Mean, Variance, and Standard Deviation

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

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

(5.20)

which implies that

$\displaystyle \langle x\rangle = \sum_{r=1,R}P_r\,x_r,$

(5.21)

where use has been made of Equation (5.1).

Let

$\displaystyle {\mit\Delta}x = x - \langle x\rangle$

(5.22)

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

$\displaystyle \langle{\mit\Delta}x\rangle = \langle x\rangle -\langle\langle x\rangle\rangle = \langle x\rangle -\langle x\rangle =0.$

(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 , is positive definite. It can only take the value 0 if all measurement of result in the mean value. Thus, the variance of measures the degree of scatter about the mean value. It follows that

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

(5.24)

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

(5.25)

Finally, the quantity

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

(5.26)

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