BioDeviceHub

Replicate Measurement Statistics (SD, %RSD, SE, CI)

Summarise replicate measurements: the mean, standard deviation, relative standard deviation (coefficient of variation), standard error and 95% confidence interval, with bias.

Formula

s=∑(x−xˉ)2n−1s = \sqrt{\dfrac{\sum (x - \bar{x})^2}{n - 1}}
%RSD=sxˉ×100\%\mathrm{RSD} = \dfrac{s}{\bar{x}}\times 100
SE=sn\mathrm{SE} = \dfrac{s}{\sqrt{n}}
95% CI=xˉ±t0.975, n−1×SE95\%\ \mathrm{CI} = \bar{x} \pm t_{0.975,\,n-1}\times\mathrm{SE}
nn
number of replicates
tt
the critical value of Student's t for n − 1 degrees of freedom

How it works

Repeated measurements of the same thing scatter. The standard deviation describes how much single measurements scatter, and the relative standard deviation (the coefficient of variation) expresses it as a percentage of the mean, which allows comparing the precision of measurements of different size. The standard error describes how well the mean is known, and shrinks with the square root of the number of replicates.

The 95% confidence interval uses Student's t distribution, not the normal, because the standard deviation is itself estimated from few values; with five or six replicates the t value is well above 1.96 and the interval correspondingly wider. If you supply a reference value, the tool reports the bias of the mean from it, and whether the interval contains it.

Worked example

Six replicate readings: 98.2, 101.5, 99.7, 100.4, 97.9 and 102.1.

  1. Mean = 99.97; SD = 1.706 (n − 1 = 5 degrees of freedom).
  2. %RSD = 1.706 / 99.97 × 100 = 1.71%; SE = 1.706 / √6 = 0.6965.
  3. t(0.975, 5) = 2.571, so the 95% CI is 99.97 ± 1.79, or 98.18 to 101.8.

Mean 99.97, %RSD 1.71%, 95% CI 98.18 to 101.8.

These are the values the calculator opens with, so you can check its output against this example.

Assumptions

  • Independent replicates of the same measurand, approximately normally distributed.
  • For %RSD, a quantity on a ratio scale with a true zero.

Common mistakes

  • Reporting the standard error as though it were the standard deviation.
  • Quoting a %RSD for values near zero or on an interval scale such as °C.
  • Drawing conclusions from three replicates. Their standard deviation is barely known.

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