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1
Notation
The following notation is used throughout this chapter unless otherwise stated:
X (^) i Value of the variable for case^ i wi Weight for case^ i N Number of cases Wi Sum of the weights for the first^ i^ cases X (^) i Mean for the first^ i^ cases
Moments
Moments about the mean are calculated recursively using a provisional means algorithm (Spicer, 1972):
W w
v w W
X X
M M v M v M W w W w v W W
M M v M
W W
w W w v
M M
W W
w v
X X v
W X M M M
j i i
j
j j j j j
j j j j j j j j j j j j j
j j j j j j j j j j
j j j j j j
j j j
โ ^
= โ + โ
= โ
โ โ โ โ โ
โ โ โ
โ โ
โ
1 1
(^4413 ) (^21) 3 (^41)
3 31 2 1 1 2 3
2 2 1 1 2
1
0 0 02 03 04
3 8
3 8
After the last observation has been processed, WN = sum of weights for all cases X (^) N = mean
M (^) Nr^ wi X (^) i Xr i
N = โ =
โ 3 8 1
Basic Statistics
Mean
X (^) N
Variance
S 2 = M (^) N^2 1 W (^) Nโ (^16)
Standard Deviation
S = S^2
Standard Error
S S
X WN
Z -Scores
Z X^ X
i (^) S = i^ โ N
If X (^) i is missing or S โค 0 , Z (^) i is set to the system missing value.
References
Bliss, C. I. 1967. Statistics in biology, Volume 1. New York: McGraw-Hill. Spicer, C. C. 1972. Algorithm AS 52: Calculation of power sums of deviations about the mean. Applied Statistics, 21: 226โ227.
