Why do we use the Greek letter μ (Mu) to denote population mean or expected value in probability and statistics

According to this Wikipedia entry, "Mu was derived from the Egyptian hieroglyphic symbol for water, which had been simplified by the Phoenicians and named after their word for water". So, my question is, why the early statisticians decided to use the letter Mu to denote the population mean / expected value? Was it because that letter was not occupied for denoting other concepts at the time or there was a deeper idea behind the choice?

• One of my students had studied hieroglyphics and so was fond of using them to represent quantities in his mathematical expressions. He, at least, was aware that especially in mathematics, symbols mean what you say they mean. Extremely few letters have near-universal meanings in mathematics: $\pi$ is about the only one and even that is readily co-opted (to refer to a probability parameter in a statistical model, for instance). – whuber Oct 17 '19 at 10:27
• Tempted to add an answer “explaining” how $\mu$ was chosen because the two peaks in the character looks like a bimodal distribution, reminding us all to be wary of blindly using measures of central tendency without considering its range and its shape. – Gaurav Oct 17 '19 at 21:51
• So far, despite having an accepted answer, every post here--although interesting--is speculative. It would be nice to see some actual historical research. – whuber Oct 17 '19 at 22:24
• The Greek for mean is $\mu \epsilon \sigma o s$. – copper.hat Oct 18 '19 at 2:09
• A deeper idea. I see what you did there. – Daniel Widdis Oct 18 '19 at 4:23

The letters that derive from $$\mu$$ include the Roman M and the Cyrillic М. Hence considering that the word "mean" starts with an $$m$$ the choice seems relatively straightforward given an already existing tradition to use greek letters in mathematical abbrevation.

To satisfy certain individuals craving for actual historical research and assuming that the webpage here is credible I can now confirm that the assumption that it comes from English turns out to be valid.

Fisher wrote the normal density with $$m$$ for the mean (see section 12 of his Statistical Methods for Research Workers) until the mid-1930s when he replaced $$m$$ with $$\mu$$. The new symbol appears in The Fiducial Argument in Statistical Inference (1935) and it went into the 1936 (sixth) edition of the Statistical Methods for Research Workers.

• As is, this answer assumes it comes from English - but luckily a lot of European languages' word for the mean also start with M (Mittel/media/moyenne...), so this holds even if it wasn't. – jkm Oct 17 '19 at 8:57
• Peter Whittle uses $A()$ tor average, which works well in his writing and sometimes fits the bill nicely when the argument is lengthy. $E()$ has similar but not identical flavour and is, evidently, immensely more common in probability and statistics. – Nick Cox Oct 17 '19 at 9:50
• I am not assuming that it "comes from" English in any historical sense of comes from. I am however assuming that English is what makes the association - $\mu$ and "mean" - straightforward. The OP asks why we use $\mu$ today? Language use today is very much determined by todays users and very many people doing statistics on a high level are english speakers if not by first then second language, so are the students and textbooks. So even for a Dane like me it makes sense to use $\mu$ for mean and $\sigma$ for standard deviation. – Jesper for President Oct 17 '19 at 11:05
• It gets even more interesting when we learn that in Persian (transliteration: mi.yan.kin) and Arabic (transliteration: mo.to.vä.set) the words denoting the mean also start with the letter m. – Dataman Oct 17 '19 at 11:08
• There's a good reason for that: 5,000 years ago all these languages (except Arabic) were the same. See indo-european.info/dictionary-translator/translate.inc.php/…. That there would be borrowings between the Semitic and Indo-European languages is also no surprise. But none of this answers your question, which is about a common habit (in the late 20th and early 21st centuries) of using $\mu$ for means. – whuber Oct 17 '19 at 11:19

There is a general rule to use Greek letters for parameters and Latin letters for statistics.

Why $$\mu$$? Well, the word 'mean' in English starts with M and $$\mu$$ sounds like M. But also, per Google translate:

• Latin: Media
• French: Moyenne
• Spanish: Media
• German: Mittel
• Dutch: Midden
• And μέσος is the Hellenic/Greek translation of mean. Mean comes from Latin Medianus. – gsamaras Oct 18 '19 at 18:57

The use of Greek letters in modern mathematics is basically a consequence of the humanist education. The normal distribution was introduced by Gauß in his celestial mechanics paper "Theoria motus corporum coelestium in sectionibus conicis solem ambientium" in 1809. The parameters $$\mu$$ and $$\sigma$$ of the normal distribution are called "Mittelwert" and "Standardabweichung" in German and their Greek letters have the phonetic equivalents "M" and "S" in Latin script.

That may well have been the motivation for giving them those letters. The equivalent Latin (rather than German) terms would likely also start with "M" and "S".