# What are bounded distributions? and can a bounded distribution hold the normality assumption?

I heard that normal distribution should be unbounded, but I want clarification about that, aren't most distributions in the real world bounded, I mean they won't go to infinity they have minimum and maximum values. From what I read bounded distributions have predefined range like scores in exams, but unbounded distributions do not have predefined range like length, although its values won't go to infinity, but it doesn't have predefined values for range. Is my understanding true?

I want also to know how will bounded data be analyzed as they are bounded, they are not normal, so can I use t-test and linear regression on them?

Here is an example of distribution of data of anxiety score which is bounded between [1:5], although it looks normal, but it is slightly light tailed as no values above 5 or below 1, can I run parametric tests on it or what should I do in this case?

• While the usual interpretation is that the support of the pdf$$f:\ x\in\mathcal X\longmapsto f(x)\in\mathbb R^+$$of a bounded distribution is compact or relatively compact, e.g. is contained within a finite interval when $\mathcal X=\mathbb R$, another interpretation is that the pdf is upper bounded$$\max_{x\in\mathcal X}f(x)<\infty$$(as for the Normal distribution). Feb 3 at 14:13
• @Xi'an Because when one speaks of "distributions" generally we cannot suppose a pdf exists, the default interpretation of a "bounded" distribution $F$ should mean there exists a compact set $\mathcal X$ for which $F(\mathcal X)=1.$ That is, any random variable with distribution $F$ is almost surely within a specified bounded set.
– whuber
Feb 3 at 15:18
• @whuber: you are right, of course. I simply thought the alternative was also of possible interest, if not directly to the question. Feb 3 at 16:39
• This data distribution often is even better than the idealized Normal distribution because it is so short-tailed (and roughly symmetric). I'm holding back a little because often the raw distribution of a response is irrelevant for many analyses, which are concerned with the response conditional on other variables.
– whuber
Feb 3 at 17:35
• You could also use an ordered regression model, where a continuous latent outcome maps to scores. There is some intuition here. Feb 3 at 17:57

First, yes, a bounded distribution has ... well, bounds. This is a case where the natural meaning of the phrase is accurate.

Second, yes, the Normal distribution has no bounds. That's part of the definition of the distribution and can be seen from the CDF or PDF.

Third, that said, a sample can be very close to Normally distributed even if it is bounded. Take human height. It can't be negative. But that's OK in all practical senses because if (say) the mean height is 170 cm and the standard deviation is 7 cm (I just made those up, but they aren't far off) then the minimum height of any population is going to be positive.

The assumption of normality in, say, OLS regression is not really a yes/no thing. The residuals will never be exactly normally distributed. They have to be close enough.

Your example has some oddities. You say it's bounded (and it is) but is it continuous? Or, rather, how close is it to continuous? Why do there seem to be some peaks in the histogram?

Further, OLS regression does not assume that the variables are normally distributed, it makes assumptions about the distribution of the errors. Still, if he assumption of normality of the residuals is violated, you could use robust regression.

• Or, rather, how close is it to continuous? It is a scale for measuring anxiety having values of (1, 1.1, 1.2, 1.3...........4.7, 4.8, 4.9, 5). I know that the scale does not have many values, but I think it is large enough to be considered as a continuous variable.
– NEA
Feb 3 at 16:22