Standard score

Standard scores are most commonly called z-scores; the two terms may be used interchangeably, as they are in this article.

Other equivalent terms in use include z-value, z-statistic, normal score, standardized variable and pull in high energy physics.

[1][2] Computing 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.

[4][5][6][7] In these cases, the z-score is given by where: Though it should always be stated, the distinction between use of the population and sample statistics often is not made.

The z-score is often used in the z-test in standardized testing – the analog of the Student's t-test for a population whose parameters are known, rather than estimated.

When scores are measured on different scales, they may be converted to z-scores to aid comparison.

Dietz et al.[9] give the following example, comparing student scores on the (old) SAT and ACT high school tests.

The table shows the mean and standard deviation for total scores on the SAT and ACT.

"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.

"[11] Standardization of variables prior to multiple regression analysis is sometimes used as an aid to interpretation.

However, Kutner et al.[13] (p 278) give the following caveat: "… one must be cautious about interpreting any regression coefficients, whether standardized or not.

Hence, it is ordinarily not wise to interpret the magnitudes of standardized regression coefficients as reflecting the comparative importance of the predictor variables."

In mathematical statistics, a random variable X is standardized by subtracting its expected value

[14][15][16] In bone density measurements, the T-score is the standard score of the measurement compared to the population of healthy 30-year-old adults, and has the usual mean of 0 and standard deviation of 1.

Comparison of the various grading methods in a normal distribution , including: standard deviations , cumulative percentages, percentile equivalents, z-scores, T-scores
The z score for Student A was 1, meaning Student A was 1 standard deviation above the mean. Thus, Student A performed in the 84.13 percentile on the SAT.
The z score for Student B was 0.6, meaning Student B was 0.6 standard deviation above the mean. Thus, Student B performed in the 72.57 percentile on the SAT.