In this video, we’re going to learn about measurement uncertainties, what they are where they come from, and how to work with them practically.
Imagine that we’re at a race. It’s the 100-meter finals for the world track championships. At the end of this race, two runners, A and B, stretch towards the finish line to try to beat out their opponent by a hair. After they break through the tape and begin to cool down after the race, the results come on screen. Runner A took 9.85 seconds to complete the race. Runner B took 9.86 seconds. The question comes up: how do we know? How can we be confident that these numbers are accurate, that we can point to runner A as the winner having run the race fastest? The answer comes down to measurement uncertainties.
Uncertainty in measurements comes down to the simple fact that whenever we measure something, whether it be a distance or a mass or a time, we can’t be infinitely precise in that measurement. For example, say we wanted to measure the mass of a chicken’s egg. We could put the egg on a scale and read out what the scale says. Say that it read the egg’s mass was 63 grams. And the question is, “Is that exactly correct? That is, is the mass of the egg 63.0000000000000 and so on to an infinitely many number of zeros? When it’s put that way, we can tell that the answer is no. The mass is not exactly 63 grams. There is a limit in the precision of our measurement that probably has a lot to do with the scale that we’re using.
Or if we were measuring the length of something, say our foot, then our precision would still be limited by the type of ruler that we used and how far apart the gradations written on it were. We’d also find some lack of precision if we measured something in terms of time. The clock that we’re using might be accurate to one one hundredth of a second. But what about one one thousandth, one ten thousandth, one one hundred thousandth? No matter what instrument we use, it will not be infinitely precise. And therefore, our measurement will be at least somewhat uncertain. Because of this, a method has been devised for quantifying how certain or uncertain a measurement that we make is.
Going back to our example of finding the mass of this egg, the scale that we use to make the measurement would be rated to a certain level of precision. So when it would read out a particular mass. When we write down that measured value, instead of writing a precise value by itself, we would first write out the measured value and then add on a symbol that means plus or minus and then add what’s called the uncertainty in that measured value, which is technically known as the measurand. When we include the uncertainty of our measured value, what we’re doing is creating a range of possible values for that actual measurand given the precision of our measuring method.
So in the case of our egg, we’re saying with these numbers that the egg’s mass could be as much as 65 grams and it could be as little. That’s the range of possible values of the mass of this egg, again, given the uncertainty of our instrument. When it comes to writing down measured values with their uncertainty, there are a few rules of thumb we can follow. The first is that uncertainty written down should be rounded to one significant figure. In our example with the egg, we had a two-significant-figure measurand but a one-significant-figure uncertainty of two grams. And also, uncertainty should be of the same order of magnitude as the last significant figure of the measurand.
So for example, if our measurand was 3.14159, then the uncertainty could be written as 0.00006, where the order of magnitude of our significant figure in our uncertainty matches the order of magnitude of the last significant figure in the measurand. With these rules of thumb in place, now let’s look at how to actually use uncertainties when we do calculations.
Imagine that you measure the height of a box, and you find that it’s 74.8 plus or minus 0.7 centimeters in height. Imagine that, next, you measure the height of a glass jar, and you find that the height of the glass jar is 32.3 plus or minus 0.2 centimeters. Now if you put the glass jar on top of the box, what would the overall height be? To find this out, we’ll clearly add together the height of the box and the jar. But what about their uncertainties? To find that out, would we simply add together the uncertainty of the height of the box with the uncertainty of the height of the jar or is there some other way we could approach this?
It turns out that there is a more accurate way to calculate the total uncertainty then simply adding together the two separate uncertainties measured. This method is called summing in quadrature. And it looks like this. Say we have one measured uncertainty, we’ll call it 𝑈 sub 𝐴, and a second measured uncertainty, we’ll call it 𝑈 sub 𝐵. We can mathematically combine these two uncertainties in the following way. The combined uncertainty of 𝑈 sub 𝐴 and 𝑈 sub 𝐵, if we call it 𝑈 sub 𝐶, is equal to the square root of 𝑈 sub 𝐴 squared plus 𝑈 sub 𝐵 squared. This may look familiar from your work with vectors or with work involving right triangles.
