Our main goal is to determine an experimental mean that is close to the true value of a particular characteristic of a population or process. In the examples from section 1, we saw that if we repeatedly measured the same sample many times with the same instrument, we could generate an experimental mean from the dataset and we could estimate the standard deviation of that dataset. The measured values are randomly distributed about the mean according to a Gaussian distribution. If the number of measurements we make is very large, the experimental mean will be almost identical to the true value. However, in real measurements we find that this is not the case. That is because measurements contain both this random uncertainty (characterized by the standard deviation in the Gaussian) as well as a bias uncertainty due to instrument calibration errors, environmental effects, and other sources. This bias uncertainty results in all measurements being biased either high or low from the mean (if the measurement instrument is mis-calibrated high, then all measurements made will be biased high regardless of the level of random wiggle in the measurements). Thus, we actually expect that all measurements we make will systematically differ from the true value by some amount (either all high or all low) depending on the particular system being considered. For an infinitely large number of measurements, the random uncertainty will shrink to zero; however, the systematic uncertainty will never change.
We expect that the measured value ( ) and the true value (T) will be related by
where ε is the difference between the true and measured values due to random errors and Δ is the difference between the true and measured values due to systematic (or bias) errors. Note that ε and Δ could be either positive or negative.
So if we knew what the expected difference between the measured and true values were, then we could determine the true value from a measurement. Of course, we don't know that (if we did, statisticians wouldn't have jobs), but we might have an expected range for these differences since they are directly related to the systematic and random uncertainties in the measurement system. Thus, understanding systematic and random uncertainties in our measurements is crucial to acquiring an estimate of the true value of a particular characteristic of a population or process.
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