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This article refers to something it calls "De Moivre's equation":

$$\sigma_{x} = \sigma / \sqrt{n}$$

Basically, it relates the observed variance in a sample size to the actual variance of the underlying distribution.

I wanted to learn more about this (I assume the actual equation is $E[\sigma_x]= \sigma/\sqrt{n}$?), but I can't find any references to "De Moivre's equation" anywhere.

Does anyone know a more standard name?

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The equation is correct as written--$\sigma_x$ is a parameter, so the expectation you inserted is unnecessary. –  Jonathan Christensen Dec 8 '12 at 2:13
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Is this the equation? en.wikipedia.org/wiki/… –  anorton Dec 8 '12 at 3:12
    
This might be relevant: en.wikipedia.org/wiki/De_Moivre%E2%80%93Laplace_theorem –  Yury Dec 8 '12 at 3:19

1 Answer 1

up vote 5 down vote accepted

$\newcommand{\var}{\operatorname{var}}$

One of the first sensible questions that a sensible person will ask in an introductory statistics course is why the root-mean-square deviation is used as a measure of dispersion rather than the mean absolute deviation. With the mean absolute deviation, you're just asking how far the average observation is from the mean of all the observations. It seems like the obvious thing to do. But instead, a measure of dispersion is used in which the deviations get squared before averaging, and then the squaring is undone after averaging. And why is that?

In the 18th century, de Moivre at least tacitly answered that question. He examined the question of the probability that the number of heads observed when a coin is tossed $1800$ times is in a specified interval---say between $885$ and $907$.

The answer to the question in the first paragraph is this with squares, we have additivity: if these random variables are independent, then $$ \var(X_1+\cdots+X_{1800}) = \var(X_1)+\cdots+\var(X_{1800}). $$ Nothing like that works with the mean absolute deviation. That explains why it's not (as often) used. The average number of heads you get in this experiment is $900$; now de Moivre enables us to know that the standard deviation is $\sqrt{450} \cong21.213$.

The number $\sqrt{450}$ is just $\sqrt{1800\cdot1/4\ {}}$ and $1/4$ is the variance of a random variable that is either $0$ or $1$, each with probability $1/2$. For the average rather than the sum, you'd have $\sqrt{(1/4)/1800}$. That's "de Moivre's equation".

De Moivre also showed that the bell-shaped curve is involved. For the probability that the sum $X$ satisfies $885\le X\le907$, or equivalently $884<X<908$, look for the probability that $884.5<X<907.5$ assuming $X$ is distributed according to a bell-shaped curve. The standard bell-shaped curve is $y=\text{constant}\cdot e^{-x^2/2}$. That is for a distribution with mean $0$ and standard deviation $1$. For mean $900$ and standard deviation $21.213$, it is $y=\text{constant}\cdot e^{-((x-900)/21.213)^2/2}$.

De Moivre found the "constant" numerically; his friend James Stirling later found it in closed form.

I doubt the author you cite intended the phrase "de Moivre's equation" to be understood as a proper name.

You might google the term "The Doctrine of Chances". That was the title of de Moivre's book on probability theory. He was a French Protestant who had fled to England to escape persecution, so he wrote in English.

For the actual derivation of the expression $\sigma/\sqrt{n}$, any of a number of theory-of-statistics books or books on probability theory go through it. There's the one by de Groot and Schervish, or the one by Bernard Lindgren.

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