Editing Standard score (section)

== Calculation==

If the population mean and population standard deviation are known, a raw score 
{{mvar|x}} is converted into a standard score by<ref>{{cite book |author=E. Kreyszig |author-link=Erwin Kreyszig |edition=Fourth |year=1979 |title=Advanced Engineering Mathematics |publisher=Wiley |isbn=0-471-02140-7 |page=880, eq. 5}}</ref>

:<math>z = {x- \mu \over \sigma}</math>

where:

: ''μ'' is the [[mean]] of the population,
: ''σ'' is the [[standard deviation]] of the population.

The absolute value of {{mvar|z}} represents the distance between that raw score {{mvar|x}} and the population mean in units of the standard deviation. {{mvar|z}} is negative when the raw score is below the mean, positive when above.

Calculating {{mvar|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 (statistics)|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.<ref name="SpiegelStephens2008">{{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 }} </ref><ref name="Mendenhall2007">{{Citation |last1= Mendenhall |first1= William |last2= Sincich
|first2= Terry |title= Statistics for Engineering and the Sciences |edition=Fifth |year=2007 |publisher= Pearson / Prentice Hall |isbn= 978-0131877061 }} </ref><ref name="GlantzSlinker2016">{{Citation |last1= Glantz |first1= Stanton A. |last2= Slinker |first2= Bryan K. |last3= Neilands |first3= Torsten B. |title= Primer of Applied Regression & Analysis of Variance |edition= Third |year=2016 |publisher= McGraw Hill |isbn= 978-0071824118 }} </ref><ref name="Aho2014">{{Citation |last1= Aho |first1= Ken A. |title= Foundational and Applied Statistics for Biologists |edition= First |year=2014 |publisher= Chapman & Hall / CRC Press
|isbn= 978-1439873380}} </ref>

In these cases, the '''z'''-score is given by

:<math>z = {x- \bar{x} \over S}</math>

where:

:<math> \bar{x} </math> 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]].