Z-Score

A z-score is obtained by subtracting the mean from the value of a variable and then dividing the result by its standard deviation. Such a standardized score is the number of standard deviations that the value of a case is removed from the mean. It shows the relative status of that score in a distribution.

In social science research, we often replace original measurement scores by new ones, a procedure that is called transformation. The original scores are called raw scores, the new ones transformed scores. We may denote these as x and x′, respectively. If the transformation is linear, so that x′ = ax + b, then it holds for the mean as well as the standard deviation that there will be a fixed relationship between the original value and the transformed value. For the mean, x¯′ = ax¯ + b, and for the standard deviation, sx′ = asx. It follows that a = sx′/sx and b = x¯′ − ax¯ =x¯′ − (sx′/sx)x¯. Hence, the transformed scores can be written as x′ = ax + b = (sx′/sx)x +x¯′ − (sx′/sx)x¯ = (sx′/sx)(x − x¯) + x¯′. In this formula, the mean and the standard deviation of the raw scores (x¯ and sx) are known from the data. Before transformation, we have to decide what the mean and standard deviation of the transformed scores will be. A popular choice is x¯′ = 0 and sx = 1. Then, the transformed score becomes x′ = (sx′/sx)(x − x¯) + x¯′ = (1/sx)(x − x¯) + 0 = (x − x¯)/sx. This transformed score is known as the standardized score or z-score, and the transformation is called z-transformation: z = (x − x¯)/sx. If we deal with the population instead of the sample, then Greek instead of Latin letters are used, so that z = (x − μ)/σ.