No single measurement can be known to be the "true value" that is sought after. With a good analytical method, each individual measurement is an approximation of the true value. The deviation between each measurement and the true value will be random. This means that all physical measurements are subject to error, so all results have some uncertainty. There are subtle differences between the meaning of the words uncertainty, error, and mistake:
An error is not a mistake, and uncertainty is not an error. Be careful of how things are worded - often scientists interchange the terms error and uncertainty even though they mean different things!
It is important to realize that error and uncertainty exists in all measurements, but mistakes should not. For example, if your measurement protocol (method) dictates water samples from Taconite Lake should be taken at a depth of 2m and one sample was collected at a depth of 3m then if an anomalous (unusual) result for lead concentration in that sample should not be considered due to error or uncertainty, but due to the mistake in measurement procedure. This sample's result should be discarded because it is the result of a blunder. The result should be discarded whether or not it appears to agree with the other measurements made at the proper depth, because we know a mistake was made.
If, say, 20 measurements were made correctly, there is still a possibility that a data point (result of a single measurement) may be an outlier as the variation from the true value is always random. If a data point looks like an outlier, statistical tests need to be applied to determine if it can be discarded (eliminated) or not. Outliers that come from samples measured correctly cannot be discarded just because they do not fit your opinion of what the data should look like. Statistical tests for outliers will be covered in a later module; for now, note the Grubb's test is a simple test for outliers when the same property has been measured multiple times.
Summary: There are only two reasons why data can be discarded:
Error and uncertainty can be minimized but never eliminated. Error leads to uncertainty, so we will focus on using the term uncertainty. There are
two sources of uncertainty: systematic and random.
Systematic Error
Systematic error is caused by a flaw in the experiment. This error is determinate, meaning that it has a concrete cause that can be detected, reproduced and corrected. Systematic uncertainty is unidirectional (all measurements are higher or lower than the actual true value). In the case of determining the concentration of lead in the lake, systematic uncertainty could be due to only taking water samples from one area of the lake rather than throughout the lake, or only at the surface of the water rather than at different depths. Both could be corrected by changing the way the experiment is done.
Random Error
Random error is present in every measurement and is due to uncontrollable non-permanent causes. This uncertainty cannot be corrected for as it does not have any single definite cause. Random uncertainty is bidirectional (individual measurements being randomly higher or lower than the actual value). The magnitude of random uncertainty can be decreased with replicate measurements but it cannot be eliminated entirely. For example, random uncertainty in measurements could be due to slight variations in amount of sample that is pipetted due to differences in reading a meniscus. Taking many pipetting multiple samples would reduce this uncertainty, because the individual variations would tend to cancel out. Table 1 gives further examples of sources of random and systematic uncertainty.
| Example Sources of Systematic Uncertainty | Example Sources of Random Uncertainty |
|---|---|
| Sampling by only taking measurements from one area of the lake | Small-scale inhomogeneity due to only a small portion of available water taken in each sample |
| Water temperature fluctuations causing solution concentration changes | Spectrophotometer light source intensity fluctuating randomly |
| Mis-calibrated instruments | Air currents affecting the recorded mass measurement from a balance |
Uncertainty Limits, Absolute and Relative Uncertainty
The total uncertainty in a measurement is always an estimate of the actual uncertainty because it comes from the sum of multiple random uncertainty sources. Similarly, error values must always be estimates because we can never perfectlyknow the true value of a measurement. The magnitude of a typical measurement uncertainty varies by instrument or by method. Measurement uncertainty can be described in two ways: absolute uncertainty and relative uncertainty.
Absolute uncertainty is the size of the uncertainty in a measurement. For example, a 50.00 mL volumetric flask might have an absolute uncertainty of 0.05 mL. This means that the flask can be filled and the volume measured to the nearest 0.05 mL. We would record this measurement as 50.00±0.05 mL.
Relative uncertainty is the size of the uncertainty compared to the actual result. It is calculated by dividing the absolute uncertainty by the measured value and given as a percentage.
$$\textrm{Relative percent uncertainty} = {\textrm{(absolute uncertainty)} \over \textrm{(measured value)}} \cdot 100 \textrm{%}$$
Using the example of the 50.00 mL volumetric flask, the absolute uncertainty is 0.05 mL, but the relative percent uncertainty is 0.05 mL/50.00 mL * 100% = 0.1%. Relative percent uncertainty is more useful than absolute uncertainty as it allows one to more easily compare uncertainties of different instruments or methods, or of measurements that have different units.
It is important to be able to estimate the order of magnitude of an uncertainty. This requires one to be able to make judgement calls on what is a reasonable uncertainty. Measuring the diameter of an atom would have a much higher relative uncertainty than measuring the diameter of a baseball. A reasonable relative uncertainty may be 1% for a volumetric flask, whereas a reasonable relative uncertainty may be 10% for measuring the concentration of trace elements in seawater.