# Combining Measurement Uncertainty

So, let’s assume that you are estimating measurement uncertainty. You have identified the influencing factors, quantified the magnitude of their contribution, and reduced them to a standard uncertainty. Now, if you are wondering what the next step is, it is to combine the independent uncertainty components to calculate ‘Combined Uncertainty.’ This is the step you will need to take before calculating ‘Expanded Uncertainty.’

The goal of combining uncertainty is to calculate the total magnitude of uncertainty from a set of independent uncertainty components, each with their own varying degrees of magnitude. It is a common process covered in the GUM and many other measurement uncertainty guides. However, I thought it would be a good idea to explain the process in a little more detail.

## What is Combined Uncertainty

Combined Uncertainty is the square-root of the linear sum of squared standard uncertainty components. This method is also known as ‘Summation in Quadrature’ or ‘Root Sum of the Squares.’ Each component is the product (i.e. result of multiplication) of the standard uncertainty and its associated sensitivity coefficient. By combining these components, we are attempting to estimate the total magnitude of uncertainty associated with our evaluated measurement system or process.

## Why is Uncertainty Combined This Way

To best explain summation in quadrature, think Vector Addition and Pythagorean’s Theorem. If we treat uncertainty contributors as orthogonal (i.e. statistically independent), each as a vector with independent quantities of displacement/magnitude, then we can calculate the net displacement/magnitude by addition in quadrature.

### Central Limit Theorem

When performing uncertainty analysis, we use a variety of probability densities/distributions to characterize each contributing factor. Some of the most common distributions used in uncertainty analysis are Gaussian (i.e. normal), uniform (i.e. rectangular), and triangular. When we combine these probability densities to calculate combined uncertainty, the resulting calculation is characterized by a normal distribution. Why? The Central Limit Theorem.

According to the Central Limit Theorem, the sum of a set of independent random variables will approach a normal distribution regardless of the individual variables distribution. This is why combined uncertainty is characterized by a normal distribution, even though we combined a several sets of data characterized by various distributions.

## How to Combine Uncertainty

As explained earlier, uncertainty is combined using a method known as summation in quadrature. Below, I provided the formula and an example of combining uncertainty.

### Example:

If we have three uncertainty components, each with a sensitivity coefficient of one (i.e. 1), the result would be:
c1 = 1
c2 = 1
c3 = 1
u(x1) = 5 ppm
u(x2) = 2 ppm
u(x3) = 3 ppm

If you are using Microsoft Excel to combine uncertainty, use the following formula to accomplish the task.

### =sqrt(sumsq(Cell 1, Cell 2, …, Cell n))

The ‘sqrt‘ function calculates the square root of the data placed in between the parentheses. The next function, ‘sumsq‘ calculates the sum of squares. This function squares the value of each cell and then adds them all together, hence, the sum of squares. When these two functions are combined as I have shown, the result is the square root of the sum of squares or the root sum of the squares. Using this equation is much simpler and easier than squaring and adding each cell independently.

Hopefully, this post has been educational to some degree. Whether you are a beginner or an expert at uncertainty analysis, I hope that I have provided some beneficial information for you to take away from this post. If you have any questions or comments, please feel free to fill in the comments section below or email me at rhogan@isobudgets.com.

http://www.isgmax.com/Articles_Papers/Estimating%20and%20Combining%20Uncertainties.pdf

http://www.isobudgets.com/pdf/uncertainty-guides/bipm-jcgm-100-2008-e-gum-evaluation-of-measurement-data-guide-to-the-expression-of-uncertainty-in-measurement.pdf

https://www.wmo.int/pages/prog/gcos/documents/gruanmanuals/UK_NPL/mgpg11.pdf

http://mathworld.wolfram.com/Vector.html

http://web.mit.edu/fluids-modules/www/exper_techniques/2.Propagation_of_Uncertaint.pdf

http://mathworld.wolfram.com/CentralLimitTheorem.html

http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/Chapter9.pdf

http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-041-probabilistic-systems-analysis-and-applied-probability-fall-2010/video-lectures/lecture-20-the-central-limit-theorem/

### Richard Hogan

Richard Hogan is the CEO of ISO Budgets, L.L.C., a U.S.-based consulting and data analysis firm. Services include measurement consulting, data analysis, uncertainty budgets, and control charts. Richard is a systems engineer who has laboratory management and quality control experience in the Metrology industry. He specializes in uncertainty analysis, industrial statistics, and process optimization. Richard holds a Masters degree in Engineering from Old Dominion University in Norfolk, VA. Connect with Richard on Google+ and LinkedIn.

1. Hi Richard,

This is a fantastic site. And I will be referencing this often and perhaps obtaining your services one day if our group can’t figure something out on their own.

But first I had a question that perhaps you can answer. We utilize piston provers (measuring device) to compute the relative error between our standard against their unit under test. So, for example, the meter runs 10 GPM, but our stand says 10.5 GPM, we know we have a 4.8% error. The error can be thought of as the accuracy if we assume that our value is the true vale.

And so we’ve always just reported that error to our customers. Now we must report the uncertainty of the UUT (unit Under Test), and I’m confused over which method we should use. I’ve seen one utilize the En value to do it, I’ve seen some say to use relative uncertainty of the mean RMS(relative error^2 of UUT + uncert. of our test stand^2), and lastly, RMS(repeatability of meter^2 + uncertainty of test stand^2).

The last one perplexes me since one is not taking into account the bias (tolerance) from the meter’s manufacturer. To complicate things more, you don’t know what into the calculation of the tolerance of a manufactured meter – did they compute the repeatability of their meter, hysteresis, linearity with in the analysis, or is that tolerance stated by the manufacture just an error one shouldn’t exceed if you measured it on a stand that is at least 4 times better in uncertainty?

Any advice will be much appreciated?

Jon

2. Just a revision from above to reduce confusion, for that was typed rather quickly.