Back in the good old days (before PCs and hand-held calculators, that is) we used to do calculations the hard way; by hand!

It seemed to give a little more significance to the actual results of a measurement or a series of them. At least I remember those early ones a heckofa lot better than some that were just series of numbers tossed into a spreadsheet and summed, averaged and “staticized”!

Take the simple Physics Lab test we were given on measurement variation, for instance. It is still with me today, more than 40 years later.

The test was a simple one. We were given a simple, short steel rod about 6 mm in diameter along with a 1inch maximum opening micrometer. It had a measurement resolution, stated on the side or 0.001 inch.

We simply had to, as carefully as possible, measure the diameter of the rod to the nearest one-thousandth of an inch and report it.

The second step was to repeat the measurement 10 times and analyze the results, i.e. take the average of the 10 reading and calculate the standard deviation.

Then we could compare our first reading with the average and explain the difference, if any.

Then all 15 or 20 sets of results from each of the lab participants were compared to each other (we ostensibly all had samples of steel rod from the same source). So, we had a new, grand average and a different, larger standard variation to consider.

Of course there was a surprising difference and some people actually got a difference in their initial reading that was well outside the range of averages and the grand average value, even when considering the standard deviations as a reasonable tolerance in expected values.

Why?

There were two or three different factors at work, but mostly the nominal diameter had variations in it larger than 0.001 inches, and the repeatability skills of the observers were also different.

It was was quite clear to me, then and now, that nearly any quantitative measurement will have an error associated with it. The only way to get some idea of the error magnitude is to repeat the measurement and analyze the results.

Then one can get to the work of making and testing hypotheses about the sources of the error. With some logical practices, then one may be able to reduce the error, if it is important to do so.

Here’s another, more contemporary example of a basic calculation on uncertainty. It’s online at the website of the National Physical Laboratory in England. The measurement is: how long is a piece of string? (Not too different to what I experienced a few years ago).

Then it goes into “Analysis of uncertainty” and a “spreadsheet model”.

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