Metrology Monday! #48 Type B Uncertainty and a few Different Distributions - Continued

Last week I started talking about some of the different types of distributions that are used during Type B uncertainty analyses.? ?One of my main points about these analyses is that we usually start with an uncertainty estimate with a very large coverage factor, often representing the 100% limits of a distribution.? In the simplest way I was trying to explain why we divide by a value for a given distribution in order to convert it to a standard uncertainty.? Fellow Metrologist Kruno Milicevic correctly noted that the value that we divide each type of distribution corresponds to its variance for that particular distribution about its expectation.? This is more technically correct, but it is a bit harder to understand.? As always, my goal is to introduce you to these concepts in a way that is not intimidating, but I also encourage you to use this as a starting point for further learning about this very important topic.? Once again, I thank Kruno for contributing to our collective knowledge.

The next distribution I would like to talk about is the uniform or rectangular distribution.? For this distribution, we know the location of the ends of the distribution, but we don’t know much more than that.? There is equal probability that a measurement can be anywhere between the limits, there is no central tendency like in a normal distribution.? So, we draw a line at the limit, a, and draw a straight line to cap the distribution, like in the picture below.?

It looks like a rectangle, hence the name.? A good example of a process that has a rectangular distribution is when we roll a single die.? The outcome can be a number from 1 to 6.? So long as the die is fair, there is equal probability of rolling any number from one to 6, given enough samples. ??Another example is when evaluating uncertainty from a digital measuring device.? For example, if a digital device displays the value “10”, we understand that it is closer to 10 than it is to 9 or 11.? We can even understand a bit more if we understand digital rounding in that if the actual measured value is less than 9.5, it the display will be 9, and if it is equal to or greater than 10.5, it will display 11.? However, we have know further knowledge about the measurement between 10 +/- 0.5 digit.

Section 4.3.7 of the GUM advises us that if we know the limits of a distribution and we have no specific knowledge about the possible values between these limits, we can only assume a worst-case situation of the measurements being a uniform distribution.? While this is true, I have observed this clause to be misinterpreted by some assessors who have compelled laboratories to use the uniform distribution in nearly all cases.? The key words of section 4.3.7 of the GUM are “no specific knowledge.”? If we do have additional knowledge about the distribution, then we should certainly apply it.?

Like the U-shaped distribution, we can convert the 100% limits to a standard uncertainty, but the conversion value is different. For a uniform distribution, the conversion factor is the square root of 3.? In the case of digital resolution, where the limits are +/- ? digit, we can divide the resolution value by the square root of 12, which is equal to 2 times the square root of three.

The last distribution that I am going to talk about is the triangular distribution.?? This is a bit of a compromise distribution.? This can be used when you have absolute limits, like in the uniform or U-shaped distribution, but you have some knowledge about central tendency, so that measurements are more likely to occur in the middle of the distribution, dropping to zero at?the limits, but you don’t know enough about the distribution to apply a normal distribution to it.? I have seen this used in Key comparisons for National Metrology Institutes where they are using all data to derive the reference value, but they want to account for the variation of each participant’s measurements.? In order to convert this distribution to a standard uncertainty we simply divide by the square root of 6.

?This is not an exhaustive list of distributions.? There are many other ones that may be more appropriate for your data.? This should give you a good start towards using the most common distributions for uncertainty analysis and may motivate you to understand an apply other distributions.? As I noted in post #38, it is always a good idea to plot your data to see what shape it looks like, and apply appropriate statistical tests that will help you validate the distribution model that you choose to use.? Often we are using our best judgement to apply the appropriate distribution, which can affect the accuracy of our analysis.? That is why it is more appropriate to refer to an uncertainty analysis as an estimate, which infers possible incomplete knowledge, rather than a computation, which infers a higher level of accuracy.? #MetrologyMonday #FlukeMetrology