# Averages and Standard Deviations, but Skewness and Kurtosis?

Suppose I explain the following table in my academic paper.

``````State     Average   St. Dev.  Skewness  Kurtosis  Fis. Info.
CA        1.23      4.56      1.35      2.46      1.47
NY        4.56      7.89      7.91      8.02      2.58
TX        7.89      1.23      3.57      4.68      3.69
``````

The table displays the averages, standard deviations, skewness (?), kurtosis (?), and Fisher information (?) of the states.

(1) I couldn't find both skewness and kurtosis in Cambridge Dictionary, but it seems they are abstract nouns and uncountable. I used averages and standard deviations above as they're countable and each column has three numbers. Should I use skewness and kurtosis, or skewnesses and kurtosises?

(2) Fisher information is another statistic as well. Though there are three numbers in the last column, I can't use Fisher informations as information is uncountable. Despite of the three numbers, should I still just use the Fisher information alongside the averages and standard deviations?

(3) Volatility is uncountable according to Cambridge Dictionary. Suppose there are historical volatility and implied volatility in my paper. Should I use historical and implied volatility, or historical and implied volatilities?

(4) I have multiple implied volatility observations (for example, 0.47, 0.13, 1.42, etc.) in my data set. Should I still use the volatility and avoid the volatilities though there are multiple numbers?

• This is discipline-specific, so you are probably best suited to ask this on a more relevant SE where those measurements are more commonly used to see what the industry standard is. That said, your statement gives no extra detail simply listing the headers of each column, and your reader should understand how to read a table. Instead of saying "these are my table columns" you should just go straight into your analysis and reference the table as needed. Sep 24, 2019 at 17:30
• There are many, many papers that list these statistics. Copying table header style isn't plagiarism. :-) Sep 24, 2019 at 17:40
• You can use moments, with the mean as first, variance as second, skew as third and kurtosis as fourth, etc. Also, you should specify whether it’s kurtosis as the fourth moment, or excess kurtosis, i.e. less 3, to more easily compare the sample kurtosis with that of the Gaussian.
– user205876
Sep 24, 2019 at 19:50

The division between countable and uncountable nouns is not so strict and arbitrary for abstract nouns as you seem to have learned. When an abstract noun refers to a specific measurable quantity or quality, it is often possible to pluralize it, even if the noun normally is not used in the plural form.

"Volatilities" is not an impossible form; you can see it used in a 2004 book by David Ruppert:

The implied volatilities vary among themselves.

Similarly, plural forms of "-ness" nouns, such as "skewnesses", can be used in this kind of context, as shown by this quote from "Robustness in ANOVA", by Rand Wilcox:

If the skewnesses corresponding to the two groups are identical,

(Applied Analysis of Variance in Behavioral Science, 1993, edited by Lynne Edwards, p. 350)

That said, "informations" does sound strange to me even in this context. (I think it's because "information", unlike "volatility" and "skewness", is not treated in ordinary speech as a noun that refers to a quality of something.)

If pluralized, "kurtosis" would most likely turn into "kurtoses" /kəɹˈtoʊsiːz/ rather than "kurtosises". Nouns ending in -sis, which mainly come from Greek, generally pluralize by replacing -sis /sɪs/ with -ses /siːz/.

You can avoid the use of any plural forms by using a phrasing like "The table displays the average, standard deviation, skewness, kurtosis, and Fisher information for each state." This is how I would recommend writing a sentence to explain a table like this.

• (1) Thanks for all the details, but is there any textbook or something that black and white states the guideline—one can pluralize an abstract noun if it refers to a specific measurable quantity—explicitly? I occasionally observed the tendency reading some papers, but couldn't see the law anywhere before. (2) Though you said the informations sound weird, can I still use Fisher informations or a Fisher information to be consistent in this context as related to a specific measurable quantity? I was just curious all the time writing sentences so would like to establish a consistent rule. Sep 24, 2019 at 19:21
• @Junyong Kim It makes far more sense to talk of 'count' and non-count' usages rather than nouns. You can read more about the convenience (for those often having to use familiar wordings) / minefield (for those less familiar with the jargon) of 'countification' by searching for previous ELU posts. But there are no binding rules; acceptability is usage-driven and hence idiosyncratic. You have to check in up-to-date dictionaries, or, for technical terms such as these, with authorities such as Professors and mandated University Science Style Guides. 'Louis XIII and Louis XIV are two very ... Sep 30, 2019 at 18:49
• different furnitures' is accepted by many in the trade if not in the ivory tower, though, so there are cases where consensus does not exist. Sep 30, 2019 at 18:51

Value is a countable noun:

a numerical quantity that is assigned or is determined by calculation or measurement

And that is what all of these uncountable numbers actually are!

So, in order to avoid vocabulary bloat, append the plural values to the class name, andyou get skewness values, kurtosis values, Fisher values, volatility values. This should be your default way to name a group of similar uncountable measurements.

• Thanks for your reading again. In the same philosophy, would "measures" or "numbers" as well as "values" be reasonable in this context? Sep 30, 2019 at 17:33
• Heh. I like this answer very much. "The laws of Moses" rather than fighting about if it should be "Moses's laws" or "Moses' laws" or something else. Sep 30, 2019 at 17:43

I'm an Applied Mathematician. This is a Mathematics orientated question. I would recommend you post in the Maths 'Sister Page'. I will happily go into detail as to how each term is interpreted mathematically.

Please note that the mathematical definition of a given word is very often quite different from the standard English use (integrate, differentiate being two examples).

• The question is about abstract nouns and uncountable nouns which is an interesting area of English grammar. Oct 1, 2019 at 12:57
• @Nigel J - you must not be familiar with mathematical texts. The use of English is vastly different from standard use. That is why this is best addressed in the Maths page. Oct 1, 2019 at 20:56
• @Marcel- It doesn't provide an answer as the question is not suited to this page and should best be put in the Maths page Oct 1, 2019 at 20:57