The Uncertainty of Calculating Uncertainty
by Steven Turvey
Enex TestLab is embarking on a new program of physical tests. Unlike most of our other testing where results are simply expressed, perhaps using standard deviation or other mathematical basics, this series relies heavily on uncertainty calculations. In this case, result measurement must be expressed with a ± uncertainty value. Without having done one previously, I expected that calculating the uncertainty of a measurement would be a fairly straight forward task. I was a little naïve.
Let me show you how challenging, and useful, it can be. The following scenario is a little unrealistic but it suffices to illustrate the point; measuring the width of a room with a tape measure. Sounds simple doesn’t it?
Let’s start looking at the variables which influence the result:
- The calibrated measuring tape is made of metal so its length will vary by expansion and contraction according to temperature.
- The room in question has a calibrated thermometer mounted 1.5m above the floor on the wall that reads 24°C.
The thermometer can measure between 0°C and 100°C and there are two components to its calibration.
1. When the thermometer was being calibrated it was found that for any precisely held given temperature the measurements taken varied by plus or minus 0.2°C-or-less in 95% of the measurements taken. This is a standard method of ascertaining uncertainty and expressing it as 95% of the standard deviation for a large number of measurements.
2. The second is the correction factor because the thermometers response to temperature across its range varies for example at 0°C to 30°C the average measurement taken by the thermometer is actually 0.1°C lower than it should be and at 80°C to 100°C the average measurement is 0.3°C lower than it should be.
So if the thermometer measures 24.5°C then the temperature would be found using the following:
T = 24.5°C (Tmeasured) +0.1°C (correction factor) ± 0.2°C (Uncertainty)
T = 24.6°C ± 0.2°C
- Without going thru the whole exercise again let’s just say the tape measures the room at 5m exactly, and from the tape’s calibration table this is actually 5.001m ± 0.5mm at 20°C (the temperature the tape was calibrated at).
Now the temperature correction, given that the room temperature is 24.6°C, is
+11 ± 1 ppm/°C
And we can calculate as follows:
Temperature coefficient (TC) variation = (24.6°C - 20°C) x (+11 ± 1 ppm)
TC = +50.6 ± 4.6 ppm
But hey don’t forget the uncertainty in the temperature measurement which is:
TC variation = ± 0.2°C x (+11 ± 1 ppm/°C)
TC = +2.2 ± 0.2 ppm
So adding all this together gives us:
(5.001m +/- 0.5mm) + 5x (+0.0000506 ± 0.0000046) + 5x (+0.000011 ± 0.000001)
So the 5m room measurement becomes (converting all to metres):
5.001264m ± 0.0050047m
Of course the above would actually be expressed as 5.001m ± 0.005m but I was just wishing to show the magnitude of some of the uncertainty calculations.
It can also be argued that we did not take into account the following factors which could also affect our measurements:
- How even is the temperature variation across the room
- Are there other thermal sources
- What about the thermal mass of the floor and its actual temperature compared to the thermometer mounted on the wall 1.5 meters above.
- <insert endless number of what ifs here>
And that was just for a simple measurement of a room using a measuring tape.
Maybe if I’m feeling sadistic I will look at some more realistic and correspondingly more difficult examples next month. Or maybe not, you see I’m “a little” uncertain.