A value x belonging to a distribution with mean 'x_mean' and standard deviation 's' can be transformed to a standard score, or z-score, in the following manner:
z = (x - x_mean)/s
The mean of standard scores is zero. When values are standardized, the units in which they are expressed are equal to the standard deviation, s. For the standardized scores, the standard deviation becomes 1. (Variance is also 1). The interpretation of the standard-score of a given value is in terms of the number of standard deviations the value is above or below the mean (of the distribution of standardized scores).
So, the standardization of a set of values involves two steps. First, the mean is subtracted from every value, which shifts the central location of the distribution to 0. Then the thus mean-shifted values are divided by the standard deviation, s. This now makes the standard deviation as 1.
normalization, mean shifting, autoscaling
Frank, H. and Althoen, S.C. "The standard score, z" žB.3 in Statistics: Concepts and Applications Cambridge, Great Britain: Cambridge University Press, pp. 92-94, 1995.
Cite This As:
Dogra, Shaillay K., "Standard Score." From QSARWorld--A Strand Life Sciences Web Resource.