(p 278) give the following caveat: "… one must be cautious about interpreting any regression coefficients, whether standardized or not. "The standardized regression slope is the slope in the regression equation if X and Y are standardized … Standardization of X and Y is done by subtracting the respective means from each set of observations and dividing by the respective standard deviations … In multiple regression, where several X variables are used, the standardized regression coefficients quantify the relative contribution of each X variable." Standardization of variables prior to multiple regression analysis is sometimes used as an aid to interpretation. In principal components analysis, "Variables measured on different scales or on a common scale with widely differing ranges are often standardized." Relative importance of variables in multiple regression: standardized regression coefficients "For some multivariate techniques such as multidimensional scaling and cluster analysis, the concept of distance between the units in the data is often of considerable interest and importance… When the variables in a multivariate data set are on different scales, it makes more sense to calculate the distances after some form of standardization." Principal components analysis Percentage of observations below a z-score Ĭontinuing the example of ACT and SAT scores, if it can be further assumed that both ACT and SAT scores are normally distributed (which is approximately correct), then the z-scores may be used to calculate the percentage of test-takers who received lower scores than students A and B.Ĭluster analysis and multidimensional scaling
X is converted into a standard score by z = x − μ σ īecause student A has a higher z-score than student B, student A performed better compared to other test-takers than did student B. If the population mean and population standard deviation are known, a raw score Ĭomputing a z-score requires knowledge of the mean and standard deviation of the complete population to which a data point belongs if one only has a sample of observations from the population, then the analogous computation using the sample mean and sample standard deviation yields the t-statistic. Other equivalent terms in use include z-value, z-statistic, normal score, standardized variable and pull in high energy physics. Standard scores are most commonly called z-scores the two terms may be used interchangeably, as they are in this article. This process of converting a raw score into a standard score is called standardizing or normalizing (however, "normalizing" can refer to many types of ratios see Normalization for more).
It is calculated by subtracting the population mean from an individual raw score and then dividing the difference by the population standard deviation. Raw scores above the mean have positive standard scores, while those below the mean have negative standard scores. In statistics, the standard score is the number of standard deviations by which the value of a raw score (i.e., an observed value or data point) is above or below the mean value of what is being observed or measured. Comparison of the various grading methods in a normal distribution, including: standard deviations, cumulative percentages, percentile equivalents, z-scores, T-scores For other uses, see Z-score (disambiguation).
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