Statistics Knowledge Base

Wednesday, January 07, 2009

Z-Score

A z-score (also known as a standard score or a normal score) is a unit-free quantity that measures the difference between a value and the arithmetic mean of the set to which the value belongs, relative to the variability of all of the values in that set. A z-score thus reflects how many standard deviations a particular value is above or below the mean of the set to which it belongs. Z-scores are frequently compared within the familiar context of a Gaussian (normal) distribution.

PROPERTIES

Some of the important properties of z-scores include:

The rank order of a set of standardized values (z-scores) is identical to the rank order of the set of unstandardized values from which the z-scores were derived.
Values that were above or below the mean when they were unstandardized will retain that characteristic after being standardized (transformed into z-scores).
After transforming a set of values into z-scores, the sum of that set of values will be zero.
After transforming a set of values into z-scores, the arithmetic mean of that set of values will also be zero.
After transforming a set of values into z-scores, the variance of that set of values will equal one.
After transforming a set of values into z-scores, the standard deviation of that set of values will also equal one.
The absolute correlation between two variables will not change if one or both of the variables are transformed into z-scores.
The shape of the distribution for an unstandardized variable will not change after the variable is standardized.

ONLINE CALCULATOR

To calculate a z-score, please click here.

FORMULAE

The formula involved in the computation of a z-score is detailed below.

Z-score formula:
Where x is the value to be standardized, μ is the arithmetic mean, and σ is the standard deviation.

REFERENCES

Cohen, J., Cohen, P., West, S.G., and Aiken, L.S. (2003) "Applied Multiple Regression/Correlation Analysis for the Behavioral Sciences (3rd edition)", Lawrence Earlbaum Associates, Mahwah, NJ

Patel, J.K. and Read, C.B. (1982) "Handbook of the Normal Distribution", Dekker, New York, NY.