Mean, Variance, and Standard Deviation

Consider some physical system $A$. Suppose that a measurement of a particular physical property of this system, $x$, can result in one of $R$ distinct outcomes. Let outcome $r$ be associated with $x$ taking the value $x_r$. Consider an ensemble of $N$ systems that are identical to system $A$. Let $N_r$ be the number of systems in the ensemble that exhibit the outcome $r$. The mean value of $x$ is, by definition, the average of a very large number of measurements of $x$. In other words,

$$\langle x\rangle = ~_{\lim N\rightarrow\infty} \sum_{r =1,R}\frac{N_r\,x_r}{N},$$ (5.20)

which implies that

$$\langle x\rangle = \sum_{r=1,R}P_r\,x_r,$$ (5.21)

where use has been made of Equation (5.1).

Let

$${\mit\Delta}x = x - \langle x\rangle$$ (5.22)

measure the deviation of an individual measurement of $x$ from the mean value. Obviously,

$$\langle{\mit\Delta}x\rangle = \langle x\rangle -\langle\langle x\rangle\rangle = \langle x\rangle -\langle x\rangle =0.$$ (5.23)

In other words, the mean deviation from the mean value is zero.

Consider $\langle({\mit\Delta} x)^2\rangle$. This quantity, which is known as the variance of $x$, is positive definite. It can only take the value 0 if all measurement of $x$ result in the mean value. Thus, the variance of $x$ measures the degree of scatter about the mean value. It follows that

$$\left\langle({\mit\Delta} x)^2\right\rangle=\left\langle \left(x ... ...angle=\left\langle x^2\right\rangle - 2\,\langle x\rangle^2+\langle x\rangle^2,$$ (5.24)

or

$$\left\langle({\mit\Delta} x)^2\right\rangle= \left\langle x^2\right\rangle -\langle x\rangle^2.$$ (5.25)

Finally, the quantity

$$\sigma_x =\left[ \left\langle({\mit\Delta} x)^2\right\rangle\right]^{1/2}$$ (5.26)

is known as the standard deviation of $x$. The standard deviation is essentially the width of the range of probable values over which $x$ is distributed around its mean value, $\langle x\rangle$.