It’s calibration time!

Published 27 February 2017 by Andy Connelly. Updated 9th May 2017.

Introduction

For most measurements in analytical chemistry some form of calibration curve is required [1]. The better the calibration of the instrument or method the more accurate and precise are the results that you can achieve. The key issues you need to consider when planning a calibration are:

  • Calibration standards: the number of calibration standards and their concentration
  • Line of best fit: shape and how it is formed
  • Quality of fit: how good the fit is to the calibration standards
  • Standard additions: the alternatives if the appropriate calibration standards are not available.

But first we need to understand how calibration curves work.

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.

How calibration curves work

A calibration curve is constructed from measurements of a number of calibration standards (normally of varying concentrations). The machine supplies the readings (e.g. signal) and you supply the calibration standard values (e.g. concentration). The curve that is produce can then be used as a reference for all other results.

Figure 1 shows the a construction of a calibration curve. A calibration curve such as this can then be used to calculate a concentration of a sample from a measured signal.

In a calibration curve the instrument reading should be on the y-axis, the calibration standard’s values on the x-axis. This is because many of the equations for calculating calibration error are based on this arrangement. They also often assume no uncertainty in the x-axis – so get your standards right!

calibration_curves.png
Figure 1: How to build a calibration curve.

Calibration standards

In general, for a linear calibration curve, the minimum number of standards is three. However, five or six is usually recommended. There are two main exceptions to this:

  1. Very well understood method for which fewer standards have been shown to be sufficient (e.g. ion chromatography, pH). Normally this will be for a highly linear system (i.e. response of detector is linear over several orders of magnitude). In this situation 3 is still normal but occasionally fewer are acceptable.
  2. A systems where there is significant variability between measurements, in which case more calibration standards are potentially required – although you may need to look at where the variability is coming from.

When making up standards you should use the same matrix that the analyte of interest is sitting in (e.g. 10% acid). You should also include “matrix blanks” which have the sample matrix as your samples but with none of the analyte. These blanks can also be included as a point on your calibration curve but only if they fit in with the other values. For example, if your standards are at 1001, 1002, 1003, etc. then a blank at 0 would not be appropriate in a calibration curve as it would act as an outlier (see Figure 3).

The concentration of the standards should be evenly spaced in composition. When running calibration standards there are two schemes that tend to be used:

  • Lowest to highest concentration – this can help reduce effects of any potential carry over between analysis as lower concentration does not have as much effect on higher concentrations. However, in this scheme carry over is very difficult to identify as it has the same effect on each subsequent sample.
  • Random order – this is considered ideal as it can help identify carry over. Any carry over present will produce a higher correlation coefficient (r^2) value.

Interpolation NOT extrapolation

You should ensure that the values (e.g. concentration) of the standards you prepare covers the complete range of your samples. This is important for two primary reasons:

  1. For most techniques you cannot guarantee that the instrument response will follow the same curve shape outside of the calibration standards you have prepared;
  2. For various mathematical reasons your results are always most precise near the centroid (middle) of the calibration range and so you want your samples be to near the centroid of the line of best fit.

Line of best fit

For most of the techniques used in physical science laboratories you will require linear calibration curves. If the technique requires a form other than linear then you will need to do some more reading because life/maths just got complicated!

A linear calibration curve (line of best fit) is normally calculated using a least squares fitting algorithm. This is what Excel does, it just minimises the different between your points and some line it draws between them and then gives you the equation of that line in the form.

You can then persuade Excel to give you a quality of fit measurement for that line which it calls R^2. This is basically a measure of how far the points you have measured are from the line it has drawn; it is sometimes called a ‘test of linearity’ – see below. Unfortunately, it does not give you an estimation of the uncertainties of any values you take from this curve. These uncertainties are important and should be calculated separately.

Measuring quality of fit (correlation coefficient)

The correlation coefficient ‘r’ is a method to check linearity. The measurement R^2 (Excel) is exactly equivalent to r^2 if the line of best fit is linear. The value r^2 is used, as opposed to r, as this removes problems of negative r values.

Equation 1 shows how to calculate r, it is not difficult but it is time consuming, luckily there is a function in Excel to make life easier. The closer to unity (one) that r^2 is the better the fit. A VERY ROUGH guide to this can be seen in Table 1.

r-eqn
Equation 1: Correlation coefficient equation.
Correlation coefficient values
Table 1. Correlation coefficient values. This is ONLY A GUIDE! It will vary from technique to technique. In some techniques achieving 70% will be an fantastic achievement.

Figure 2 show what various values that r^2 takes for different deviations of data from linearity.

r2 values
Figure 2: Graphs showing the effect of different degrees of uncertainty on r2 values.

Figure 3 highlights the importance of not having outliers in your choice of calibration standards. Calibration standards need to be evenly spaced as any outliers have a significant effect on the line. It also shows that r^2 is a very crude measure of correlation. An outlier such as the point at x=25 can make a random set of points into what looks like a good correlation. This should be avoided.

Outliers and r2
Figure 3: Graphs showing the effect of outliers on r^2 value. The first graph shows a random set of data points with a low r^2. The second shows the same data points but with an outlier added. In this way the r^2 has go to a respectable 0.98.

Standard Additions

In some circumstances it is not possible to construct a calibration curve from standards [2]. This is often because of a very variable matrix which would mean using different standards for every sample or due to an unusual matrix causing some issues with the analysis.

In this technique a calibration curve is constructed by spiking your sample with known amounts of a standard. A calibration curve can then be constructed (Figure 4).

How to construct a calibration curve using standard additions method.
Figure 4: How to construct a calibration curve using standard additions method. A, B, and C are known amounts of standard.

Summary

Getting your calibration right is key to producing good data. Do not take it for granted – make sure you think it through before you start.

Glossary

  • Analyte: a substance whose chemical constituents are being identified and measured – i.e. thing you are interested in.
  • Carry over: a small amount of sample/analyte from the previous sample that remains in the system during the next analysis. This can be a small amount of sample/analyte stuck to the sampling tube or sample stuck in the instrument. Blank samples (e.g. ultra pure water) can be used to detect carry over.
  • Matrix: the solution/material that the analyte sits in/is dissolved in. For example, 10% HCl, granite rock, or peat.

References

[1] For more information on calibration curves see: Preparation of Calibration Curves: A Guide to Best Practice, September 2003, LGC/VAM/2003/032.

[2] See http://en.wikipedia.org/wiki/Standard_addition for more details.

Further reading

  1. Statistics and Chemometrics for Analytical Chemistry, Miller & Miller, 5th ed. Pearson (2005)
  2. Data analysis for chemistry: An introductory guide for students and laboratory scientists, Hibbert & Gooding, 2006
  3. Statistics: A guide to the use of statistical methods in the physical sciences. Roger Barlow, John Wiley & Sons, 1989.
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