User blog:BunsenH/Uncertainties

No measurement is perfectly exact.

Imagine that you're measuring the length of a metal rod with a ruler. If the ruler was marked only in centimeters, you would have to guess at the length of the rod to the extent that it was in between the markings. If the ruler was marked in millimeters, you would be able to get a more precise measurement; if it was marked in even finer divisions, you'd be able to measure even more precisely. But no matter how fine the divisions were, they could (in theory) be finer still, and you could (in theory) measure even better.

Of course, on a real ruler, the markings aren't absolutely perfectly positioned, and they aren't absolutely thin. That contributes to the degree that you couldn't get a perfect measurement. Also, the bar itself can't be perfectly regular; no matter how carefully it was cut and polished, it's always going to be at least a bit irregular, if only at the atomic level. If you used a measurement device based on light, so you didn't have to worry about markings, no matter how good your system was, you couldn't be sure of the length more precisely than the wavelength of the light. There are fundamental limits to the possible precision of such a measurement, based on quantum mechanics.

Even if your measurement is a whole number, say from counting objects, there may be the possibility of the number being not exactly right -- counting errors, losing track and missing an object or double-counting an object. This is especially true if the number of objects is large, or the objects are irregularly placed.

The extent to which you can't be sure of the measurement is called the uncertainty, or the precision.

There are a number of ways of calculating it, and a number of ways of writing it. It's often written in the form of a "plus or minus", or "+/-", or "±", followed by the amount of uncertainty. For example, "13 ± 2" would mean that the number was within 2 of 13 -- that is, beteween 11 and 15. "13.3 ± 2.1" would mean that the number was within 2.1 of 13.3, i.e. between 11.2 and 15.4. If you understand "significant figures", the uncertainty is written to one (or at most two) decimal places, and the measurement and the uncertainty are written to the same number of decimal places. That gives you the number of significant figures in the measurement.

Sometimes the uncertainty is written as an absolute number, for example 45.6 mm ± 0.2 mm, or (45.6 ± 0.2) mm. Sometimes it is written as a proportion of the measurement, most often as a percentage: 123.4 mm ± 2.0%, or (123.4 ± 2.0%) mm. There's possible confusion if the measurement itself is a percentage: if you see something like 32.0% ± 8.0%, is that "8.0%" an absolute value, or a proportion of the measurement? Especially in cases like this, it's best written in the form (32.0 ± 8.0)%.

Note that uncertainty/precision isn't the same as accuracy (or error). Accuracy assumes that you know a correct value, and can compare your measurement to that. This is a very important distinction. A measurement can be precise but not accurate -- consider the possibility of a ruler with very fine markings, but the ruler is distorted in some way. (Or you were using a good ruler incorrectly, but in a consistent way, such as looking at it from an odd angle.)  Or a measurement can be accurate but not precise; for example, think of a ruler with very large divisions, and the object being measured happens to exactly match one of those divisions.

One way of reducing the uncertainty of a measurement is to repeat the measurement lots of times, and average the results. For this kind of repetition, where the measurements are independent and the uncertainty is due only to the above kind of possible measurement problems, the reduction in uncertainty is often calculated to be the original uncertainty divided by the square root of the number of tries. Obviously, that does help, but there is a point of diminishing returns -- as the number of tries increases, the uncertainty does decrease, but more and more slowly as you go on.

If you do many measurements, you might notice a pattern. Many of the measurements are close to the average value, only a few are far away, and a moderate number of measurements are in between -- this is probably not very surprising if you think about it! If you draw a chart of the measurements, with the measurement values along the x axis and the number of measurements on the y axis, if the uncertainties are due to the kind of difficulties described above, you will probably get a graph that is shaped like a simple bell. This is the so-called "bell curve". The pattern of results is sometimes called a "Gaussian distribution" after mathematician Carl Gauss, who described it, or a "normal distribution" (because it's a very common pattern). If the pattern of results isn't that kind of curve, there's probably something going on in your measurement process that isn't that kind of basic uncertainty.

http://upload.wikimedia.org/wikipedia/commons/0/0d/Gaussian_Function.png

That bell curve may be very broad, compared with its center (which is the average value); this is the case of a large relative uncertainty. Or it can be quite narrow, in the case of a small uncertainty. There is a statistical term, "standard deviation", which describes this distribution. Without getting into how it is calculated, it can be seen in graphical form on the bell chart below. The symbol used to write the standard deviation is the greek letter 'σ' (the lower-case Greek 's')  For large numbers of measurements, 68.3% would be within one σ of the average value. 95.5% would be within two σ; that is, about 19/20. This is the "19 times out of 20" that you hear about with regard to polls. 99.7% of the measurements would be within three standard deviations.

 https://upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Standard_deviation_diagram.svg/640px-Standard_deviation_diagram.svg.png

Often, the standard deviation is used to describe the accuracy of a value. Assuming that the measurement errors are the kind of random thing described above rather than a repeated problem with the measurement process, one would expect that the real/correct value would match the average measurement value, which is the middle of that bell curve. So if you see something like poll results written as "75 ± 2", it means that the real value is about 68.3% likely to be within 1 σ of 75, where σ is 2... that is, between 73 and 77. The real value is 95% likely to be within 2 σ of 75, i.e. between 71 and 79 -- "19 times out of 20". The real value is 99.7% likely to be between 69 and 81. And so on.

I've written this for the MSM wiki because uncertainties come up in discussions of how likely things are to happen, such as breeding a Ghazt. It was pretty clear from the start that the chance of breeding a Ghazt was around 1%, but we wanted to collect enough data that the uncertainty of that value was below 0.1% absolutely (which is pretty small), which is a 10% relative uncertainty (i.e. 0.1% ÷ 1% -- which is rather large).