# When is it OK to write "we assumed a normal distribution" of an empirical measurement?

It is ingrained in the teaching of applied disciplines, such as medicine, that measurements of bio-medical quantities in the population follow a normal "bell curve." A Google search of the the string "we assumed a normal distribution" returns $\small 23,900$ results! They sound like, "given the small number of extreme data points, we assumed a normal distribution for the temperature anomalies" in a study on climate change; or "we assumed a normal distribution of chick hatching dates" on a possibly less contentious document on penguins; or "we assumed a normal distribution of GDP growth shocks", referring to macroeconomic changers in markets (bringing up to memory this book, ... and other things).

Recently, I found myself questioning the treatment of count data as normally distributed due to their strictly positive nature. Of course, count data are discrete, making their normality all the more artificial. But even leaving this latter point aside, why should continuous empirical measures such as weight, height or concentration of glucose, deemed prototypically "continuous", be considered normal? They can't have negative realized observations any more than counts do!

I understand that when the standard deviation is substantially lower than the mean, indicating few negative values ("95% range check") it may be a practical assumption, and frequency histograms may support it if not too skewed. But the question didn't seem trivial, and a quick search yielded interesting stuff.

In Nature we can find the following statement on a letter by DF Heath: "I wish to point out that for the statistical analysis of certain types of data the assumption that the data are drawn from a normal population is usually wrong, and that the alternative assumption of a log-normal distribution is better. This alternative is widely used by statisticians, economists and physicists, but for some reason is often ignored by scientists of some other disciplines."

Limpert notes that "the log-normal model may serve as an approximation in the sense that many scientists perceive the normal as a valid approximation now", while noting the low power of goodness-of-fit tests of normality, and the difficulty in selecting the right distribution empirically when dealing with small samples.

Therefore the question is, "When is it acceptable to assume a normal distribution of an empirical measurement in the applied sciences without further supportive evidence?" And, Why other alternatives, such as the log-normal, have not, and probably are just not going to take hold?

• The answer would depend on what sort of thing you're doing, and the sensitivity it has to potential deviations from normality (i.e. if you're testing equality of variances using an F test of the ratio, you'd better have distributions that are very close to normal ... but if you were constructing a t-interval for the difference in means, with large samples, you might not need to have them very close to normality at all). ... and on your tolerance (or your audience's) for the kind of impacts it would have on the inference you're doing. Mar 13 '16 at 9:18

I find your question really interesting. Let's have some things into account:

1. To say that an observed variable is continuous in real life is going to be always kind of wrong, because it's very difficult to measure really continuously.
2. Now add the properties of a normal random variable $N(\mu, \sigma^2)$: range $(-\infty; +\infty)$, symmetrical distribution (mean = mode = median), the probability density function $f_X(x)$ has inflection points at $x = \mu - \sigma$ and $x = \mu + \sigma$.
3. To say that a random variable $X$ follows a Log-Normal distribution implies that the variable $Y=log(X)$ follows a normal distribution.

With that said, to say that any observed variable follows a normal or a Log-Normal distribution sounds kind of crazy. In practice, what's done is that you measure deviations of the observed frequencies from the expected frequencies, if that variable came from a normal (or any other distribution) population. If you can say that those deviations are just random, because you are sampling, then you can say something like there's not enough evidence to reject the null hypothesis that this variable comes from a normal population, which is translated into we will work as if (assuming that) the variable follows a normal distribution.

Answering to your first question, I don't think that there's someone as bold to say that a variable is assumed to be normally distributed without further evidence. To say something like that, you need at least a qq-plot, an histogram, a goodness-of-fit test or a combination of those.

To answer the second question, the particular interest in the normal distribution is that many of the classical tests are based on an assumption of normality of the variable, like the t-test, or the $\chi^2$-test for the variance. So, normality simplifies work, that's all.

