Asitgoes

Variation in random samples

Variations of nine return period curves of 50-year samples from a theoretical 1000 year record (base line), data from Benson

Binomial confidence belts

• Different picture formats
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  jpg file
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  tif file, less clear

Probability paper

• Plotting on probability paper
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  Plot of data on normal probability paper
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  Plot of data and confidence belt on normal probability paper
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  Plot of data and confidence belt on Gumbel probability paper

Inversion of skewness

(A) Gumbel probability distribution skew to right and (B) Gumbel mirrored skew to left

Skewed distributions can be inverted (or mirrored) by replacing in the mathematical expression of the cumulative distribution function (F) by its complement: F'=1-F, obtaining the complementary distribution function (also called survival function) that gives a mirror image. In this manner, a distribution that is skewed to the right is transformed into a distribution that is skewed to the left and vice versa.

Example. The F-expression of the positively skewed Gumbel distribution is: F=exp[-exp{-(X-u)/0.78s}], where u is the mode (i.e. the value occurring most frequently) and s is the standard deviation. The Gumbel distribution can be transformed using F'=1-exp[-exp{-(x-u)/0.78s}] . This transformation yields the inverse, mirrored, or complementary Gumbel distribution that may fit a data series obeying a negatively skewed distribution.