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, such as but not limited to physics and chemistry. 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,
T = 2 π L g [ 1 + 1 4 sin 2 ( θ 2 ) ] E q ( 1 ) {\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,
g ^ = 4 π 2 L T 2 [ 1 + 1 4 sin 2 ( θ 2 ) ] 2 E q ( 2 ) {\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.
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. Finally, the initial angle could be measured with a simple protractor. It is difficult to position and read the initial angle with high accuracy (or precision, for that matter; this measurement has poor reproducibility). Assume that the students consistently mis-position the protractor so that the angle reading is too small by, say, 5 degrees. Then all the initial angle measurements are biased by this amount.
Sensitivity errors However, biases are not known while the experiment is in progress. If it was known, for example, that the length measurements were low by 5 mm, the students could either correct their measurement mistake or add the 5 mm to their data to remove the bias. Rather, what is of more value is to study the effects of nonrandom, systematic error possibilities before the experiment is conducted. This is a form of sensitivity analysis. The idea is to estimate the difference, or fractional change, in the derived quantity, here g, given that the measured quantities are biased by some given amount. For example, if the initial angle was consistently low by 5 degrees, what effect would this have on the estimated g? If the length is consistently short by 5 mm, what is the change in the estimate of g? If the period measurements are consistently too long by 0.02 seconds, how much does the estimated g change? What happens to the estimate of g if these biases occur in various combinations? One reason for exploring these questions is that the experimental design, in the sense of what equipment and procedure is to be used (not the statistical sense; that is addressed later), depends on the relative effect of systematic errors in the measured quantities. If a 5-degree bias in the initial angle would cause an unacceptable change in the estimate of g, then perhaps a more elaborate, and accurate, method needs to be devised for this measurement. On the other hand, if it can be shown, before the experiment is conducted, that this angle has a negligible effect on g, then using the protractor is acceptable. Another motivation for this form of sensitivity analysis occurs after the experiment was conducted, and the data analysis shows a bias in the estimate of g. Examining the change in g that could result from biases in the several input parameters, that is, the measured quantities, can lead to insight into what caused the bias in the estimate of g. This analysis can help to isolate such problems as measurement mistakes, problems with apparatus, incorrect assumptions about the model, etc.
Direct (exact) calculation of bias The most straightforward, not to say obvious, way to approach this would be to directly calculate the change using Eq(2) twice, once with theorized biased values and again with the true, unbiased, values for the parameters:
Δ g ^ = g ^ ( L + Δ L , T + Δ T , θ + Δ θ ) − g ^ ( L , T , θ ) E q ( 3 ) {\displaystyle \Delta {\hat {g}}\,\,\,=\,\,\,\,{\hat {g}}\left({L+\Delta L,\,\,\,T+\Delta T,\,\,\,\theta +\Delta \theta }\right)\,\,\,-\,\,\,{\hat {g}}\left({L,\,\,T,\,\,\theta }\right){\mathbf {\,\,\,\,\,\,\,\,\,Eq(3)} }}
where the ΔL etc. represent the biases in the respective measured quantities. (The carat over g means the estimated value of g.) To make this more concrete, consider an idealized pendulum of length 0.5 meters, with an initial displacement angle of 30 degrees; from Eq(1) the period will then be 1.443 seconds. Suppose the biases are −5 mm, −5 degrees, and +0.02 seconds, for L, θ, and T respectively. Then, considering first only the length bias ΔL by itself,
Δ g ^ = g ^ ( 0.495 , 1.443 , 30 ) − g ^ ( 0.500 , 1.443 , 30 ) = − 0.098 m / s 2 {\displaystyle \Delta {\hat {g}}\,\,\,=\,\,\,{\hat {g}}\left({0.495,\,\,\,1.443,\,\,\,30}\right)\,\,\,-\,\,\,{\hat {g}}\left({0.500,\,\,1.443,\,\,30}\right)\,\,\,=\,\,\,-0.098{\rm {\,\,\,m/s^{2}}}}
and for this and the other measurement parameters T and θ the changes in g are recorded in Table 1. It is common practice in sensitivity analysis to express the changes as fractions (or percentages). Then the exact fractional change in g is
Δ g ^ g ^ = g ^ ( L + Δ L , T + Δ T , θ + Δ θ ) − g ^ ( L , T , θ ) g ^ ( L , T , θ ) E q ( 4 ) {\displaystyle {{\Delta {\hat {g}}} \over {\hat {g}}}\,\,\,=\,\,\,\,{{{\hat {g}}\left({L+\Delta L,\,\,\,T+\Delta T,\,\,\,\theta +\Delta \theta }\right)\,\,\,-\,\,\,{\hat {g}}\left({L,\,\,T,\,\,\theta }\right)} \over {{\hat {g}}\left({L,\,\,T,\,\,\theta }\right)}}{\mathbf {\,\,\,\,\,\,\,\,Eq(4)} }}
The results of these calculations for the example pendulum system are summarized in Table 1.
