The Trouble with Numbers

I have had several people comment about numerical data. I think this is the most important subject in all science. Precision and accuracy are different aspects of any measurement. Understanding this is paramount to understanding how nature works.

Accuracy in any measurement is how close it is to a true value. If, for example, you say that the temperature is 21 degrees Centigrade. How close is this to the actual temperature? Accuracy depends upon the quality of the measurement tool.

Precision relates to how repeatable a measurement is. In other words, if you use an instrument to measure a temperature, how close will the result be when the measurement is repeated?

A measurement can be precise but not accurate. And, a measurement can be accurate but not precise. The goal in science is to have a measurement be both accurate and precise.

Accuracy depends upon the quality of the measurement tool. Accuracy is often determined by using a tool to measure a standard. This would often be done to calibrate the tool. In the case of a thermometer, the standards would be the freezing point and the boiling point of pure water.

Precision, on the other hand, is determined by a using a mathematical procedure. First you take several readings. Let's say you do ten measurements of a temperature. The first step is to find the average of the ten numbers. This is often called the mean. Then you find the deviations from the mean by subtracting each measurement from the mean. Each of these results is squared to get the deviations squared. Then you add the squared deviations and divide by the number of readings minus 1. in this case it would be 9. This gives you the variance. You take the square root of this and this would be the deviation.

Let's say you measured the temperature ten times and obtained the following data: 22, 21, 20, 19, 23, 21, 20, 19, 18, 21. The average of these is 22 + 21 + 20 + 19 + 23 +21 + 20 + 19 +18 + 21 = 204. Divide this by 10 and you get a mean of 20.4. Now we subtract each of readings from the mean to get the deviations. (20.4 - 22) = (-1.6^2 or 2.56) (20.4 - 21) = (-0.6^2 or 0.96) (20.4- 20) = (0.4^2 or 0.16) (20.4 - 19) = (1.4^2 or 1.96) (20.4 - 23) = (2.6^2 or 6.76) (20.4 - 21) = (-0.6^2 or 0.96) (20.4 - 20) = (0.4^2 or 0.16) (20.4 -19) = (1.4^2 or 1.96) (20.4 -18) = (2.4 or 5.76) (20.4 - 21) = (-0.6^2 or 0.96). Add these together and you get 22.2 divided by 9 (mean -1) gives 2.46 The square root of this is the standard deviation or 1.57.
In other words you would say that you should be able to measure this same temperature to + or - 1.57 degrees precision or standard deviation.

This is a very simple example of a much more complicated concept. The only reason that I discuss this is because it's very important to understanding that there is no such thing as an absolute. All measurements have the two problems of accuracy and precision. And believe me, this is something that many people don't realize about how nature works.

Thanks for reading.