How to use
- Paste the measurements into the box. A column copied from a spreadsheet is fine, and commas, spaces or line breaks all work as separators.
- n, the mean, the standard deviation, CV (%), the minimum, maximum and range appear as you paste.
- The standard deviation is the sample SD (n−1) by default. You can switch to the population SD (n), and the screen says which one is in use.
- Anything that could not be read is reported. Nothing is dropped in silence.
Exponents can be entered with e — for example, 1.5×10⁻⁵ is entered as 1.5e-5.
Formula & notes
The coefficient of variation expresses scatter as a proportion of the mean. It exists because a standard deviation on its own cannot be compared across quantities of different size. An SD of 2 around a mean of 100 and an SD of 2 around a mean of 10 mean quite different things; as CV they become 2% and 20% and compare directly.
- Mean
- x̄ = Σx ÷ n
- Sample SD
- s = √[ Σ(xᵢ − x̄)² ÷ (n − 1) ]
- Population SD
- σ = √[ Σ(xᵢ − x̄)² ÷ n ]
- Coefficient of variation
- CV(%) = s ÷ x̄ × 100
n−1 and n
This calculator uses the sample standard deviation (n−1) by default, because that is the standard approach for laboratory precision.
The reasoning runs like this. The twenty measurements you took are a sample of every measurement the method could produce. The mean calculated from that sample is fitted to it, so dividing the scatter by n comes out smaller than the truth. Dividing by n−1 corrects that bias.
The population SD (n) is for when the values you have are all the values there are — uncommon in experimental data.
The smaller n is, the further apart the two run. At n=5 the difference is about 11.8%; by n=30 it is under 2%.
Practical notes
- Record which basis you used. Mixing n−1 and n makes later comparison impossible, which is why this calculator states the basis on screen.
- CV loses its meaning as the mean approaches zero, because the denominator shrinks and the value runs away. At a mean of exactly zero it cannot be calculated.
- CV does not suit data that can go negative. On a scale whose zero is arbitrary — temperature in °C, or a difference — the ratio carries no meaning.
- With a small n the CV is itself unstable. A CV from three to five values is indicative only; precision studies normally use twenty or more.
- Keep within-run and between-run apart. A CV from consecutive measurements on one day and a CV across several days are different quantities, and the second is usually larger. Decide which one you are measuring first.