Getting the measure of uncertainty

The work of the laboratory is dependant on taking measurements. But are you certain you know the importance of uncertainty? Here, John Hurll of UKAS gives us an overview of measurement uncertainty

IN everyday life, we are accustomed to the doubt that arises when estimating how large or small things are. If somebody asks, “what do you think the temperature of this room is?” we might say, “it is about 23 degrees Celsius”. The use of the word “about” implies that we know it is not exactly 23ËšC, but is somewhere near it. We intuitively recognise that there is some doubt about our estimate.

We could, of course, be a bit more specific. We could say, “it is 23 degrees Celsius give or take a couple of degrees”. The term “give or take” suggests that there is still doubt, but now we are assigning limits to its extent. We have given quantitative information about the doubt, or uncertainty, of our estimate.

It is also reasonable to assume that we may be more sure that our estimate is within, say, 5ËšC of the true room temperature than we are that it is within 2ËšC. The larger the uncertainty, the more confident we are that it encompasses the true value. So the uncertainty is related to the level of confidence.

So far, our estimate has been based on a subjective evaluation. This is not entirely a guess, as we may have experience of exposure to similar and known environments. However, in order to make a more objective measurement it is necessary to make use of a measuring instrument of some kind; in this case we can use a thermometer.

Even if we use a thermometer, there will still be some doubt, or uncertainty, about the result. For example we could ask:

“Is it accurate?”
“How well can I read it?”
“Is the reading changing?”
“I am holding the thermometer in my hand. Am I warming it up?”
“The humidity in the room can vary. Will this affect my results?”
“Does it matter where in the room I make the measurement?”

All these factors, and others, may contribute to the uncertainty of our measurement of the room temperature.

In order to quantify the uncertainty of the room temperature measurement we will have to consider all the factors that could affect the result. We will have to make estimates of the possible variations associated with these influences. Let us consider the questions posed above.

1. Is the thermometer accurate?
In order to find out, we have to compare it with a more accurate thermometer. This thermometer, in turn, will have to be compared with an even better one, and so on. This leads to the concept of traceability of measurements, whereby measurements at all levels can be traced back to agreed references. This is usually achieved by an unbroken chain of comparisons to a national metrology institute, which maintains measurement standards that are directly related to SI units.

In other words, we need a traceable calibration. This calibration itself will provide a source of uncertainty, as the calibrating laboratory will assign a calibration uncertainty to the reported values. When used in a subsequent evaluation of uncertainty, this is often referred to as the imported uncertainty.

In terms of the thermometer accuracy, however, a traceable calibration is not the end of the story. Measuring instruments drift. This, of course, is why regular recalibration is needed. It is therefore important to evaluate the likely change since the instrument was last calibrated.

If the instrument has a reliable history it may be possible to predict what the reading error will be at a given time in the future, based on past results, and apply a correction. This prediction will not be exact and so there will be uncertainty on the corrected value. Sometimes the past data may not indicate a reliable trend, and limit values may have to be assigned for the likely change since the last calibration. This can be estimated from past history. Evaluations made using these methods yield the uncertainty due to secular stability, or changes with time, of the instrument.

There are other influences relating to the thermometer accuracy. Suppose we have a traceable calibration, but only at 15°C, 20°C and 25°C. What does this tell us about its indication error at 23°C?

In such cases we will have to make an estimate of the error, perhaps using interpolation between the calibrated points. This is not always possible as it depends on the measured data being such that accurate interpolation is practical. It may then be necessary to use other information, such as the manufacturer’s specification, to evaluate the additional uncertainty that arises when the reading is not directly at a calibrated value.

2. How well can I read it?
There will be a limit to which we can resolve the thermometer reading. If it is a liquid-in-glass thermometer, this limit will depend on our ability to interpolate between the scale graduations. If it is a thermometer with a digital readout, the number of digits in the display will define the limit.

There will always be an uncertainty of ± half of the change represented by one increment of the last displayed digit. This does not only apply to digital displays; it applies every time a number is recorded. If we write down a rounded result of 123.45, an identical effect occurs and an uncertainty of ± 0.005 will arise.

3. Is the reading changing?
Almost certainly! This may be due to variations in the temperature itself, variations in the performance of the thermometer and in other influence quantities, such as the way we hold the thermometer.

So what can be done about this? We could, of course, just record one reading and call that the result. It would have little meaning, because the next reading, a few seconds later, may be different. So which is correct?

We will probably take an average of several measurements to smooth out these effects. The average of several readings can be closer to the true value than any individual reading is.

However, we can only take a finite number of measurements. This means that we will never obtain the true mean value that would be revealed if we could carry out a very large number of measurements. There will be an unknown error - and therefore an uncertainty - represented by the difference from the calculated mean value and the underlying true mean value.

