The model used to convert the measurements into the derived quantity is usually based on fundamental principles of a science or engineering discipline.
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, 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.
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.
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.
It is difficult to position and read the initial angle with high accuracy (or precision, for that matter; this measurement has poor reproducibility).
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.
(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.
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).
The values are reasonably close to those found using Eq(3), but not exact, except for L. That is because the change in g is linear with L, which can be deduced from the fact that the partial with respect to (w.r.t.)
This information is very valuable in post-experiment data analysis, to track down which measurements might have contributed to an observed bias in the overall result (estimate of g).
To illustrate, Figure 1 shows the so-called Normal PDF, which will be assumed to be the distribution of the observed time periods in the pendulum experiment.
Ignoring all the biases in the measurements for the moment, then the mean of this PDF will be at the true value of T for the 0.5 meter idealized pendulum, which has an initial angle of 30 degrees, namely, from Eq(1), 1.443 seconds.
In the figure there are 10000 simulated measurements in the histogram (which sorts the data into bins of small width, to show the distribution shape), and the Normal PDF is the solid line.
For the variance (actually MSe), which differs only by the absence of the last term that was in the exact result; since σ should be small compared to μ, this should not be a major issue.
Let so that This expression could remain in this form, but it is common practice to divide through by z2 since this will cause many of the factors to cancel, and will also produce in a more useful result: which reduces to Since the standard deviation of z is usually of interest, its estimate is where the use of the means (averages) of the variables is indicated by the overbars, and the carats indicate that the component (co)variances must also be estimated, unless there is some solid a priori knowledge of them.
For simplicity, consider only the measured time as a random variable, so that the derived quantity, the estimate of g, amounts to where k collects the factors in Eq(2) that for the moment are constants.
Next, to find an estimate of the variance for the pendulum example, since the partial derivatives have already been found in Eq(10), all the variables will return to the problem.
It must be stressed that these "sigmas" are the variances that describe the random variation in the measurements of L, T, and θ; they are not to be confused with the biases used previously.
Rather than the variance, often a more useful measure is the standard deviation σ, and when this is divided by the mean μ we have a quantity called the relative error, or coefficient of variation.
The second partial for the angle portion of Eq(2), keeping the other variables as constants, collected in k, can be shown to be[8] so that the expected value is and the dotted vertical line, resulting from this equation, agrees with the observed mean.
As was calculated for the simulation in Figure 4, the bias in the estimated g for a reasonable variability in the measured times (0.03 s) is obtained from Eq(16) and was only about 0.01 m/s2.
This dependence of the overall variance on the number of measurements implies that a component of statistical experimental design would be to define these sample sizes to keep the overall relative error (precision) within some reasonable bounds.
Returning to the Type II bias in the Method 2 approach, Eq(19) can now be re-stated more accurately as where s is the estimated standard deviation of the nT T measurements.
In Figure 6 is a series PDFs of the Method 2 estimated g for a comparatively large relative error in the T measurements, with varying sample sizes.
In Figure 7 are the PDFs for Method 1, and it is seen that the means converge toward the correct g value of 9.8 m/s2 as the number of measurements increases, and the variance also decreases.
The Taylor-series approximations provide a very useful way to estimate both bias and variability for cases where the PDF of the derived quantity is unknown or intractable.
[18] In summary, the linearized approximation for the expected value (mean) and variance of a nonlinearly-transformed random variable is very useful, and much simpler to apply than the more complicated process of finding its PDF and then its first two moments.
Finding the PDF is nontrivial, and may not even be possible in some cases, and is certainly not a practical method for ordinary data analysis purposes.
These expressions are based on "Method 1" data analysis, where the observed values of x are averaged before the transformation (i.e., in this case, raising to a power and multiplying by a constant) is applied.