Linearized approximation; introduction Next, suppose that it is impractical to use the direct approach to find the dependence of the derived quantity (g) upon the input, measured parameters (L, T, θ). Is there an alternative method? From calculus, the concept of the total differential is useful here:
d z = ∂ z ∂ x 1 d x 1 + ∂ z ∂ x 2 d x 2 + ∂ z ∂ x 3 d x 3 + ⋯ = ∑ i = 1 p ∂ z ∂ x i d x i E q ( 5 ) {\displaystyle dz={{\partial z} \over {\partial x_{1}}}dx_{1}\,\,\,+\,\,\,{{\partial z} \over {\partial x_{2}}}dx_{2}\,\,\,+\,\,\,{{\partial z} \over {\partial x_{3}}}dx_{3}\,\,\,+\,\,\,\cdots \,\,\,\,\,=\,\,\,\sum \limits _{i\,\,=\,\,1}^{p}{\,{{\partial z} \over {\partial x_{i}}}dx_{i}}{\mathbf {\,\,\,\,\,\,\,\,\,\,\,\,\,Eq(5)} }}
where z is some function of several (p) variables x. The symbol ∂z / ∂x1 represents the "partial derivative" of the function z with respect to one of the several variables x that affect z. For the present purpose, finding this derivative consists of holding constant all variables other than the one with respect to which the partial is being found, and then finding the first derivative in the usual manner (which may, and often does, involve the chain rule). In functions that involve angles, as Eq(2) does, the angles must be measured in radians. Eq(5) is a linear function that approximates, e.g., a curve in two dimensions (p=1) by a tangent line at a point on that curve, or in three dimensions (p=2) it approximates a surface by a tangent plane at a point on that surface. The idea is that the total change in z in the near vicinity of a specific point is found from Eq(5). In practice, finite differences are used, rather than the differentials, so that
Δ z ≈ ∂ z ∂ x 1 Δ x 1 + ∂ z ∂ x 2 Δ x 2 + ∂ z ∂ x 3 Δ x 3 + ⋯ = ∑ i = 1 p ∂ z ∂ x i Δ x i E q ( 6 ) {\displaystyle \Delta z\approx {{\partial z} \over {\partial x_{1}}}\Delta x_{1}\,\,\,+\,\,\,{{\partial z} \over {\partial x_{2}}}\Delta x_{2}\,\,\,+\,\,\,{{\partial z} \over {\partial x_{3}}}\Delta x_{3}\,\,\,+\,\,\,\cdots \,\,\,\,\,=\,\,\,\sum \limits _{i\,\,=\,\,1}^{p}{\,{{\partial z} \over {\partial x_{i}}}\Delta x_{i}}{\mathbf {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,Eq(6)} }}
and this works very well as long as the increments Δx are sufficiently small. Even highly curved functions are nearly linear over a small enough region. The fractional change is then
Δ z z ≈ 1 z ∑ i = 1 p ∂ z ∂ x i Δ x i E q ( 7 ) {\displaystyle {{\Delta z} \over z}\,\,\,\approx \,\,\,{1 \over z}\,\,\sum \limits _{i\,\,=\,\,1}^{p}{\,{{\partial z} \over {\partial x_{i}}}\Delta x_{i}}{\mathbf {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,Eq(7)} }}
