Chebyshev's Inequality - Haldane's Transformation

Haldane's Transformation

One use of Chebyshev's inequality in applications is to create confidence intervals for variates with an unknown distribution. Haldane noted, using an equation derived by Kendall, that if a variate (x) has a zero mean, unit variance and both finite skewness (γ) and kurtosis (κ) then the variate can be converted to a normally distributed standard score (z):

This transformation may be useful as an alternative to Chebyshev's inequality or as an adjunct to it for deriving confidence intervals for variates with unknown distributions.