# Do test scores really follow a normal distribution?

I've been trying to learn which distributions to use in GLMs, and I'm a little fuzzled on when to use the normal distribution. In one part of my textbook, it says that a normal distribution could be good for modeling exam scores. In the next part, it asks what distribution would be appropriate to model a car insurance claim. This time, it said that the appropriate distributions would be Gamma or Inverse Gaussian because they're continuous with only positive values. Well, I believe that exam scores would also be continuous with only positive values, so why would we use a normal distribution there? Doesn't the normal distribution allow for negative values?

• If you're worried about the bounds on scores, you could try en.wikipedia.org/wiki/Truncated_normal_distribution – J.G. Nov 12 '18 at 6:39
• In the real world, of course, exam score distributions often don't look anything like a normal distribution anyway. As an example from my math undergrad days, I remember the Topology I class as having been notorious for its highly bimodal "dumbbell curve" grade distribution: you either understood the key concepts and got a nearly perfect score, or you didn't and were lucky to get any points at all. Very few people ended up scoring anywhere in the middle between those two extremes. – Ilmari Karonen Nov 12 '18 at 10:44
• No. Next question. – Carl Witthoft Nov 12 '18 at 16:24

Height, for instance, is often modelled as being normal. Maybe the height of men is something like 5 foot 10 with a standard deviation of 2 inches. We know negative height is unphysical, but under this model, the probability of observing a negative height is essentially zero. We use the model anyway because it is a good enough approximation.

All models are wrong. The question is "can this model still be useful", and in instances where we are modelling things like height and test scores, modelling the phenomenon as normal is useful despite it technically allowing for unphysical things.

• In this particular case, it's questionable whether the normal distribution is even a useful approximation. Nearly every grade distribution I've seen resembled the bimodal curve Ilmari Karonen mentioned in the comments to some degree. (Although usually with modes around 0.6 and 0.9, rather than on the extreme ends) However, a linear combination of two normal distributions with different means wouldn't be a bad approximation. – Ray Nov 13 '18 at 4:13
• I wasn't arguing that the normal is THE BEST approximation. The entire point of my comment is really made in that last paragraph. Comments about bimodality of actual grade distributions, at least at this level of abstraction, are really not helpful. – Demetri Pananos Nov 13 '18 at 4:16
• I agree with everything you said in your answer, but part of the question concerns whether the normal distribution is specifically applicable to modeling grade distributions. And the answer to that is usually "No". All models are wrong and some models are useful, but some are more wrong and less useful than others. The fact that the normal distribution in particular is an especially bad fit for this problem is important, and the answer as it is seems to suggest that the normal is only wrong because the tails go negative and infinite, when there are actually much deeper problems. – Ray Nov 13 '18 at 4:23
• You're being a little pedantic here. OP's problem was that the normal allows for negative scores. Bimodality wasn't the issue. You are not seeing the forest for the trees with respect to this question. Modelling details aren't relevant right now. – Demetri Pananos Nov 13 '18 at 4:29

Doesn't the normal distribution allow for negative values?

Correct. It also has no upper bound.

In one part of my textbook, it says that a normal distribution could be good for modeling exam scores.

In spite of the previous statements, nevertheless this is sometimes the case. If you have many components to the test, not too strongly related (e.g. so you're not essentially the same question a dozen times, nor having each part requiring a correct answer to the previous part), and not very easy or very hard (so that most marks are somewhere near the middle), then marks may often be reasonably well approximated by a normal distribution; often well enough that typical analyses should cause little concern.

We know for sure that they aren't normal, but that's not automatically a problem -- as long as the behaviour of the procedures we use are close enough to what they should be for our purposes (e.g. standard errors, confidence intervals, significance levels and power - whichever are needed - do close to what we expect them to)

In the next part, it asks what distribution would be appropriate to model a car insurance claim. This time, it said that the appropriate distributions would be Gamma or Inverse Gaussian because they're continuous with only positive values.

Yes, but more than that -- they tend to be heavily right skew and the variability tends to increase when the mean gets larger.

Here's an example of a claim-size distribution for vehicle claims:

https://ars.els-cdn.com/content/image/1-s2.0-S0167668715303358-gr5.jpg

(Fig 5 from Garrido, Genest & Schulz (2016) "Generalized linear models for dependent frequency and severity of insurance claims", Insurance: Mathematics and Economics, Vol 70, Sept., p205-215. https://www.sciencedirect.com/science/article/pii/S0167668715303358)

This shows a typical right-skew and heavy right tail. However we must be very careful because this is a marginal distribution, and we are writing a model for the conditional distribution, which will typically be much less skew (the marginal distribution we look at if we just do a histogram of claim sizes being a mixture of these conditional distributions). Nevertheless it is typically the case that if we look at the claim size in subgroups of the predictors (perhaps categorizing continuous variables) that the distribution is still strongly right skew and quite heavy tailed on the right, suggesting that something like a gamma model* is likely to be much more suitable than a Gaussian model.

* there may be any number of other distributions which would be more suitable than a Gaussian - the inverse Gaussian is another choice - though less common; lognormal or Weibull models, while not GLMs as they stand, may be quite useful also.

[It's rarely the case that any of these distributions are near-perfect descriptions; they're inexact approximations, but in many cases sufficiently good that the analysis is useful and has close to the desired properties.]

Well, I believe that exam scores would also be continuous with only positive values, so why would we use a normal distribution there?

Because (under the conditions I mentioned before -- lots of components, not too dependent, not to hard or easy) the distribution tends to be fairly close to symmetric, unimodal and not heavy-tailed.

Exam scores might be better modeled by a binomial distribution. In a highly simplified case, you might have 100 true/false questions each worth 1 point, so the score would be an integer between 0 and 100. If you assume no correlation between the test-taker's correctness from problem to problem (dubious assumption though), the score is a sum of independent random variables, and the Central Limit Theorem applies. As the number of questions increases, the fraction of correct problems converges to a normal distribution.

You ask a good question about the values less than 0. You could also ask the same question about the values greater than 100%. As the number of test questions increases, the variance of the sum decreases, so the peak gets pulled towards the mean. Similarly, the best fit normal distribution will have smaller variance and the weight of the pdf outside the [0, 1] interval tends towards 0, although it will always be nonzero. The space between possible values of "fraction correct" will also decrease (1/100 for 100 questions, 1/1000 for 1000 questions, etc.), so informally, the pdf begins to behave more and more like a continuous pdf.