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{{Short description|How many standard deviations apart from the mean an observed datum is}}{{Use American English|date = January 2019}}{{redirect|Standardize|industrial and technical standards|Standardization}}{{redirect|Z-score}}File:The Normal Distribution.svg|thumb|upright=1.5|Comparison of the various grading methods in a normal distribution, including: standard deviations, cumulative percentages, percentile equivalents, z-scores, T-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. Raw scores above the mean have positive standard scores, while those below the mean have negative standard scores. It is calculated by subtracting the population mean from an individual raw score and then dividing the difference by the population standard deviation. 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). 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.BOOK,weblink 2015 European School of High-Energy Physics: Bansko, Bulgaria 02 - 15 Sep 2015, 2017, CERN, 978-92-9083-472-4, Mulders, Martijn, CERN Yellow Reports: School Proceedings, Geneva, Zanderighi, Giulia, JOURNAL, Gross, Eilam, 2017-11-06, Practical Statistics for High Energy Physics,weblink CERN Yellow Reports: School Proceedings, en, 4/2017, 165–186, 10.23730/CYRSP-2017-004.165, 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.

Calculation

If the population mean and population standard deviation are known, a raw score x is converted into a standard score byBOOK, E. Kreyszig, Erwin Kreyszig, Fourth, 1979, Advanced Engineering Mathematics, Wiley, 0-471-02140-7, 880, eq. 5, where: The absolute value of z represents the distance between that raw score x 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 use of the population mean and the population standard deviation, not the sample mean or sample deviation. However, knowing the true mean and standard deviation of a population is often an unrealistic expectation, 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 estimated by using the sample mean and sample standard deviation as estimates of the population values.{{Citation |last1= Spiegel |first1= Murray R. |last2= Stephens |first2= Larry J |title= Schaum's Outlines Statistics |edition=Fourth |year=2008 |publisher= McGraw Hill |isbn= 978-0-07-148584-5 }} {{Citation |last1= Mendenhall |first1= William |last2= Sincich

title= Statistics for Engineering and the Sciences

year=2007

isbn= 978-0131877061 }} {{Citation

first1= Stanton A.

first2= Bryan K.

first3= Torsten B.

edition= Third

publisher= McGraw Hill

last1= Aho

title= Foundational and Applied Statistics for Biologists

year=2014 |publisher= Chapman & Hall / CRC Press|isbn= 978-1439873380}} In these cases, the z-score is given by z = {x- bar{x} over S} where: bar{x} is the mean of the sample, S is the standard deviation of the sample. Though it should always be stated, the distinction between use of the population and sample statistics often is not made. In either case, the numerator and denominator of the equations have the same units of measure so that the units cancel out through division and z is left as a dimensionless quantity. 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 {{anchor|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 gamma, i.e. P(L