BioDeviceHub

Accuracy and Precision Calculator

From repeated measurements of a reference, calculate the bias (trueness), the standard deviation and %RSD (precision), and the total error |bias| + 1.96 SD.

Formula

bias=xˉ−xref\text{bias} = \bar{x} - x_{\text{ref}}
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
total error=∣bias∣+1.96 s\text{total error} = |\text{bias}| + 1.96\,s
bias\text{bias}
the systematic difference from the reference (trueness)
ss
the spread of repeated results (precision)
total error\text{total error}
an estimate of the error 95% of single results stay within

How it works

Accuracy has two parts. Trueness is how close the mean of many measurements is to the true value, measured by the bias; precision is how close the measurements are to each other, measured by the standard deviation. An instrument can be precise and biased, or unbiased and imprecise. ISO 5725 and the VIM define accuracy as the combination of the two.

A single measurement carries both. A common way to combine them is the total error, the absolute bias plus 1.96 standard deviations, a figure that about 95% of single measurements stay within if the errors are normal. It is used in clinical laboratory method evaluation, where it is compared with an allowable total error.

Worked example

Eight repeats of 9.78, 9.83, 9.80, 9.85, 9.79, 9.82, 9.81 and 9.84, against a reference of 9.81.

  1. Mean = 9.815; bias = +0.005.
  2. SD = 0.02449; %RSD = 0.25%.
  3. Total error = 0.005 + 1.96 × 0.02449 = 0.0530.

The bias is 0.005, the precision 0.25% RSD, and the total error 0.053.

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

Assumptions

  • The reference is accurate and the repeats are independent.
  • Roughly normal errors, for the 1.96 factor.

Common mistakes

  • Using the repeatability SD as the total uncertainty without the bias.
  • Drawing conclusions from three or four repeats, where the SD itself is uncertain.
  • Mistaking a small SD for accuracy.

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