How many repeats?

Change in theoretical uncertainty with the number of times (n) a single sample is analysed.

Written by Andy Connelly. Published 16th May 2017.

When carrying out any experiment if you only measure two samples you will not know whether one or both are, by random chance, a long way from the ‘true’ value. As such, if you only have two measurements do not take the mean, report them both. The better way is to carry out the experiment 5-6 times.

DISCLAIMER: I am not an expert in analytical chemistry. The content of this blog is what I have discovered through my efforts to understand the subject. I have done my best to make the information here in as accurate as possible. If you spot any errors or admissions, or have any comments, please let me know.

Figure 1 shows the decrease in uncertainty for an increasing number of measurements. As you can see, after around five measurements the return on time invested is negligible in most circumstances. It is normally suggested to carry out six measurements as then if there is a with one of the measurements problem (e.g. you drop the tube) then you will still have five! In reality you may have to balance time, sample and money with the statistical ideal.

Figure 1 may not be true in systems with large amounts of random uncertainty (i.e. very high standard deviation) and in these systems other approaches may be required with much higher numbers of repeats. Where repeats measurements are not possible other methods of uncertainty estimation must be used.

In a general case, if you measure a sample 5 times to achieve an uncertainty you will need to measure 15 more times (a total of 20 times) to half that uncertainty. Equally, improvement by a factor of 10 implies 100 times more experiments. This relationship of decreasing uncertainty to increasing number of samples also is applicable to repeated collection of spectra (e.g. at a beam time).

uncertainty

There are also formulas that you can use to calculate the ideal number of samples to measure to achieve some target uncertainty (see [1]).

Change in theoretical uncertainty with the number of times (n) a single sample is analysed.
Figure 4: Change in theoretical uncertainty with the number of times (n) a single sample is analysed. The first 9 data points have been included to highlight the large initial decrease in uncertainty.

Summary

It is often a daunting prospect to repeat measurements but if you are clever with your experimental design it can be achieved with minimal extra effort. you don’t need to repeat every sample 6 times just key, representative samples.

References and further reading

[1] Data Analysis for Chemistry, Hibbert & Gooding, 2006.

  • Statistics and Chemometrics for Analytical Chemistry, Miller & Miller, 5th ed. Pearson (2005)
  • Statistics: A guide to the use of statistical methods in the physical sciences. Roger Barlow, John Wiley & Sons, 1989.
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