How Small Errors Lead to Big Variances

In science and engineering, it is rare to measure a single quantity with a single source of error. I’ll talk about sources of error, and the kinds of errors, in another posting, but for now I want to talk a bit about statistical errors that are compounded by combined quantities, for example, differences. The examples below are really intended to be done as “hands on” exercises to put yourself “face to face” with the effects of computing on measured quantities.

Differences

One of the hardest problems in experimental work occurs when the desired quantity is the difference of two measured quantities of nearly equal magnitude. Unfortunately, the rule of thumb, “Never divide a small quantity into large parts,” is almost always violated in the real world of experimental analysis.

A simple example can come from a real problem from my own research into the energetic properties of atoms and molecules. In such a case one typically is trying to find the total binding energy of an atom in a liquid, but what you are able to measure is the separate parts, that is, the kinetic energy and the potential energy, separately. In many cases, each of these quantities is quite large, but of opposite sign, so the total energy is really a difference. Here are some realistic numbers from typical experiments and calculations:

Kinetic Energy: 12.5 +/- 0.5 K (degrees Kelvin)

Potential Energy: -14.7 +/- 0.5 K

The relative errors for the kinetic and potential energies (due to statistical fluctuations in the experiment) are approximately 4% and 3.4%, respectively, quite small and rather good for measurements of this sort that require rather elaborate experimental procedures. The question is, what is the resultant error in the total energy, which is what one really is looking for?

Now the sum of the means of these two quantities would be -2.2 K (making the usual mistake of ignoring the errors). But what would be the range of values if one takes the errors into account? If one compares the range of likely values consistent with these measurements, one obtains total energies ranging from -3.2 K to -1.2K! That is, the error could be fairly reported as -2.2 +/- 1 K, or a relative error of 45%!!

In a number of the engineering exercises in this course, you will find a similar situation, for instance the difference of two length measurements, or the difference of weight measurements.

Try the same kind of analysis with ratios.

In science and engineering, it is rare to measure a single quantity with a single source of error. I’ll talk about sources of error, and the kinds of errors, in another posting, but for now I want to talk a bit about statistical errors that are compounded by combined quantities, for example, differences. The examples below are really intended to be done as “hands on” exercises to put yourself “face to face” with the effects of computing on measured quantities.

Differences

One of the hardest problems in experimental work occurs when the desired quantity is the difference of two measured quantities of nearly equal magnitude. Unfortunately, the rule of thumb, “Never divide a small quantity into large parts,” is almost always violated in the real world of experimental analysis.

A simple example can come from a real problem from my own research into the energetic properties of atoms and molecules. In such a case one typically is trying to find the total binding energy of an atom in a liquid, but what you are able to measure is the separate parts, that is, the kinetic energy and the potential energy, separately. In many cases, each of these quantities is quite large, but of opposite sign, so the total energy is really a difference. Here are some realistic numbers from typical experiments and calculations:

Kinetic Energy: 12.5 +/- 0.5 K (degrees Kelvin)

Potential Energy: -14.7 +/- 0.5 K

The relative errors for the kinetic and potential energies (due to statistical fluctuations in the experiment) are approximately 4% and 3.4%, respectively, quite small and rather good for measurements of this sort that require rather elaborate experimental procedures. The question is, what is the resultant error in the total energy, which is what one really is looking for?

Now the sum of the means of these two quantities would be -2.2 K (making the usual mistake of ignoring the errors). But what would be the range of values if one takes the errors into account? If one compares the range of likely values consistent with these measurements, one obtains total energies ranging from -3.2 K to -1.2K! That is, the error could be fairly reported as -2.2 +/- 1 K, or a relative error of 45%!!

In a number of the engineering exercises in this course, you will find a similar situation, for instance the difference of two length measurements, or the difference of weight measurements.

Try the same kind of analysis with ratios.