When to average...?

This applet is intended to be used in conjunction with the Wikipedia article Experimental Uncertainty Analysis.

In that article there is some mathematical background and discussion of the so-called "propagation of error." The example used is the estimation of the gravitational acceleration constant "g" using a pendulum. At issue here is whether the N observed pendulum-period T observations (1) should be averaged and then used once in the physics model, Eq(2) of the article, to find a single estimate of g, or (2) should each T observation be converted using the physics model, and those N estimates of g subsequently averaged. Does it matter?

Yes, it does matter. It is discussed in the article that Method 1 has a bias that is reduced with increasing sample size N, while Method 2 has a bias that is not affected by N. This applet illustrates the situation. The PDF of the "g" estimates was derived (see Meyer, Data Analysis for Scientists and Engineers, Wiley (1975) ISBN 0-471-59995-6, Chapter 20) and those are plotted here. The (approximate) expected value of the "g" PDFs is found using Eq(14) of the Wiki article, and those are also shown (vertical dotted lines). See Rohatgi, Statistical Inference, Dover (2004 reprint) ISBN 0-486-42812-5, pp 269-70 for more on this topic.

Use the sliders to adjust the scatter (standard deviation) of the observed T (pendulum periods) and N, the number of replicated measurements. It helps to click on the slider and then use the arrow keys, for better control. Note that as N increases the Method 1 PDF becomes narrower and shifts to the left, closer to the correct value for "g" which is 9.8 m/s2. Thus, its bias is being reduced. Meanwhile the Method 2 PDF also becomes narrower, but it does not shift- its bias is unchanged with N.

In both cases, if the scatter in the T measurements is small, then there is little bias for either method. The bias is displayed as the difference between 9.8 and the respective mean value for the PDF. The nominal period of this 0.5 meter pendulum is 1.44 seconds, so that a sigma-T of 0.1 second  represents roughly a 10 percent uncertainty in the time measurements, which should be attainable by students using stopwatches, without any heroic effort.

Bottom line: if the measurement scatter is small, then either processing method is acceptable. If the scatter is larger, then   Method 1 is clearly to be preferred. What "large" or "small" measurement scatter represents depends on the problem at hand.

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WCEvans (12/09), Created with GeoGebra