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This question is a bit left-field, but I figured that the community here probably has strong views on the subject!

I am writing up my PhD thesis. Consistently, when talking about quantities which are formally related to a Gaussian distribution, I have capitalised the "N" in "Normal" to refer to them. For example, "[...Under such circumstances] the resulting distribution is not Normal, but rather described by [...]".

My supervisor has read through the relevant chapter, and replaced every single one of these with a lower-case 'n'. I can't find any definitive literature on the subject -- Springer apparently wanted names properly capitalised, and according to another random dude on the internet, capitalising distribution names is a Good Idea.

Lacking a definitive style guide for my thesis, I thought I'd turn to the community of experts -- what is commonly done, and why?

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    $\begingroup$ I tend to capitalize "Normal" to emphasize that no member of this family of distributions is "normal." $\endgroup$
    – whuber ♦
    Sep 21, 2015 at 21:54
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    $\begingroup$ Why not just call it a Gaussian distribution and avoid fussing about whether or not to capitaliise the first letter in the word "normal"? If you feel the need to be complete and to cross your eyes and dot your teas, just say at the very first occurrence in your thesis something like, "Throughout this thesis, a random variable with density function ...... shal be referred to as a Gaussian random variable with mean .... and variance .... and the density function shown shall be referred to as a Gaussian density" $\endgroup$
    – Dilip Sarwate
    Jun 22, 2020 at 14:38

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For what it's worth, Wikipedia says this on the origin of the name:

Since its introduction, the normal distribution has been known by many different name... Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than "usual". However, by the end of the 19th century some authors had started using the name normal distribution, where the word "normal" was used as an adjective...

https://en.wikipedia.org/wiki/Normal_distribution#Naming

It is also not capitalized in the Wikipedia article, nor have I seen it capitalized in general as an American English speaker. For all intents and purposes normal IS an adjective, though not one that's meant to imply all other distributions are 'abnormal'.

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    $\begingroup$ (+1) this should be the accepted answer. I was mislead by my non-English background. After reviewing multiple sources it seems that in vast majority use lowercase names are used except names based on surnames. $\endgroup$
    – Tim ♦
    Sep 21, 2015 at 15:17
  • $\begingroup$ @Tim that's the standard I've always seen and used $\endgroup$
    – shadowtalker
    Sep 22, 2015 at 3:07
  • $\begingroup$ @ssdecontrol I've seen different usages, check my edited answer. But yes, it seems that lowercase is the standard. $\endgroup$
    – Tim ♦
    Sep 22, 2015 at 5:52
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On one hand, "Normal" seems not to be an adjective, nor a feature of some distribution that it is more normal than any other (or more "beta", more "binomial"). "Normal" is a name of a distribution and can to be considered as a proper noun, and so be capitalized. As @Scortchi noticed in his comment, this is also a general term and people seem to capitalize such terms. If you look into the literature, you'll see that some authors capitalize all the names of distributions, while some seem to never do so.

On another hand, currently (e.g. by Forbes et al., Krishnamoorty, Fisher, Cox et al. and others), it seems that most commonly names of distributions are written in lowercase (e.g. normal, beta, binomial) and are capitalized if they come from surnames (e.g. Cauchy, Gaussian, Poisson). There are also some names that are always written in lowercase as $t$-distribution (example here). While Halperin et al. (1965) in their recommendations do not mention distribution names, in their text they write about chi-squared and standarized normal distributions in lowercase.

This convention may be confusing since in formulas names of distributions are almost always written capitalized (e.g. $X \sim \mathrm{Normal}(\mu, \sigma)$ or $X \sim \mathcal{N}(\mu, \sigma)$) and also because many names come from surnames. However, contrary to my initial answer, it seems that lowercase names are used more commonly and so can be considered as a current convention.

enter image description here

(image source: Freeman, 2006)

Halperin, M., Hartley, H.O., and Hoel, P.G. (1965). Recommended Standards for Statistical Symbols and Notation. COPSS Committee on Symbols and Notation. The American Statistician, 19(3): 12–14.

Freeman, A. (2006). A visual comparison of normal and paranormal distributions. J Epidemiol Community Health, 60(1): 6.

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    $\begingroup$ I'd say "normal" in this context is an adjective and therefore should not be capitalised. However, "Gaussian" would be capitalised. This seems to be the accepted usage on the Wiki page for the normal distribution. $\endgroup$
    – babelproofreader
    Sep 21, 2015 at 10:36
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    $\begingroup$ Shouldn't it be "Normal Distribution" then, if it's a proper name? Seems to me it's rather like the "Mountain Bluebird" example in that Wikipedia article. $\endgroup$
    – Scortchi - Reinstate Monica ♦
    Sep 21, 2015 at 10:42
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    $\begingroup$ It's a bit deceptive to quote Wikipedia in support of capitalization, when they uniformly lowercase "normal distribution": en.wikipedia.org/wiki/Normal_distribution $\endgroup$
    – Charles
    Sep 21, 2015 at 15:07
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    $\begingroup$ I down-voted not because the answer is bluntly wrong but to steer people towards Grace's answer. (Otherwise your answer is quite nice!) $\endgroup$
    – usεr11852
    Sep 21, 2015 at 15:56
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    $\begingroup$ @David "Just wrong" seems to be going too far. No part of the meaning of "proper noun" concerns the singularity or plurality of the referent. Plenty of proper nouns refer to families, such as the Obamas, the Beatles, or a Gaussian (which is even a synonym for "normal distribution"!). $\endgroup$
    – whuber ♦
    Sep 22, 2015 at 12:24

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