Experimental uncertainty analysis is a technique that analyses a derived quantity, based on the uncertainties in the experimentally measured quantities that are used in some form of mathematical relationship ("model") to calculate that derived quantity. The model used to convert the measurements into the derived quantity is usually based on fundamental principles of a science or engineering discipline.
The uncertainty has two components, namely, bias (related to accuracy) and the unavoidable random variation that occurs when making repeated measurements (related to precision). The measured quantities may have biases, and they certainly have random variation, so what needs to be addressed is how these are "propagated" into the uncertainty of the derived quantity. Uncertainty analysis is often called the "propagation of error."
Introduction
For example, an experimental uncertainty analysis of an undergraduate physics lab experiment in which a pendulum can estimate the value of the local gravitational acceleration constant g. The relevant equation for an idealized simple pendulum is, approximately,
where T is the period of oscillation (seconds), L is the length (meters), and θ is the initial angle. Since θ is the single time-dependent coordinate of this system, it might be better to use θ0 to denote the initial (starting) displacement angle, but it will be more convenient for notation to omit the subscript. Solving Eq(1) for the constant g,
This is the equation, or model, to be used for estimating g from observed data. There will be some slight bias introduced into the estimation of g by the fact that the term in brackets is only the first two terms of a series expansion, but in practical experiments this bias can be, and will be, ignored.
The procedure is to measure the pendulum length L and then make repeated measurements of the period T, each time starting the pendulum motion from the same initial displacement angle θ. The replicated measurements of T are averaged and then used in Eq(2) to obtain an estimate of g. Equation (2) is the means to get from the measured quantities L, T, and θ to the derived quantity g.
Note that an alternative approach would be to convert all the individual T measurements to estimates of g, using Eq(2), and then to average those g values to obtain the final result. This would not be practical without some form of mechanized computing capability (i.e., computer or calculator), since the amount of numerical calculation in evaluating Eq(2) for many T measurements would be tedious and prone to mistakes.
Systematic error / bias / sensitivity analysis
Introduction
There are three quantities that must be measured: (1) the length of the pendulum, from its suspension point to the center of mass of the “bob;” (2) the period of oscillation; (3) the initial displacement angle. The length is assumed to be fixed in this experiment, and it is to be measured once, although repeated measurements could be made, and the results averaged.
The initial displacement angle must be set for each replicate measurement of the period T, and this angle is assumed to be constant. Often the initial angle is kept small (less than about 10 degrees) so that the correction for this angle is considered to be negligible; i.e., the term in brackets in Eq(2) is taken to be unity. For the experiment studied here, however, this correction is of interest, so that a typical initial displacement value might range from 30 to 45 degrees.
Suppose that it was the case, unknown to the students, that the length measurements were too small by, say, 5 mm. This could be due to a faulty measurement device (e.g. a meter stick), or, more likely, a systematic error in the use of that device in measuring L. This could occur if the students forgot to measure to the center of mass of the bob, and instead consistently measured to the point where the string attached to it. Thus, this error is not random; it occurs each and every time the length is measured.
Next, the period of oscillation T could suffer from a systematic error if, for example, the students consistently miscounted the back-and-forth motions of the pendulum to obtain an integer number of cycles. (Often the experimental procedure calls for timing several cycles, e.g., five or ten, not just one.) Or perhaps the digital stopwatch they used had an electronic problem, and consistently read too large a value by, say, 0.02 seconds. There will of course also be random timing variations; that issue will be addressed later. Of concern here is a consistent, systematic, nonrandom error in the measurement of the period of oscillation of the pendulum.