Transforming To A Uniform Distribution
If we observe a set of n values X1, ..., Xn with no ties (i.e. there are n distinct values), we can replace Xi with the transformed value Yi = k, where k is defined such that Xi is the kth largest among all the X values. This is called the rank transform, and creates data with a perfect fit to a uniform distribution. This approach has a population analogue. If X is any random variable, and F is the cumulative distribution function of X, then as long as F is invertible, the random variable U = F(X) follows a uniform distribution on the unit interval .
From a uniform distribution, we can transform to any distribution with an invertible cumulative distribution function. If G is an invertible cumulative distribution function, and U is a uniformly distributed random variable, then the random variable G−1(U) has G as its cumulative distribution function.
Read more about this topic: Data Transformation (statistics)
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