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Variation in random samples

Reference

^ Benson, M.A. 1960. Characteristics of frequency curves based on a theoretical 1000 year record. In: T.Dalrymple (Ed.), Flood frequency analysis. U.S. Geological Survey Water Supply Paper, 1543-A, pp. 51-71.

Binomial confidence belts

Different picture formats
jpg file
tif file, less clear

Probability paper

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

Inversion of skewness

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.

[1] Benson, M.A. 1960. Characteristics of frequency curves based on a theoretical 1000 year record. In: T.Dalrymple (Ed.), Flood frequency analysis. U.S. Geological Survey Water Supply Paper, 1543-A, pp. 51-71.

[1]