Standard score

Applications

Z-test

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. As it is very unusual to know the entire population, the t-test is much more widely used.

Prediction intervals

The standard score can be used in the calculation of prediction intervals. A prediction interval [L,U], consisting of a lower endpoint designated L and an upper endpoint designated U, is an interval such that a future observation X will lie in the interval with high probability ${\displaystyle \gamma }$, i.e.

$${\displaystyle Pr(L<X<U)=\gamma ,}$$

For the standard score Z of X it gives: $${\displaystyle Pr\left(\frac{L-\mu }{\sigma }<Z<\frac{U-\mu }{\sigma }\right)=\gamma .}$$ By determining the quantile z such that $${\displaystyle Pr\left(-z<Z<z\right)=\gamma }$$ it follows: $${\displaystyle L=\mu -z\sigma ,U=\mu +z\sigma }$$

Process control

In process control applications, the Z value provides an assessment of the degree to which a process is operating off-target.

Comparison of scores measured on different scales: ACT and SAT

When scores are measured on different scales, they may be converted to z-scores to aid comparison. Dietz et al. 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. Suppose that student A scored 1800 on the SAT, and student B scored 24 on the ACT. Which student performed better relative to other test-takers?

The z-score for student A is ${\displaystyle z=\frac{x-\mu }{\sigma }=\frac{1800-1500}{300}=1}$

The z-score for student B is ${\displaystyle z=\frac{x-\mu }{\sigma }=\frac{24-21}{5}=0.6}$

Because student A has a higher z-score than student B, student A performed better compared to other test-takers than did student B.

Percentage of observations below a z-score

Continuing 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.

Cluster analysis and multidimensional scaling

"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 Component Analysis

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

Standardization of variables prior to multiple regression analysis is sometimes used as an aid to interpretation. (page 95) state the following.

"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."

However, Kutner et al. (p 278) give the following caveat: "… one must be cautious about interpreting any regression coefficients, whether standardized or not. The reason is that when the predictor variables are correlated among themselves, … the regression coefficients are affected by the other predictor variables in the model … The magnitudes of the standardized regression coefficients are affected not only by the presence of correlations among the predictor variables but also by the spacings of the observations on each of these variables. Sometimes these spacings may be quite arbitrary. Hence, it is ordinarily not wise to interpret the magnitudes of standardized regression coefficients as reflecting the comparative importance of the predictor variables."

Standardizing in mathematical statistics

In mathematical statistics, a random variable X is standardized by subtracting its expected value ${\displaystyle \mathrm{E}[X]}$ and dividing the difference by its standard deviation $\sigma (X)=\sqrt{\mathrm{Var}(X)}$:

$${\displaystyle Z=\frac{X-\mathrm{E}[X]}{\sigma (X)}}$$

If the random variable under consideration is the sample mean of a random sample ${\displaystyle X_1,\dots ,X_n}$ of X:

$${\displaystyle \overline{X}=\frac{1}{n}\sum _{i=1}^n{X}_i}$$

then the standardized version is

$${\displaystyle Z=\frac{\overline{X}-\mathrm{E}[\overline{X}]}{\sigma (X)/\sqrt{n}}}$$

Where the standardised sample mean's variance was calculated as follows:

$${\displaystyle \begin{array}{c}\mathrm{Var}\left(\sum _i{x}_i\right)=\sum _i\mathrm{Var}(x_i)=n\mathrm{Var}(x_i)=n{\sigma }^2\\ \mathrm{Var}(\overline{X})=\mathrm{Var}\left(\frac{\sum _i{x}_i}{n}\right)={\displaystyle \frac{1}{n^2}}\mathrm{Var}\left(\sum _i{x}_i\right)={\displaystyle \frac{n{\sigma }^2}{n^2}}={\displaystyle \frac{{\sigma }^2}{n}}\end{array}}$$

T-score

In educational assessment, T-score is a standard score Z shifted and scaled to have a mean of 50 and a standard deviation of 10. It is also known as hensachi in Japanese, where the concept is much more widely known and used in the context of high school and university admissions.

In bone density measurement, 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.