Z statistic

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A commonly used transformation of a normal distribution.

The resulting standardised normal distribution has a mean of 0 and a standard deviation of 1.


It is used extensively in hypothesis testing.


The Z statistic is also known as the 'Z score'.


So - for example - if a data point has a Z score (or Z statistic) of -1.64, then it lies 1.64 standard deviations below the mean.

The Z score is calculated as the difference between the data point (X) and the mean E[x], all divided by the standard deviation of the population (SD).


For example if:

the mean (E[x]) of a population = 100;

the standard deviation (SD) = 10; and

a given observation (or data point) = 83.6;

then the Z score (Z) is calculated as: Z = (X - E[x])/SD

= (83.6 - 100 = -16.4)/10

= - 1.64 standard deviations.


In this case the Z score is negative, indicating that the data point (83.6) lies below the mean (of 100).


See also