Summing in quadrature is a helpful method to use whether we’re adding, subtracting, multiplying, or dividing values with uncertainties. If we apply it to our example of the box in the jar, we can say that 0.7 centimeters is 𝑈 sub 𝐴 and 0.2 centimeters is 𝑈 sub 𝐵. When we plug in those values into our square root sign and calculate 𝑈 sub 𝐶, we find that, to one significant figure, 𝑈 sub 𝐶 rounds to 0.7 centimeters. So that’s what we would write for our uncertainty in the total combined height of these two objects.
So for all the mathematical operations we’d use — addition, subtraction, multiplication, and division — remember the rule of summing in quadrature, that the total or overall or combined uncertainty is equal to the square root of the sum of the squares of the individual uncertainties associated with each measurement. So far, we’ve really been talking about one kind of source of uncertainty in our measurements. And that has to do with the measuring device itself and our ability to read off values from it. Importantly, there is another way of understanding uncertainty. It doesn’t have to do with limits in reading values or making measurements, but comes down to variability in the objects being measured themselves. Here is an example.
Remember our chicken egg when we measured its mass. If we ignore the uncertainty of the device itself, we found a result of 63 grams. But what if we were to take another chicken egg and measure its mass? Say this egg is slightly smaller. And when we put it on the scale, it reads out 59 grams. Say we got a third egg and measured the mass of that. And it came out to 61 grams. And a fourth egg and a fifth egg. Say after we make all these measurements, someone comes to us and poses the question, “What is the mass of a chicken egg?” Well, we have five different masses. So how do we answer the question of what the mass of a chicken egg is.
Now you might think, “well, what if we take the average of all five of our measured values and report that as the mass of a chicken egg?” Good idea! When we calculate that average, we find it’s 61.8 grams. So we can report that as the mass of a chicken egg. But wait a second! Is that an exactly precise value? It’s not because it’s the result of averaging a series of different values together. But how do we find the uncertainty of this measure? Well, we can use a statistical function called standard deviation. It’s the average amount a set of values deviates or differs from the average of the set.
In the case of our egg example, we’ve calculated the average of our data points. That’s 61.8 grams. That’s the average mass of a chicken egg. To find out how much each of our five measured values differs from our average, we can start one by one to compare them against that average or that mean. The difference between 63 grams and our average 61.8 grams is 1.2 grams. The difference between 59 grams and our average is 2.8 grams, and so on down the line through all of our measured data.
To go on to the next steps, we can look at the precise mathematical formulation of standard deviation. This formulation says that if we calculate the average of our dataset — 𝑥 bar, which we have in our case, it’s 61.8 grams — and then find the difference between that average and each of our measured values individually. We’ve also done that. Those are the differences between our measured values and our average. Then following the formula, it says if we square each of these five values and then add them all together and then finally divide by the number of values there are. In our case, 𝑛 is equal to five. And then finally take the square root of that result. We will have calculated the standard deviation or standard uncertainty of our data set. When we calculate the standard deviation of this data set of five points, we find it’s equal to approximately 1.9 grams.
So in answer to the question what’s the mass of a chicken egg, we could reply that it’s 61.8 plus or minus 1.9 grams. And remember that our calculated uncertainty is not due to instrument imprecision or our inability to accurately read data, but is a result of the natural variation of the object that we’re measuring. So let’s summarise measurement uncertainties.
Uncertainties in measurements are caused by instrument limits, human error, and variation in the object being measured. Whenever we make a measurement, there is some level of uncertainty in that measurement. When performing calculations involving uncertainties to find that the total uncertainty, what we’ve called 𝑈 sub 𝑇, use summation in quadrature where 𝑈 sub 𝑇 is equal to the square root of the sum of the squares of the individually measured uncertainties.
And when uncertainty on the other hand comes from variation from within a dataset, it can be calculated through the standard deviation function. This function involves first calculating the mean or average of the dataset then finding the difference between that average and each one of the data points individually, squaring that difference, and then dividing by the number of points in the dataset 𝑛, and finally taking the square root of this result. Because it’s easy to make arithmetic mistakes when using this function by hand, using a calculator or computer to calculate standard deviation is often best unless the dataset is very simple.
And finally, measurement uncertainty tells us within what range a measured value falls. And it lets us compare different measurements accurately. Think back to the example of the two track runners. An understanding of measurement uncertainty is what lets us determine which one crossed the finish line first.