide to test for.

* or a higher chance to produce a set of values within some small $$\delta x$$ of each observed value if you want an event with an actual non-zero probability, rather than comparing relative likelihoods say.

• (+1) I think this question is complicated by the way we talk about "what a population can tell us about a sample" (suggests we "know" the population & are interested in how samples from it behave, treating them as unrealised random variates) vs "what a sample can tell us about the population" (where the population is mysterious & unknown, but sample is now just a bunch of numbers in our dataset). I think answering with this perspective shift is necessary, as otherwise there's not much to say "population $X\sim N(\mu,\sigma^2)$ if and only if each randomly sampled $X_i \sim N(\mu, \sigma^2)$ Jul 31, 2022 at 13:58
• +1 I agree. We do have some information going back the other way, of a limited kind. We can't hope to pick out 'the' distribution unless we narrow the scope of the question dramatically (e.g. a choice from a short list of possibilities - but generally they'll all be wrong). If we leave it general we have the progress of various nonparametric estimators to the corresponding population distributions (e.g. the Dvoretzky–Kiefer–Wolfowitz–Massart inequality, results on convergence rates of kernel density estimators, histogram estimators and so on) but that doesn't pick out a specific choice. ...ctd Jul 31, 2022 at 22:49
• ... ctd I think (and more generally than with distribution choice) that it's necessary to give up on "the" model and focus on "a" model, one that is satisfactory for some purpose. This will involve very different considerations than looking at goodness of fit tests -- we know the model is wrong, so the test is not answering the question we should worry about, which is more about how it behaves for what we're doing, under a range of circumstances that we might plausibly experience. Jul 31, 2022 at 22:53
• Re: "Once you observe it, a sample is a collection of numbers, not a collection of random variables (with corresponding distribution(s))." One interpretation is trivial: yes, a dataset consists of numbers. Alternatively, it seems you would like the reader to conclude that no probability-based analysis of the sample would be legitimate. It certainly precludes assigning any relevant meaning to "in the limit," if the sample is just a bunch of numbers. I understand you mean something subtler, but my concern is with those many readers who will misunderstand these statements.
– whuber
Aug 15, 2022 at 19:45
• I agree there's a potential issue. Do you see a good way of correcting that without me bogging it down too much? Aug 15, 2022 at 23:09

A sample only contains some information about the underlying distribution. And statistics gives you tools to check some properties of the underlying distribution. E.g., you can use hypothesis tests which tell you whether you can confidently reject the hypothesis that a sample originated from a normal distribution.

Thus, if tests like Shapiro-Wilk reject normality, you can be reasonably certain that the underlying distribution is not normal. If the test does not reject normality, you cannot really deduce anything.

If you have, say, an i.i.d. sample, and you compute certain characteristics of it, like skewness and excess kurtosis, and they are close to the Normal's such, and you perform Normality tests that do not reject Normality, then you are reasonably justified in proceeding with the working hypothesis that the distribution of the population from which the sample came, follows a Normal distribution.

But don't try to coat this with more philosophical certainty than that.

• Not disputing the general gist, but checking the skewness and kurtosis (as some normality tests do) isn't necessarily much help (even aside from the bias in sample kurtosis). There are very non-normal distributions with moment-skewness 0 and moment-kurtosis 3 (0 excess kurtosis). One of my favourite examples is a 50-50 mixture of a gamma and its negative, with shape parameter $\alpha=(\sqrt{13}+1)/2$. It has density 0 at the mean and is distinctly bimodal with noticeably heavier tails than the normal with the same mean and variance. There are other examples that are not even symmetric. Jul 31, 2022 at 3:49