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# Normal transform (z-transform)

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# Normal transform (z-transform)

This transform normalizes or standardizes the data so that its distribution has a zero mean and unit variance. In principle, if the population mean and variance are known, then the transform is of the form:

However, in practice the mean and variance are estimated from the sample data, and thus the expression is amended to:

where the divisor, s, is generally taken to be the unbiased estimator of the population standard deviation (hence computed with n-1 degrees of freedom). Transforming data simply by subtracting the mean (or mode or median in some instances) is known as centering.

If {xi} is a set of n independent sample mean values from the same probability distribution with mean μ and variance σ2>0 then the z-transform shown below as z2 will be distributed as a standard Normal distribution, N(0,1) for large n (this result is known as the Central Limit Theorem). The divisor in this instance is the standard error:

Because many data sets are actually comprised of broadly independent samples that are added together to provide the individual data values, approximation to the Normal distribution is extremely common.