# Relative standard deviation

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In probability theory and statistics, the relative standard deviation (RSD or %RSD) is the absolute value of the coefficient of variation. It is often expressed as a percentage. A similar term that is sometimes used is the relative variance which is the square of the coefficient of variation.[1] Also, the relative standard error is a measure of a statistical estimate's reliability obtained by dividing the standard error by the estimate; then multiplied by 100 to be expressed as a percentage.

The relative standard deviation is widely used in analytical chemistry to express the precision and repeatability of an assay. It is also commonly used in fields such as engineering or physics when doing quality assurance studies and ANOVA gauge R&R.

The equation of the relative standard deviation, given as a percentage is as follows:

${\%RSD} = \frac{s}{\bar{x}} \times 100$

where ${s}$ is equal to the standard deviation, and $\bar{x}$ is equal to the mean.[2] A lower percentage indicates a lower variability in the data set. Equally, a higher percentage indicates the data set is more varied.

## Examples

A data set of [100, 100, 100] has constant values. Its standard deviation is 0 and average is 100:

100% × 0 / 100 = 0%

A data set of [90, 100, 110] has more variability. Its standard deviation is 8.16 and its average is 100:

100% × 8.16 / 100 = 8.16%

A data set of [1, 5, 6, 8, 10, 40, 65, 88] has more variability again. Its standard deviation is 32.90 and its average is 27.875:

100% × 32.90 / 27.875 = 118.04%

## Notes

1. ^ Dodge, Y. (2003) The Oxford Dictionary of statistical Terms, OUP. ISBN 0-19-920613-9
2. ^ "Calculations for Assessing Data Quality". Retrieved 13 September 2013.