This uncertainty cannot be evaluated using methods like those we have already considered. Up until now, we have looked for evidence, such as calibration uncertainty and secular stability. We have considered what happens with resolution by logical reasoning. The effects of variation between readings cannot be evaluated like this, because there is no background information available to use.

All we have is a series of readings and an average value. We therefore have to use a statistical approach to determine how far our calculated mean could be away from the true mean. This provides the uncertainty associated with the repeatability of our measurements and is referred to as the standard deviation of the mean.

4. I am holding the thermometer in my hand. Am I warming it up?
There may be heat conduction from the hand to the temperature sensor or radiated heat from the body affecting it. These effects may or may not be significant, but we will not know until an evaluation is performed. Special experiments may be required to determine the significance of the effect.

How could we do this? Some simple methods come to mind. For example, we could set up the thermometer in a temperature-stable environment and read it remotely, without the operator nearby. We could then compare this result with that obtained when the operator is holding it in the usual manner, or in a variety of manners. This would yield empirical data on the effects of heat conduction and radiation. If it turns out to be significant, we could either improve the method so that operator effects are eliminated, or we could include a contribution to measurement uncertainty based on the results of the experiment.

This reveals some important issues. First, that the measurement may not be independent of the operator and that special consideration may have to be given to operator effects. We may have to train the operator to use the equipment in a particular way. Special experiments may be necessary to evaluate particular effects. Uncertainty evaluation may therefore reveal ways to improve the method, thus giving more reliable results. This is a positive benefit of uncertainty evaluation.

5. The relative humidity in the room can vary considerably. Will this affect my results?
Maybe. If it is a liquid in glass thermometer, it is difficult to see how the humidity could significantly affect the expansion of the liquid. However, if we are using a digital thermometer the humidity could affect the electronics that process the signal from the sensor, or the sensor itself.

We need ways of evaluating any such effects. In this case, we could expose the thermometer to a constant temperature and change the humidity. This will reveal how sensitive the thermometer is to this quantity.

This raises a general point that applies to all measurements. Every measurement we make has to be carried out in an environment of some kind. So could any aspect of the environment affect the result?

The following environmental effects are amongst those most commonly encountered when considering measurement uncertainty:

-Temperature   
-Relative humidity
-Barometric pressure
-Electric or magnetic fields
-Gravity
-Electrical supplies to measuring equipment
-Air movement
-Vibration
-Light and optical reflections

Some of these influences may have little effect if they remain constant, but could affect measurement results when they vary. Rate of change of temperature can be particularly important.

It is clear that understanding of the system is important in order to identify and quantify the uncertainties that can arise in a measurement situation. Conversely, analysis of uncertainty can often yield a deeper understanding of the system and reveal ways in which the process can be improved. This leads on to the next question…

6. Does it matter where in the room I make the measurement?
It depends what we are trying to measure! Are we interested in the temperature at a specific location? Or the average of the temperatures encountered at any location within the room? Or the average temperature at bench height? Do we require the temperature at a particular time of day, or the average over a specific period of time?

Such questions have to be asked, and answered, so that we can devise an appropriate measurement method that gives us the information we require. Until we know the details of the method, we cannot evaluate the associated uncertainties.

This leads to what is perhaps the most important question of all, one that should be asked before we even start with our evaluation of uncertainty:

“What exactly is it that I am trying to measure?”

Until this question is answered, we cannot carry out a proper evaluation of the uncertainty. The particular quantity subject to measurement is known as the measurand. In order to evaluate the uncertainty we must define the measurand otherwise we cannot know how a particular influence quantity affects the value we obtain for it.

This means that there has to be a specified relationship between the influence quantities and the measurand. This relationship is known as the mathematical model. This is an equation that describes how each influence quantity affects the value assigned to the measurand. In effect, it is a description of the measurement process. A proper analysis of this process also answers another important question:

“Am I actually measuring the quantity that I thought I was measuring?” 

Some measurement systems are such that the result is only an approximation to the true value because of assumptions and approximations inherent in the method. The model should include any such assumptions and therefore associated uncertainties will be accounted for in the analysis.

This overview has given some insights into how measurement uncertainties might arise. It has shown that we have to know our measurement system and the way in which the various influences can affect the result. It has also shown that analysis of uncertainty can have positive benefits in that it can reveal where enhancements can be made to measurement methods, hence improving the reliability of measurement results.

More information, including worked examples, can be found in M3003, The Expression of Uncertainty and Confidence in Measurement, published by the United Kingdom accreditation service. M3003 is available as a free download from http://www.ukas.com/Library/downloads/publications/M3003.pdf.

By John Hurll, Assessment Manager United Kingdom Accreditation Service (UKAS)