# Why statistical moments are called moments?

According to the Jeff Miller's Earliest Known Uses of the Words of Mathematics "Moment was taken into Statistics from Mechanics by Karl Pearson when he treated the frequency-curve (or observation curve) as the sheet enclosed by the curve and the horizontal axis. See his "Asymmetrical Frequency Curves," Nature October 26th, 1893: "Now the center of gravity of the observation curve is found at once, also its area and its first four moments by easy calculation."

This implies that the term moment was taken from mechanics. On the other hand, a Wikipedia reference Robertson, D.G.E.; Caldwell, G.E.; Hamill, J.; Kamen, G.; and Whittlesey, S.N. (2004) Research Methods in Biomechanics. Champaign, IL: Human Kinetics Publ., p. 285 says that the concept of moments was taken from mathematics

I was wondering about the analogy of mechanical moment (force into the distance) with the statistical moment. What would be a mechanical analog of the zeroth moment (area), first moment (center of gravity), and second moment (variance) or there is none? Thanks.

The Oxford English Dictionary shows moment of a force appearing in 1830 in A Treatise on Mechanics by Henry Kater and Dionysius Lardner.

So perhaps it is reasonable to guess that Stieltjes and/or Pearson took the term from mechanics.

• This is as claimed at Earliest Uses, who as often belie their title: Google finds moment of a force earlier in Marrat’s Mechanics (1810, p. 25). – Francois Ziegler Aug 6 '18 at 20:16

This seems to depend on who you call a statistician, mathematician, or mechanician. Certainly Pearson sounds like he’s using, as a matter of course, a term also found in e.g. Stieltjes (1894, p. 48; 1885), Wittenbauer (1881), Reye (1870), Poinsot (1806), Euler (1752, p. 192), etc.

I hadn’t heard of Commandino (1565): it would be interesting to see what Greek word he translated into moment[um] — and whether it was from Archimedes or Pappus.

(Mechanical versions of 0th and 2nd moment are total mass and moment of inertia.)

• Thanks. It implies that credit to the first usage of moments to Pearson is not entirely correct. Perhaps he was the first one to use it on distributions. Wittenbauer seems to show the modern (physics) definition of the moment. The book by Commandino is online, books.google.com/books?id=2jgPAAAAQAAJ. On page 10 momentorum occurs. He shows a Greek paragraph on the same page. Perhaps it is about the center of gravity. Now I realize why French and German are still required to do a PhD in mathematics. I am a chemist with an interest in scientific terminologies. – M. Farooq Aug 4 '18 at 23:38
• @M.Farooq The Greek quotation on that page is from Pappus of Alexandria's Collection, probably book VIII, books.google.com/books?id=FSlOCc_QjiIC, but Wikipedia says Commandino took it from Archimedes – Endy Aug 6 '18 at 21:02
• @M.Farooq Pearson is definitely predated by non-“statistician” Stieltjes (1885, p. 851), who introduces the word in a “quasi-mechanical interpretation”. Note that Pearson’s 1893 paper was his first in statistics: prior to that he was publishing on elasticity, e.g. editing Todhunter, which is probably where he picked up the word. (I replaced my broken Commandino link with yours.) – Francois Ziegler Aug 9 '18 at 18:03

Moments in mechanics and statistics are defined by the same formula: $$\int x \rho(x)dx,$$ for the first moment. In mechanics, $x$ is distance, and $\rho$ is the mass density. In statistics, $x$ is anything (whatever your random variable represents) and $\rho$ is the probability density. So it is not surprising that the name is the same. Moments in mechanics were of course considered much earlier (since Archimedes at least).

• Is there a physical meaning for the third moment or nth moment in general in mechanics? The zeroth moment as suggested by Dr. Francois is total mass, the first one is the standard moment, and the second one is the moment of inertia. Thanks. – M. Farooq Aug 6 '18 at 14:22

@ Francois, I can see the usage in Stieltjes book but I can't read French. One can see the word moment but the context is missing. I checked Oxford English Dictionary for earliest usage, it turns that moment was first used in calculus as "Mathematics. In Newtonian calculus: the increment in the value of a time-varying quantity that occurs in an infinitesimal period of time; = differential n. 1. Cf. momentane n. Now hist. 1706 Phillips's New World of Words (new ed.) (at cited word) In Mathematicks, Moments are such indeterminate and uncertain Parts of Quantity, as are supposed to be in a perpetual Flux, i.e. either continually encreasing or decreasing."

Then another meaning is given "Mathematics. Any of various functions describing torsional effects, generally having the form of the product of a force and a distance; spec. the turning effect produced by a force; the magnitude of this, equal to the product of the force and the perpendicular distance from its line of action to the point about which rotation may occur. moment of a couple n. [compare French moment d'un couple (1869)] the product of either of the two equal forces comprising the couple and the perpendicular distance between their lines of action. moment of inertia n. [compare French moment d'inertie (1786)] the product of the mass of a particle and the square of its distance from a given axis; the sum of such products for all the particles of a body. moment of momentum n. the vector product of the momentum of a particle and its radius vector from a given point; the sum of such products for all the particles of a body; also called angular momentum.

bending, pitching, rolling moment: see the first element. 1830 H. Kater & D. Lardner Treat. Mechanics x. 135 The moment of a force is therefore found by multiplying the force by its leverage."

OED credits Pearson for statistical moments.