# Standard score

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In statistics, the **standard score** is the (signed) number of standard deviations an observation or datum is *above* the mean. Thus, a positive standard score indicates a datum above the mean, while a negative standard score indicates a datum below the mean. It 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 (statistics) for more).

Standard scores are also called **z-values, z-scores, normal scores,** and

**standardized variables;**the use of "Z" is because the normal distribution is also known as the "Z distribution". They are most frequently used to compare a sample to a standard normal deviate, though they can be defined without assumptions of normality.

The z-score is only defined if one knows the population parameters; if one only has a sample set, then the analogous computation with sample mean and sample standard deviation yields the Student's t-statistic.

The standard score is not the same as the z-factor used in the analysis of high-throughput screening data though the two are often conflated.

## Contents

## Calculation from raw score

The standard score of a raw score *x* ^{[1]} is

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.

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.

It measures the sigma distance of actual data from the average.

The Z value provides an assessment of how off-target a process is operating.

## Applications

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.

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Also, 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:^{[2]}

By determining the quantile z such that

it follows:

## Standardizing in mathematical statistics

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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

See normalization (statistics) for other forms of normalization.

## See also

## References

- Kreyszig, E (fourth edition 1979).
*Applied Mathematics*, Wiley Press.

## Further reading

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- Richard J. Larsen and Morris L. Marx (2000)
*An Introduction to Mathematical Statistics and Its Applications, Third Edition,*ISBN 0-13-922303-7. p. 282.