Unbiased plotting positions — A review

Abstract

The existing attitude that the criterion for choice of plotting position is arbitrary is rebuked and it is shown that a worthwhile criterion can be based on desired statistical properties of the plot, rather than on comparison of plotting positions with estimates of probability for individual sample values. These properties are that any quantile estimate made from the plot should be unbiased and should have smallest mean square error among all such estimates. This leads to specification of plotting position initially in terms of reduced variate rather than probability value. The unbiased plotting position is E( y ( i) ), the mean of the ith order statistic in samples from the reduced variate population, which differs from one distribution to another. A good approximation for each distribution is available in the probability domain. These take the general form (i − α) (N + 1 − 2α) with α = 3 8 in the normal case and α = 0.44 in the extreme-value type-1 (EV1) and exponential cases. The Weibull formula, α = 0, is correct for the uniform distribution alone and is shown to be biased for other distributions. Hazen's formula α = 1 2 shows up much better in terms of bias than many would expect. If a single simple distribution free formula were required then α = 2 5 would be the best compromise. The plotting position postulates which have supported the Weibull formula for many years are examined and some are seen to be unreasonable in view of statistical facts.