下面的推導(dǎo)過程中,也用x代替平均數(shù)．
(x1-x)^2+(x2-x)^2+ … +(xn-x)^2
= x1^2 - 2x1x + x^2 + x2^2 - 2x2x + x^2 + ...+ xn^2 - 2xn*x + x^2
= x1^2 + x2^2 +...+ xn^2 - 2(x1x + x2x + ...+ xn*x) + nx^2
= x1^2 + x2^2 +...+ xn^2 - 2x(x1 + x2 + ...+ xn) + nx^2
= x1^2 + x2^2 +...+ xn^2 - 2x * nx + nx^2
= x1^2 + x2^2 +...+ xn^2 - nx^2
將此結(jié)果代入s^2=1/n[(x1-x)^2+(x2-x)^2+ … +(xn-x)^2：
s^2=(1/n)[(x1-x)^2+(x2-x)^2+ … +(xn-x)^2
= (1/n)(x1^2 + x2^2 +...+ xn^2 - nx^2)
= (x1^2 + x2^2 +...+ xn^2 )/n - x^2