Some tools are so useful and intuitive that they achieve widespread acceptance without recommendation. Percent relative standard
deviation (%RSD) is one such tool. By formula, it is the standard deviation of a data set divided by the average of the data
set multiplied by 100. Conceptually, it is the variability of a data set expressed as a percentage relative to its location.
Statisticians know it as the coefficient of variation (CV) (1).
Lynn D. Torbeck
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Analytical laboratories measure and understand variability to estimate product quality. For most analytical laboratory staff
and management, the %RSD is used extensively as a universal measure of that variability. It is a general yardstick for chemists
for analysis and interpretation.
The understandable appeal lies in its simplicity and in its expression as a percentage. People interpret percentages more
easily than the standard deviation in units of, for example, mg/mL.
An advantage of the %RSD is that comparisons can be made across dissimilar results. In fact, the first use was to compare
men and women. In 1895, the geneticist and statistician Karl Pearson calculated the averages and standard deviations of male
and female internal organs. The men were larger and had larger variability. To compare them on the same basis, he adjusted
the standard deviation of each by dividing by the average of each. Multiplying by 100 then yielded a percentage.
%RSD is most beneficial for comparisons where the units of the measurements are different, for example mg/mL and absorbance.
It provides a common base for comparison.
Table I: Example of a misleading standard deviation (SD).
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While %RSD is clearly useful, care is needed in the application and interpretation because of its statistical properties.
For example, it should only be used where zero for the measurement has real physical meaning such as length, weight, or area
under the curve. Don't use it where zero is arbitrary, such as pH.
Table II: Example of a standard deviation (SD) that is proportional to the average.
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The %RSD is also useful when the standard deviation is proportional to the average. If the standard deviation doesn't change
much over the range of the group averages, it can be misleading as shown in Table I. Note that the %RSD is changing because
the average is changing, not the standard deviation. In Table II, the standard deviation is proportional to the average and
the %RSD is unchanged.
Good documentation requires reporting the sample size, average, and standard deviation with the %RSD. The %RSD is misleading
when used with data that are expressed as a percentage such as percent recovery. The standard deviation of the data is already
in percent. To then find the %RSD is to make the results dependent on the random variation of the average. This defeats the
original purpose of estimating the variation of the data values.
How to work with the percent relative standard deviation tool (%RSD).
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The %RSD is not useful for data with a very small average. As the average gets smaller, the %RDS gets larger, approaching
infinity as the average approaches zero, which is not a desirable result. It is not useful for limit of quantitation and limit
of detection for example.
Estimates of the standard deviation and thus the %RSD are very poor for small sample sizes. There is considerable variation
in the estimates of variation when the sample sizes are small. For example, for a sample of 10 data points with 95% confidence,
the maximum percent error in estimating the standard deviation can be as large as 50% (2). This variation also illustrates
that a calculated standard deviation or %RSD is not a fixed value. Another sample would yield a different value for both.
Following the dictum of the right tools for the job, use the %RSD but do it correctly.
Lynn D. Torbeck is a statistician at Torbeck and Assoc.,2000 Dempster Plaza, Evanston, IL 60202, tel. 847.424.1314, Lynn@Torbeck.org
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http://www.torbeck.org/.
References
1. K. Pearson, Phil. Tans. Royal Soc. London (A), 187, 253–318 (1896).2. L. S. Nelson, Jrnl. of Quality Technol., 8 (3), 179–180 (1976).
