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I was researching the etymology of "average" to enlighten the debate as to whether "average" can in fact mean a "median" (or if the presenter was merely trying to sweep up their tracks so to speak). I was surprised by the etymology: unknown possible from Arabic meaning losing an eye, and via Italian/French meaning damage to shipping goods according to wiktionary.org

The first use for a statistical sense was 1735 (I am guessing it was De Moivre measures of central tendency). Was there any justification for the use of the word in this sense? The median does in fact have a sampling distribution which tends to normal in most cases, but my suspicion was that this early work focused on arithmetic quantities.

My questions are:

  1. What is the document attributed to the 1735 usage for the statistical of an "average"?
  2. Is the definition "(statistics) Any measure of central tendency, especially any mean, the median, or the mode. [from 1735]" actually reflective of what the 1735 treatise reflected?
  3. If not, at what point (if ever) did average start to reflect other statistical quantities than the arithmetic mean?

1: https://en.wiktionary.org/wiki/average

gung - Reinstate Monica
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AdamO
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    I recommend (1) Churchill Eisenhart's 1971 paper unpublished in his lifetime, but public at https://galton.uchicago.edu/~stigler/eisenhart.pdf (2) a 2003 paper by Bakker at http://ww2.amstat.org/publications/jse/v11n1/bakker.html The idea of the mean and median are both ancient; it's when they become statistical that is hard to track. It's arguable that e.g. cutting food or any total amount into equal shares or picking the value in the middle of a set of ordered values is statistical, but to me entirely plausible that both were invented many times over before anyone documented them. – Nick Cox Jul 02 '18 at 17:17
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    @NickCox I agree that the concepts are ancient, and that those terms (mean and median) are rigorous and well accepted. My question is moreover how or if "average" ever came to represent some blurry, umbrella concept relating them. Maybe I'm missing your point. – AdamO Jul 02 '18 at 17:38
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    You're not missing my point because I am just commenting obliquely. Your question is a challenge if only because (1) much of the literature is in Latin, French, German, etc.; English-language literature is just one strand (2) statistical textbooks in the modern sense as encoding practice don't really start before the late 19th century (3) you need to cover informal or semi-formal senses in law, insurance, surveying, etc. as well as mathematics and the sciences. Sometimes people aimed at estimating averages but wondered whether using medians would work fine. The concepts remained distinct! – Nick Cox Jul 02 '18 at 17:52
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    I can't help with the etymology, but there are times when "average" is used to refer to medians. Someone might say "The average American household earns 50,000 per year," and mean that the median household income is 50,000. [Compare the headline and data presented here](https://www.cnbc.com/2017/08/24/how-much-americans-earn-at-every-age.html). (Also later in the article, use of "average man".) I feel like this use was common in news sources a few decades ago, and has been replaced more now with more precise language using the word "median", but I have no evidence for this. – Sal Mangiafico Jul 05 '18 at 14:58
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    @SalMangiafico an anecdotal counter, the `AVG` cell operator in Excel calculates a mean. – AdamO Jul 05 '18 at 16:19
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    @AdamO , Oh, yes I think _average_ **usually** indicates the mean. But to the original question, it's possible the speaker was using _average_ in a way that implies a median value.... A slightly different example, I wonder if when we say, "man of average height" if we are thinking of the median and not the mean. Probably neither per se. But I could see how speaking this way, I might use the median as the criterion. – Sal Mangiafico Jul 05 '18 at 17:19
  • @SalMangiafico Good point. The English language always accepts the conventional usage as the *de novo* definition and etymology. – AdamO Jul 05 '18 at 17:21
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    I found another example of interest. From Grissom and Kim, 2012, Effect Sizes for Research. The first sentence of chapter 2. `Recall the the *location* of a population is a parameter that is usually defined as a measure of its "average," commonly its mean or median.` – Sal Mangiafico Jul 16 '18 at 13:19

2 Answers2

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Consolidation of my comments to an answer.

This doesn't address the etymology, but the underlying concern of the OP as to whether "average" can mean median.

There are times when "average" is used to refer to medians. It's common to say "The average American household earns 50,000 per year," and intend that the median household income is 50,000.

Compare, for example, the headline and data presented here. The headline:

Here’s how much the average American earns at every age

and the data presented:

Below, check out the median earnings for Americans at every age bracket, according to data from the Bureau of Labor Statistics for the second quarter of 2017.

(Also later in the article, use of "average man".)

As a different example from a statistical source, Grissom and Kim, 2012, Effect Sizes for Research, 2nd ed. The first sentence of chapter 2:

Recall that the location of a population is a parameter that is usually defined as a measure of its "average," commonly its mean or median.

To be clear, I think that "average" usually indicates the mean. But to the original question, it's possible the speaker was using average in a way that implies a median value.

As a final note, I wonder if when we say, "man of average height" if we are thinking of the median and not the mean. Probably neither per se. But I could see how speaking this way, I might use the median as the criterion.

Sal Mangiafico
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    Agreed. Many people interpret "average" to mean, in a sense, typical. One reason to prefer median over mean is that (depending on the algorithm used): median will reflect an actual observable value. That's the source of the statistical -ism that (in the US) a young family might say they want to have 2.5 kids. – AdamO Jul 16 '18 at 13:48
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Average had for at least two centuries since its induction in European languages implied damage to shipping goods, Words with similar roots are noted among shipping peoples (English, Dutch) and other traders in the Southern Mediterranean (Italian, French). The root awar- is from Arabic meaning damaged.

King Louis the XIV reissued and enforced a set of pan-European maritime laws after nearly a millenia of lawlessness. This was in 1681. The Maritime Law of General Average states that damage to shipping goods should result in a proportionate share of losses to all parties: receiver and shipper and among the shipper all the crew.

It seems like no coincidence that the usage of average to imply a mean in a general sense is attributed only 50 years later to another French writer, I guess it would be De Moivre.

Since the Maritime Law refers to a sum being divided with equal weights, if there is any bearing on the conversation, I think we can probably say it is the arithmetic mean.

AdamO
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