# Experimental uncertainty analysis

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.”

## . . . Experimental uncertainty analysis . . .

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[1] for an idealized simple pendulum is, approximately,

${displaystyle T,=,2,pi ,{sqrt {L over g}},,left[{1,,,+,,,{1 over 4}sin ^{2}left({theta over 2}right),}right]{mathbf {,,,,,,,,,Eq(1)} }}$

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,

${displaystyle {hat {g}},=,{{4,pi ^{2}L} over {T^{2}}},,left[{,1,,,+,,,{1 over 4}sin ^{2}left({theta over 2}right),}right]^{2}{mathbf {,,,,,,,,,,,,Eq(2)} }}$

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.