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1
Notation
Let y 1 < y 2 < L (b) if Ci < 1
R (^) i = CCi− 1 if TIES = LOW
R (^) i = CCi if TIES = HIGH
R (^) i = CC (^) i − 1 +Ci 2 if TIES = MEAN
R (^) i = i if TIES = CONDENSE
Note: CC 0 = 0.
RFRACTION ( RFi )
Fractional rank: RFi = R (^) iW , i = 1, K,m
PERCENT ( P i )
Fractional rank as a percentage:
P
R
i (^) W = i× 100 , i = 1, K,m
PROPORTION ( Fi ): Estimate for Cumulative Proportion
The proportion is calculated for each case based on four different methods of estimating fractional rank:
Fi = 4 Ri − (^38) 9 4W +^149 (BLOM)
Fi = 4 Ri − 129 W (RANKIT)
Fi = 4 Ri − (^13) 9 4W +^139 (TUKEY)
Fi = R (^) i 1 W + 16 (Van der Waerden)
Note: Fi will be set to SYSMIS if the calculated value of Fi by the formula is negative.
where
i CC i CC W
W W
W W
g CC i g CC i
i i
i i i i
1 1 2
1 1 2
1 2
% & ' = − = −
−
∗
−
if is an integer if is not an integer
and l 1 , K ,l (^) w∗ are defined as the expected values of the order statistics from an exponential distribution; that is
l W K
j K
j
= ∗−^ +
∑
References
Blom, G. 1958. Statistical estimates and transformed beta variables. New York: John Wiley & Sons, Inc.
Chambers, J. M., Cleveland, W. S., Kleiner, B., and Tukey, P. A. 1983. Graphical methods for data analysis. Belmont, Calif.: Wadsworth International Group; Boston: Duxbury Press.
Lehmann, E. L. 1975. Nonparametrics: Statistical Methods Based on Ranks. San Francisco: Holden-Day.
Tukey, J. W. 1962. The future of data analysis. The Annals of Mathematical Statistics, 33: 1–67 (Correction: 33: 812)
