등차수열 값의 표준편차

① 평균 $\displaystyle m = \frac{( a - 15 d ) + ( a - 14 d ) + \cdots + a + \cdots + ( a + 15 d )}{31}$$\displaystyle = \frac{31 a}{31} = a$ ② 분산$= E ( X ^{2} ) - E ^{2} ( X )$에서 $\displaystyle \frac{( a - 15 d ) ^{2} + ( a - 14 d ) ^{2} + \cdots + a ^{2} + \cdots + ( a + 15 d ) ^{2}}{31}$$- a ^{2}$ $\displaystyle = \frac{31 a ^{2} + 2 ( 1 + 2 ^{2} + \cdots 15 ^{2} ) d ^{2}}{31} - a ^{2}$ $\displaystyle = \frac{2}{31} ( 1 ^{2} + 2 ^{2} + \cdots + 15 ^{2} ) d ^{2} = 80 d ^{2}$ ③ 표준편차 $\displaystyle \sigma = 4 \sqrt{5} d$에서 $\displaystyle \frac{\sigma}{d} = 4 \sqrt{5}$