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There is a sentence in Sokal and Rohlf's classic text [R.R.Sokal F.J.Rohlf; Biometry, 3rd ed., 1994: p.132, chapter 7 Estimation and Hypothesis Testing]:

The variance of means is therefore partly a function of the sample size on which the means are based.

In this specific case, I understand that the greater the sample size the lesser the standard deviation of sample means. But in general what does it mean to be a function of something? Relationship or dependence?

(Please provide answers/comments in plain language.)

abc
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2 Answers2

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"$A$ is partly a function of $B$" means that if you change $B$, even if you don't change anything else, the chances are that $A$ will also change.

Gerry Myerson
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  • Unless the function is constant, of course :-) – Siminore Dec 09 '12 at 11:42
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    @Sim, if $A$ is constant, Sokal et al would not write $A$ is partly a function of $B$ --- and you know it --- so why confuse OP? – Gerry Myerson Dec 09 '12 at 11:49
  • @Siminore That is what the phrase "chances are" is there for. – Arthur Dec 09 '12 at 11:49
  • @GerryMyerson You should know that constant functions are functions. By the the way, a function is something that must be defined without any reference to "chances", "good luck" or equivalent words. – Siminore Dec 10 '12 at 09:24
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    @Siminore, you should know that a teacher answers a question using language the student can be expected to understand. – Gerry Myerson Dec 10 '12 at 11:47
  • @GerryMyerson Yes, but we are currently living in 2012. Functions are no longer "dark things", nowadays. And, sincerely, a teacher should answer correctly, no matter how difficult the answer is. Here it is clear that the authors of that book use an old-fashioned language, which has almost nothing to do with mathematics. Everything is a function of everything, you know. – Siminore Dec 10 '12 at 14:35
  • @Siminore, let me encourage you to post your own answer to stan's question. – Gerry Myerson Dec 10 '12 at 22:52
  • Well, as a mathematician I should say that the sentence has no meaning at all. Period. It is a vague sentence that the authors write to do some "conversation", or probably (I cannot find the book in my library) to use scientific words and pave the way to a mathematical model. – Siminore Dec 11 '12 at 12:20
  • @Siminore, good. Post that as an answer. – Gerry Myerson Dec 11 '12 at 23:33
  • @GerryMyerson - quick question, when authors say something like “the position of a ball is a function of time”, is it correct to think about this mathematically as: we’re considering a function $f$ such that $f(t)$ is interpreted to be the position of the ball at time $t$? – Taylor Rendon Jun 25 '22 at 02:31
  • @Taylor, yes, that would be how I would interpret it. – Gerry Myerson Jun 25 '22 at 06:40
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In layman terms as requested, a function takes input values and gives output values. It is like a blender. Put in apples and we get apple juice. Put in orange and we get orange juice. In your context, it means that the variance depends on sample size, just like what juice you get depends on what fruits you throw in!