• Thank you for your answer, which touches upon many key points. However, I tend to think that things in the "real-world" of applied sciences are less structured, and a direct tangent is often taken to assume normality. Mar 13 '16 at 0:11
• Something that I didn't mention is the other part of the history if the normal distribution: it is the limit distribution of the standardization of a sum of iid random variables, as it's stated in the theorem of the central limit. If you can say that your variable is a sum of many iid random variables, like in the reasoning behind brownian motion, then you can say that it is a normal random variable. That's the only valid shortcut I know. I can include this in the answer if you want. Mar 13 '16 at 0:22

#### This largely depends on the robustness of your inferences to errors in the distribution

When you are dealing with quantities that are either directly observable, or for which there is some close estimator (e.g., residuals for error terms), you can use the data to make non-parametric estimates of the distribution. Assuming you have sufficient data to do this, it is generally best to avoid making a distributional assumption about those quantities, unless you are subjecting these assumptions to empirical tests/diagnostics to confirm the plausibility of the assumed form. The reason for this is obvious --- if the values are observable (or closely estimable) then with a reasonable amount of data you can estimate the distribution, so there is no need to blindly assume its form.

Now, it is of course that case that parametric statistical models assume a parametric distributional form of some kind. In most statistical analysis in a regression context, we specify a model form that has an assumed form for the "error terms". For example, in a standard Gaussian regression we assume that the error terms are normally distributed. Once we have fit our model to the data we can then construct diagnostic plots of residuals and use these to determine whether there are any obvious departures from the assumed distributional form. If the assumed distribution is dubious we have two options: (1) we can either change our model assumptions and rinse-and-repeat; or (2) we can keep our existing model and use it only for inferences, etc., that are known to be robust to errors in the assumed error distribution in the model. In many cases the latter approach is fine (so long as other assumptions in the model are okay), since many of the conclusions from a regression model are robust to the distribution of the error term.$$^\dagger$$

More generally, the importance of the distributional form is going to depend on what kind of inferences you are making and how robust these inferences are to errors in the assumed form. This is contextual; in some projects you will want inferences about model coefficients, whereas in others you will want predictions of observable variables, and in others you may want something else. Generally speaking, the following advice applies:

• If you are using a parametric model (which assumes a parametric distribution of a particular kind), always make an effort to choose a form that is as close to the data as is reasonable. This will usually involve conducting diagnostic tests on observable quantities in the model (e.g., residuals in a regression) and then varying the model with transformations, changes in form, etc., until you feel that you have a good representation of the data. It is generally bad practice to make "bald assumptions" that are not scrutinised against diagnostic plots, etc.

• Always consider the kinds of inferences/predictions you actually want to make in your analysis, and consider how sensitive/robust each of these are to errors in the assumed parametric distributional form. If you need to make inferences/predictions that are highly sensitive to the assumed distributional form then you need to ensure that you apply diagnostic tests and you are satisfied that the assumed distribution is a reasonable representation of the data.

• For some proposed transformations, like taking the logarithms of quantities under analysis, there are natural theoretical considerations that apply. For example, variables are generally transformed onto a log-scale (i.e., taking logarithms) if they are positive quantities that tend to change by a percentage that has a roughly fixed distribution over time. This is true of many quantities studied in economics and physics. Transformation of variables should be informed by theoretical considerations and empirical analysis.

• In some cases you will be limited by the available knowledge on model forms (i.e., the extent of the statistical literature on the forms of interest to you). You might want to vary your model in some way but find that there is little or no research on the properties of the ideal model. In such cases, you may need to fall back on suboptimal models, and include appropriate caveats in your analysis.

$$^\dagger$$ Specifically, assuming you have a reasonable amount of data, the estimated coefficients are highly robust to the assumed error distribution, but predicted values of the response variable are not.

Gnedenko and Hinchin in their textbook "Elementary introduction in the theory of probability" claim that there is a tedious theoretical proof of the following claim. If the value of a variable is influenced by numerous statistically independent factors, then it is appropriate to assume that the values of the variable is normally distributed. For instance, if one shuts bullets from the gun into the target, then the final position of the bullet is influenced by numerous independent factors (wind, shaking of the hands, the mass of a bullet, the friction between the hand and the gun, etc).

• They are probably referring to the central-limit-theorem Jul 23 at 19:14
• This is a good comment. However, bear in mind that "influenced by" in this textbook context must be interpreted in terms of additive factors. The point of the question is that in many disciplines, those factors are not all additive: many are multiplicative. That leads to approximate lognormal distributions rather than normal distributions.
– whuber
Jul 23 at 19:50
• CLT relates to the distribution of the means. In this case, we formulate the conditions that are enough for the variable to be normally distributed. The factors do need to be additive. Besides, the contribution of each factor must be much smaller than the total contribution of all factors. Jul 24 at 9:37