Standard score
In statistics, the standard score is the signed number of standard deviations by which the value of an observation or data point differs from the mean value of what is being observed or measured. Observed values above the mean have positive standard scores, while values below the mean have negative standard scores. The standard score is a dimensionless quantity obtained by subtracting the population mean from an individual raw score and then dividing the difference by the population standard deviation. This conversion process is called standardizing or normalizing (however, "normalizing" can refer to many types of ratios; see normalization for more).
Standard scores are also called zvalues, zscores, normal scores, and standardized variables. They are most frequently used to compare an observation to a standard normal deviate, though they can be defined without assumptions of normality.
Computing a zscore requires knowing 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 with sample mean and sample standard deviation yields the tstatistic.
Contents
 1 Calculation from raw score

2 Applications
 2.1 Ztest
 2.2 Prediction intervals
 2.3 Process control
 2.4 Comparison of scores measured on different scales: ACT and SAT
 2.5 Percent of observations below a zscore
 2.6 Cluster analysis and multidimensional scaling
 2.7 Principal components analysis
 2.8 Relative importance of variables in multiple regression: Standardized regression coefficients
 3 Standardizing in mathematical statistics
 4 Tscore
 5 See also
 6 References
 7 Further reading
 8 External links
Calculation from raw score
If the population mean and population standard deviation are known, the standard score of a raw score x^{[1]} is calculated as
where:
 μ is the mean of the population.
 σ is the standard deviation of the population.
The absolute value of z represents the distance between the raw score and the population mean in units of the standard deviation. z is negative when the raw score is below the mean, positive when above.
Calculating z using this formula requires the population mean and the population standard deviation, not the sample mean or sample deviation. But knowing the true mean and standard deviation of a population is often unrealistic except in cases such as standardized testing, where the entire population is measured.
When the population mean and the population standard deviation are unknown, the standard score may be calculated using the sample mean and sample standard deviation as estimates of the population values.^{[2]}^{[3]}^{[4]}^{[5]}
In these cases, the z score is
where:
 is the mean of the sample.
 S is the standard deviation of the sample.
Applications
Ztest
The zscore is often used in the ztest in standardized testing – the analog of the Student's ttest for a population whose parameters are known, rather than estimated. As it is very unusual to know the entire population, the ttest 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 , i.e.
For the standard score Z of X it gives:^{[6]}
By determining the quantile z such that
it follows:
Process control
In process control applications, the Z value provides an assessment of how offtarget a process is operating.
Comparison of scores measured on different scales: ACT and SAT
When scores are measured on different scales, they may be converted to zscores to aid comparison. Diez et al. ^{[7]} give the following example comparing student scores on the SAT and ACT high school tests. The table shows the mean and standard deviation for total score 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 testtakers?
SAT  ACT  

Mean  1500  21 
Standard deviation  300  5 
The zscore for student A is
The zscore for student B is
Because student A has a higher zscore than student B, student A performed better compared to other testtakers than did student B.
Percent of observations below a zscore
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 zscores may be used to calculate the percent of testtakers who received lower scores than students A and B. The R function pnorm() and the Excel function NORM.DIST() give the probability that a random observation from a normal distribution will have a z score less than a specified z score.
For Student A, with a z score of 1 on the ACT, NORM.DIST(1,0,1,TRUE) = 0.84, indicating that 84% of students taking the ACT scored lower than Student A. For Student B, with a z score of 0.6 on the SAT, NORM.DIST(0.6,0,1,TRUE) = 0.73, indicating that 73% of students taking the SAT scored lower than Student 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." ^{[8]}
Principal components analysis
In principal components analysis, "Variables measured on different scales or on a common scale with widely differing ranges are often standardized." ^{[9]}
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. Affif et al ^{[10]} (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. ^{[11]} (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 and dividing the difference by its standard deviation
If the random variable under consideration is the sample mean of a random sample of X:
then the standardized version is
 .
Tscore
In educational assessment, Tscore is a standard score Z shifted and scaled to have a mean of 50 and a standard deviation of 10.^{[12]}^{[13]}^{[14]}
In bone density measurements, the Tscore is the standard score of the measurement compared to the population of healthy 30yearold adults.^{[15]}
See also
References
 ^ E. Kreyszig (1979). Advanced Engineering Mathematics (Fourth ed.). Wiley. p. 880, eq. 5. ISBN 0471021407.
 ^ Spiegel, Murray R.; Stephens, Larry J (2008), Schaum's Outlines Statistics (Fourth ed.), McGraw Hill, ISBN 9780071485845
 ^ Mendenhall, William; Sincich, Terry (2007), Statistics for Engineering and the Sciences (Fifth ed.), Pearson / Prentice Hall, ISBN 9780131877061
 ^ Glantz, Stanton A.; Slinker, Bryan K.; Neilands, Torsten B. (2016), Primer of Applied Regression & Analysis of Variance (Third ed.), McGraw Hill, ISBN 9780071824118
 ^ Aho, Ken A. (2014), Foundational and Applied Statistics for Biologists (First ed.), Chapman & Hall / CRC Press, ISBN 9781439873380
 ^ E. Kreyszig (1979). Advanced Engineering Mathematics (Fourth ed.). Wiley. p. 880, eq. 6. ISBN 0471021407.
 ^ Diez, David; Barr, Christopher; ÇetinkayaRundel, Mine (2012), OpenIntro Statistics (Second ed.), openintro.org
 ^ Everitt, Brian; Hothorn, Torsten J (2011), An Introduction to Applied Multivariate Analysis with R, Springer, ISBN 9781441996497
 ^ Johnson, Richard; Wichern, Wichern (2007), Applied Multivariate Statistical Analysis, Pearson / Prentice Hall
 ^ Afifi, Abdelmonem; May, Susanne K.; Clark, Virginia A. (2012), Practical Multivariate Analysis (Fifth ed.), Chapman & Hall/CRC, ISBN 9781439816806
 ^ Kutner, Michael; Nachtsheim, Christopher; Neter, John (204), Applied Linear Regression Models (Fourth ed.), McGraw Hill, ISBN 9780073014661
 ^ John Salvia; James Ysseldyke; Sara Witmer (29 January 2009). Assessment: In Special and Inclusive Education. Cengage Learning. pp. 43–. ISBN 0547134371.
 ^ Edward S. Neukrug; R. Charles Fawcett (1 January 2014). Essentials of Testing and Assessment: A Practical Guide for Counselors, Social Workers, and Psychologists. Cengage Learning. pp. 133–. ISBN 9781305161832.
 ^ Randy W. Kamphaus (16 August 2005). Clinical Assessment of Child and Adolescent Intelligence. Springer. pp. 123–. ISBN 9780387262994.
 ^ "Bone Mass Measurement: What the Numbers Mean". NIH Osteoporosis and Related Bone Diseases National Resource Center. National Institute of Health. Retrieved 5 August 2017.
Further reading
 Carroll, Susan Rovezzi; Carroll, David J. (2002). Statistics Made Simple for School Leaders (illustrated ed.). Rowman & Littlefield. ISBN 9780810843226. Retrieved 7 June 2009.
 Larsen, Richard J.; Marx, Morris L. (2000). An Introduction to Mathematical Statistics and Its Applications (Third ed.). p. 282. ISBN 0139223037.
External links
 Interactive Flash on the zscores and the probabilities of the normal curve by Jim Reed