Significant Figures and Uncertainty
There are certain basic concepts in analytical chemistry that are helpful to the analyst when treating analytical data. The last section (part 14) addressed accuracy, precision, mean and deviation as related to chemical measurements in the general field of analytical chemistry. This section will address significant figures and uncertainty.
When working with analytical data it is important to be certain that you are using and reporting the correct number of significant figures. The number of significant figures is dependent upon the uncertainty of the measurement or process of establishing a given reported value. In a given number, the figures reported, i.e. significant figures, are those digits that are certain and the first uncertain digit. It is confusing to the reader to see data or values reported without the uncertainty reported with that value.
A sample is measured using ICP-OES and reported to contain 0.00131 ppm of Fe. This value implies with certainty that the sample contains 0.0013 ppm Fe and that there is uncertainty in the last digit (the 1). However, we know how difficult it is to make trace measurements to 3 significant figures and may be more than a little suspicious. If the value is reported as 0.00131 ± 0.00006 ppm Fe this indicates that there was an estimation of the uncertainty. A statement of how the uncertainty was determined would add much more value to the data in allowing the user to make judgments as to the validity of the data reported with respect to the number of significant figures reported.
- You purchase a standard solution that is certified to contain 10,000 ± 3 ppm boron prepared by weight using a 5-place analytical balance. This number contains 5 significant figures. However, the atomic weight of boron is 10.811 ± 5. It is, therefore, difficult to believe the data reported in consideration of this fact alone.
- The number 0.000013 ± .000002 contains two significant figures. The zeros to the left of the number are never significant. Scientific notation makes life easier for the reader and reporting the number as 1.3 x 10-5 ± 0.2 x 10-5 is preferred in some circles.
- A number reported as 10,300 is considered to have five significant figures. Reporting it as 1.03 x 104 implies only three significant figures, meaning an uncertainty of ± 100. Reporting an uncertainty of 0.05 x 104 does not leave the impression that the uncertainty is ± 0.01 x 104, i.e., ± 100.
- A number reported as 10,300 ± 50 containing four significant figures. If the number is reported as 10,300 ± 53, the number of significant figures is still 4 and the number reported this way is acceptable, but the 3 in the 53 is not significant.
Mathematical calculations require a good understanding of significant figures. In multiplication and division, the number with the least number of significant figures determines the number of significant figures in the result. With addition and subtraction, it is the least number of figures to the left or right of the decimal point that determines the number of significant figures.
- The number 1.4589 (five significant figures) is multiplied by 1.2 (two significant figures). The product, which is equal to 1.75068, would be reported as 1. 8 (two significant figures).
- The number 1.4589 (five significant figures) is divided by 1.2 (two significant figures). The dividend, which is equal to 1.21575, would be reported as 1.2 (two significant figures).
- The addition of 5.789 (four significant figures) to 105 (three significant figures) would be reported as 111.
The International Vocabulary of Basic and General Terms in Metrology (VIM) defines uncertainty as:
"A parameter associated with the result of a measurement, that characterizes the dispersion of the values that could reasonably be attributed to the measurand."
NOTE 1: The parameter may be, for example, a standard deviation (or a given multiple of it), or the width of a confidence interval.
NOTE 2: Uncertainty of measurement comprises, in general, many components. Some of these components may be evaluated from the statistical distribution of the results or series of measurements and can be characterized by standard deviations. The other components, which also can be characterized by standard deviations, are evaluated from assumed probability distributions based on experience or other information The ISO Guide refers to these different cases as Type A and Type B estimations respectively.
There are numerous publications concerning uncertainty calculations. I am concerned that many presentations on the topic are written in a language that may be difficult for the beginner to easily grasp. However, there is a clear and complete guide that I highly recommend.
Whether you're a beginner or an experienced student of the subject, I strongly encourage you to read Quantifying Uncertainty in Analytical Measurement, published by Eurachem.
Of the numerous volumes of publications on this topic I have seen over the years, this one stands out above all others. It is quite thorough, written in an understandable manner and it includes several good examples.