- Standard score
In

statistics , a**standard score**is a dimensionless quantity derived by subtracting thepopulation mean from an individual raw score and then dividing the difference by the populationstandard deviation . This conversion process is called**standardizing**or**normalizing**.Standard scores are also called

**z-values, "z"-scores, normal scores,**and**standardized variables**.The standard score indicates how many

standard deviation s an observation is above or below the mean. It allows comparison of observations from different normal distributions, which is done frequently in research.The standard score is not the same as the

z-factor used in the analysis ofhigh-throughput screening data, but is sometimes confused with it.**Formula**The quantity "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.

A key point is that calculating "z" requires the population mean and the population standard deviation, not the sample mean or sample deviation. It requires knowing the population parameters, not the statistics of a sample drawn from the population of interest. But knowing the true standard deviation of a population is often unrealistic except in cases such as

standardized testing , where the entire population is measured. In cases where it is impossible to measure every member of a population, the standard deviation may be estimated using a random sample. For example, a population of people who smokecigarette s is not fully measured.When a population is normally distributed, the

percentile rank may be determined from the standard score and statistical tables.**tandardizing in mathematical statistics**In

mathematical statistics , arandom variable "X" is**standardized**using the theoretical (population) mean and standard deviation::$Z\; =\; \{X\; -\; mu\; over\; sigma\}$

where μ = E("X") is the

mean and σ = thestandard deviation of theprobability distribution of "X".If the random variable under consideration is the

sample mean ::$ar\{X\}=\{1\; over\; n\}\; sum\_\{i=1\}^n\; X\_i$

then the standardized version is

:$Z=\{ar\{X\}-muoversigma/sqrt\{n.$

**References**Richard J. Larsen and Morris L. Marx: "An Introduction to Mathematical Statistics and Its Applications, Third Edition," p. 282.

**External links*** [

*http://www.stats4students.com/Essentials/Standard-Score/Overview.php A Guide to Understanding & Calculating the Standard Score (Z-Score)*]

* [*http://www.acposb.on.ca/conversion.htm Z-Score to percentile conversion table*] With a given Z-Score, calculate the value's percentile rank.

* [*http://www.measuringusability.com/pcalcz.php Z-Score to percentile calculator*] Converts Z-Scores into percentiles (1 & 2 Sided).

* [*http://www.uark.edu/misc/lampinen/tutorials/normal.htm Normal Distribution & calculation of Z-Scores and percentile rank with Excel functions*]**ee also***

Z-Score Financial Analysis Tool

*Z-test

*Z-factor

*moment (mathematics)

*central moment

*sampling distribution

*Student's t-test

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