Order r

Moment M 0 ( r )

Central Moment: M μ ( r )

1

e m + 1 2 s 2

0

2

e 2 m + 2 s 2

e 2 m ( e 2 s 2 e s 2 )

3

e 3 m + 9 2 s 2

e 3 m ( e 9 2 s 2 3 e 5 2 s 2 + 2 e 3 2 s 2 )

4

e 4 m + 8 s 2

e 4 m ( e 8 s 2 4 e 5 s 2 + 6 e 3 s 2 3 e 2 s 2 )

5

e 5 m + 25 2 s 2

e 5 m ( e 25 2 s 2 5 e 17 2 s 2 + 10 e 11 2 s 2 10 e 7 2 s 2 + 4 e 5 2 s 2 )

6

e 6 m + 18 s 2

e 6 m ( e 18 s 2 6 e 13 s 2 + 15 e 9 s 2 20 e 6 s 2 + 15 e 4 s 2 5 e 3 s 2 )

7

e 7 m + 49 2 s 2

e 7 m ( e 49 2 s 2 7 e 37 2 s 2 + 21 e 27 2 s 2 35 e 19 2 s 2 + 35 e 13 2 s 2 21 e 9 2 s 2 + 6 e 7 2 s 2 )

8

e 8 m + 32 s 2

e 8 m ( e 32 s 2 8 e 25 s 2 + 28 e 19 s 2 56 e 14 s 2 + 70 e 10 s 2 56 e 7 s 2 + 28 e 5 s 2 7 e 4 s 2 )

Statistic

Symbol

Expression

Standard Deviation

σ

e m ( e 2 s 2 e s 2 )

Skewness

Sk

( e 3 s 2 3 e s 2 + 2 ) / ( e s 2 1 ) 3 2

Kurtosis

K

e 4 s 2 + 2 e 3 s 2 + 3 e 2 s 2 3

Variance of Skewness

Var (Sk)

V a r ( M μ ( 3 ) ) S 6 + 3 M μ ( 3 ) 2 V a r ( M μ ( 2 ) ) 4 S 13

Variance of Kurtosis

Var (K)

V a r ( M μ ( 4 ) ) S 8 + 2 M μ ( 4 ) 2 V a r ( M μ ( 2 ) ) S 16