An alternate, useful, way to write Eq(6) uses vector-matrix formalism:
Δ z ≈ ( ∂ z ∂ x 1 ∂ z ∂ x 2 ∂ z ∂ x 3 ⋯ ∂ z ∂ x p ) ( Δ x 1 Δ x 2 Δ x 3 ⋮ Δ x p ) E q ( 8 ) {\displaystyle \Delta z\,\,\approx \,\,{\begin{pmatrix}{\partial z \over \partial x_{1}}&{\partial z \over \partial x_{2}}&{\partial z \over \partial x_{3}}&\cdots &{\partial z \over \partial x_{p}}\end{pmatrix}}{\begin{pmatrix}{\Delta x_{1}}\\{\Delta x_{2}}\\{\Delta x_{3}}\\{\vdots }\\{\Delta x_{p}}\end{pmatrix}}{\mathbf {\,\,\,\,\,\,\,\,\,\,\,\,\,\,Eq(8)} }}
In the application of these partial derivatives, note that they are functions that will be evaluated at a point, that is, all the parameters that appear in the partials will have numerical values. Thus the vector product in Eq(8), for example, will result in a single numerical value. For bias studies, the values used in the partials are the true parameter values, since we are approximating the function z in a small region near these true values.
Linearized approximation; absolute change example Returning to the pendulum example and applying these equations, the absolute change in the estimate of g is
Δ g ^ ≈ ∂ g ^ ∂ L Δ L + ∂ g ^ ∂ T Δ T + ∂ g ^ ∂ θ Δ θ E q ( 9 ) {\displaystyle \Delta {\hat {g}}\,\,\approx \,\,{{\partial {\hat {g}}} \over {\partial L}}\Delta L\,\,\,+\,\,\,{{\partial {\hat {g}}} \over {\partial T}}\Delta T\,\,\,+\,\,\,{{\partial {\hat {g}}} \over {\partial \theta }}\Delta \theta {\mathbf {\,\,\,\,\,\,\,\,\,\,\,Eq(9)} }}
and now the task is to find the partial derivatives in this equation. It will considerably simplify the process to define
α ( θ ) ≡ [ 1 + 1 4 sin 2 ( θ 2 ) ] 2 {\displaystyle \alpha (\theta )\,\,\equiv \,\,\left[{\,1\,\,\,+\,\,\,{1 \over 4}\sin ^{2}\left({\theta \over 2}\right)\,}\right]^{2}}
Rewriting Eq(2) and taking the partials,
g ^ = 4 π 2 L T 2 α ( θ ) ∂ g ^ ∂ L = 4 π 2 T 2 α ( θ ) ∂ g ^ ∂ T = − 8 L π 2 T 3 α ( θ ) ∂ g ^ ∂ θ = L π 2 T 2 α ( θ ) sin ( θ ) E q ( 10 ) {\displaystyle {\begin{aligned}{\hat {g}}&={{4\pi ^{2}L} \over {T^{2}}}\alpha (\theta )\\\\{{\partial {\hat {g}}} \over {\partial L}}\,\,&=\,\,\,{{4\,\pi ^{2}} \over {T^{2}}}\alpha (\theta )\\\\{{\partial {\hat {g}}} \over {\partial T}}\,\,&=\,\,{{-8\,L\,\pi ^{2}} \over {T^{3}}}\alpha (\theta )\\\\{{\partial {\hat {g}}} \over {\partial \theta }}\,\,&=\,\,{{L\,\pi ^{2}} \over {T^{2}}}\,\,{\sqrt {\alpha (\theta )}}\,\,\sin(\theta )\\\\{\mathbf {\,\,\,\,Eq(10)} }\end{aligned}}}
Plugging these derivatives into Eq(9),
Δ g ^ ≈ [ 4 π 2 T 2 α ( θ ) ] Δ L + [ − 8 L π 2 T 3 α ( θ ) ] Δ T + [ L π 2 T 2