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If I plot two different parameters a and b which are both dependent on the same variable x, which of these is correct?

The plot shows the dependence of a and b on x.

or

The plot shows the dependences of a and b on x.

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    Did you mean dependencies (as in your title)? Or dependences (as in your second example)? Jan 1 at 23:03
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    The plot shows a and b as functions of x. This is rather formal. The distributed singular is very common here - the plot shows a and b as a function of x.
    – Phil Sweet
    Jan 2 at 4:49

3 Answers 3

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It is correct to use a plural.

(Controllable Electrorheological and Magnetorheological Materials - Seung-Bok Choi, ‎Norman M. Wereley, ‎Weihua Li · 2019) dependences of the current density and shear stress ERF-1 and ERF-2 on the electric field strength

(Constitutive Models for Rubber VI - Gert Heinrich, ‎Michael Kaliske, ‎Alexander Lion · 2009) dependences of the relaxation part of the storage modulusa polymer (G′−G network eq ) and do of the loss modulus (G′′) for not change with elongation and coincide with these dependences for soft Gaussian in the experiment networks

(Atmospheric and Oceanic Optics - Volume 15,Numéros 1 à 6 - 2002 ) Dependences of the amplification N and variation V coefficients on the wind speed . Figure 3 depicts the dependences of the amplification and variation coefficients on the solar zenith angle ( or , in other words , on the angle between

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    It’s pretty hard to defend dependences as word, at least outside of papers using non-native-English, translations, or jargon. And there is no need for a plural anyway. Jan 2 at 2:01
  • @TinfoilHat Apparently, a new acceptation has come into use in various scientific fields, where "dependence" has to mean "dependence relation"; otherwise, "dependence" is only uncountable, that is true.
    – LPH
    Jan 2 at 10:53
  • @TinfoilHat Its news to me too, but there are plenty of examples that I'm not prepared to write-off as typos or OCR errors. The word has appeared sporadically since at least the 1830s. It may have been a shortening of interdependences, which seems to be a recognized plural - merriam-webster.com/dictionary/…
    – Phil Sweet
    Jan 2 at 19:09
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The accepted answer shows correct use of the plural.

But it's also correct to use the distributive singular (arguably the singular-form noncount usage):

  • The dependence of the pressure and sample thickness on the permeation properties ... [NIH/NLM]
  • Poppenberg and Stephan engaged in the dependence of the pressure and the temperature on the bullet trajectory ... [CEON; courtesy of Google]
  • This paper reports theoretical models, Eqs. (9-2) and (12), used to describe the dependence of density and thermal conductivity on temperature. [Science Direct; Vol 171; May 2020]

As usual, the better choice depends on whether the commonality of dependence on a single factor (which in a mathematical model will appear as the independent variable), or the plurality (here duality) of properties so dependent, is judged the better to stress. I'd say that usually, the distributive singular is the better choice:

  • The dependence of both the volume and the pressure of a fixed mass of gas on its temperature is well known.
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  • This. And in terms of usage, the use of dependence in such a context is, I think, much more common than the use of dependences - and probably so in all (at least most) contexts.
    – Drew
    Jan 2 at 22:53
  • +1. Comparing "the dependence of both" vs. "the dependences of both" using Google Books Ngram Viewer, I find that the former is almost 50x as common: books.google.com/ngrams/…
    – ruakh
    Jan 3 at 3:34
  • Upvoted for suggesting “both”, which would be my preferred way to resolve this without introducing an esoteric plural form. Okay, “both” means adding an additional word, but I’ve had to edit stuff like this in the past, and I despair of scientific authors who think that being concise is more important than being properly understood.
    – KrisW
    Jan 3 at 12:34
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Either use can be correct in some contexts, and my recommendation is that

  • if a and b depend on x in the same ways (mathematically or semantically), use “dependence” or “dependency”.
  • if the relationships are different in a significant way (one is linear and the other exponential; one is positive, the other negative; one causal, the other co-causal) and especially if you intend to elaborate on the difference, use “dependences” or “dependencies”.

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