In courses on experimentation, propagated errors are typically treated through the use of a Taylor series expansion to evaluate the total contribution of individual measurement uncertainties to a final calculated result. As an example, suppose we wish to experimentally determine the acceleration of a body due to gravity. We could take an object and drop it a measured distance while recording the elapsed time. From basic physics we know that the distance traveled is proportional to the time squared and that the proportionality constant is or,
Solving for the acceleration, we obtain,
which is a function of two measured variables, the distance travelled and the elapsed time. Both of these measurements, no matter how carefully obtained, will have some uncertainty. Suppose we measure the distance travelled with a ruler that has graduations every 1 inches and the time with a stopwatch with a resolution of 0.1 seconds. With both of these instruments it is evident that we cannot measure the quantity to a higher resolution than the instrument provides, therefore it is typical to take the total uncertainty in the measurement as the least significant digit in the scale, centered on the measurement value. This would equate to uncertainties in the measurements of ± .5 inches and ± .05 seconds.
Let’s say we dropped the object from a height of 36 feet and measured the elapsed time as 1.5 seconds. From the above equation we would find the acceleration to be 32 . But how carefully did we measure? Was the distance exactly 36 feet (432 inches), or was it 432.3 inches? Was the time 1.48 seconds? As long as these uncertainties are within the predefined ranges established above, we can calculate the total uncertainty in the measurement of the acceleration.
In this post we discussed the approximation of any function by a Taylor series expansion about a specific point. We can apply that technique to determine how the value of the acceleration may vary with perturbations in the input values of time and distance about the measured point. For the simplest implementation, we restrict ourselves to the first order terms of the expansion.
Recal that the Taylor series of a function about the point is given as
Expanding this to the first order terms yields
We can now see that the left side of the equation evaluates to the change in the function corresponding to a perturbation of and by a small amount and . Examining the partial derivative terms, we can see that we are multiplying the rate of change of the function in a single variable to a change in that variable from the interested point . Since we are interested in small perturbations of and about the point , we will denote these changes and . The change of the function under these perturbations we will denote .
Substituting we obtain
Lastly, since there should be no preference for the uncertainty to be in the positive or negative direction, we take the absolute value of the derivative terms and require that our perturbations be defined as positive,
This can be generalized to a function in any number of variables as
Returning to our example, to find the total uncertainty in the calculated acceleration, we simply need to determine the partial derivatives of the function in the independent variables,
and insert them into our formula
Evaluating with our collected data
Therefore our calculated acceleration should be given as .
It should be noted that the calculated uncertainty is the worst case situation that is possible under the individual assumptions in the treatment. Alternate treatments based on statistical treatments may be more realistic. Also, another useful application of these methods is to define the sensitivity of the dependant variable to inputs to the function. These will be address in subsequent posts.
Caveat lector — All work and ideas presented here may not be accurate and should be verified before application.
2nd order and higher terms include the square and higher powers of the perturbation amount. Assuming that the perturbation is small, the square of this small perturbation is much smaller still, and higher order terms become negligible. If the relative size of the perturbation to the curvature of the function is in doubt, the magnitude of the 2nd and higher order terms should be checked.