# Video: Combining Uncertainties

In this video, we will learn how to find the uncertainties in the values of quantities that can be calculated from two or more measured values.

15:20

### Video Transcript

In this video, our topic is combining uncertainties. In this sketch, we see a person checking in for their flight at the airport. But in response to the gate agent’s questions, this person doesn’t know where they’re going. They don’t know when they’re leaving. And they don’t have their ID with them. All these factors combine to create a greater overall uncertainty about who this person is and where they’re going to.

As we’ll see, something similar takes place when we combine measured values. And in this lesson, we’ll learn rules for doing that. As we get started, we can recall that all measured values have some uncertainty about them. This is because any device we use to make a measurement will not be infinitely precise. For example, let’s say that we measure the height of this person here. We might use a tape measure to do that. And even if we were very careful in the measurement process, lining up the top of the person’s head with a tape measure. Still, if we looked closely at the tape measure, we would see that the markings on it limit the precision to which we can report this person’s height.

Let’s say that each one of these hash marks on our ruler indicated a difference in length of one centimeter. In looking at our dotted line representing the height of the top of the person’s head, we could go one decimal place beyond that precision to estimate this person’s height to the nearest tenth of a centimeter. Say that when we do this, we come up with a value of 165.4 centimeters for this person’s height. But this number is a bit uncertain. The true height of the person could be a little bit higher or a little bit lower than this value. Specifically, it could be one-tenth of a centimeter above or one-tenth of a centimeter below the value we’ve reported.

It’s important to include this overall uncertainty in our reported value. We would say this person’s measured height is 165.4 plus or minus 0.1 centimeters. We see then that uncertainties and measurements come from limits in the precision of our measurement devices. Since no measurement device is infinitely precise, there will always be some uncertainty in a measured value. Now, let’s say that using the same ruler, we measure the height of a second person, and that this person’s height comes out to be 148.6 plus or minus 0.1 centimeters. At this point, we have these two measured values and they’re separate. They separately indicate the height of these respective people.

But what if we wanted to combine these heights? What if, through an impressive feat of balance and strength, the second person was able to stand on the head of the first? In that case, to get the combined height of these two people, we’ll need to combine these two measured values. Now, if these were simply height values with no uncertainties, that would be straightforward enough. But let’s think for a moment about how we do this, including the uncertainties in our measured values.

We know that the measured height of the first person has an uncertainty of one-tenth of a centimeter. This means that their true hight could be a tenth of a centimeter higher or lower than this reported value. And then the same thing is true for the height of the second person. Their height also could be a tenth of a centimeter higher or lower than 148.6. Knowing this, we can write out maximum and minimum possible height values for these two people’s heights. Now, if we were to add together these two maximum possible heights, then we would effectively be adding the heights of these two people along with 0.1 centimeters of additional height for each one.

Or on the other hand, if we were to consider adding together the minimum possible heights of these two people, then that value would be equal to the reported heights minus 0.1 centimeters again for each person. What we’re seeing then is that, in combining the heights of these two people, not only do we add together the measured values of their heights, but we also add together the uncertainties in those heights. So then if we take the combined heights of these two people and say we call that height 𝑇 sub 𝐻, that total height will be equal to the height of the first person plus the height of the second person. And we find that total height by adding together the measured values of the two people as well as those measured value uncertainties.

Doing this gives us a result of 314.0 plus or minus 0.2 centimetres. What we’re seeing here is a specific example that can be expanded into a general rule. We can clear a bit of space and then write out that general rule using words. We can say that when adding or subtracting measured values, the uncertainties in the values are added together. Then, here’s how we can write this mathematically. And we need to be a bit careful to understand this notation.

If we have a value — we’ll call it 𝑣 one, and that’s equal to 𝑎 plus or minus the uncertainty in 𝑎 — and then if we have a second measured value — and that’s equal to 𝑏 plus or minus the uncertainty in 𝑏. Then, if we were either to add 𝑣 one and 𝑣 two together or subtract 𝑣 two from 𝑣 one, that would lead to a result of 𝑎 plus or minus 𝑏, depending on whether we’re adding or subtracting between 𝑣 one and 𝑣 two. And then that value has an uncertainty plus or minus of the uncertainty of 𝑎 plus the uncertainty of 𝑏. And by the way, this rule of adding uncertainties together when we add or subtract measured values applies even if we have more than two measured values.

For example, if we had three or even four people standing on one another’s heads, to calculate the total height, we would add together the heights of each person plus the uncertainties in each of those heights. Now, what we’ve covered so far is adding and subtracting measured values. But we know we can also multiply and divide.

For example, let’s say that we had an empty room and we wanted to measure the volume of this room. So we make measurements of the width, the height, and the depth of the room. And say that we come up with these values. In each case, we have an uncertainty of one-tenth of a meter based on the way we’ve measured these distances. Now, we know that to find the volume of the room we’ll need to multiply the height and width and depth together. But then, as we do that, is finding the total uncertainty in this volume as simple as multiplying together the uncertainties for each dimension? As it turns out, no. If that were true, then our uncertainty in volume would be 0.1 meters times itself three times. This is 0.1 meters quantity cubed, which comes out to 0.001 cubic meters.

But this is much too small an uncertainty. We can’t justify a number this precise using the measured dimensions of the room. So this approach is not correct for calculating the uncertainty in our volume. We can’t simply multiply the individual uncertainties together. Instead, before we multiply the room dimensions together, we take one extra step to modify the uncertainties in our measured values. What we do is we convert them to percents. That is, we calculate what percent of 3.0 meters 0.1 meters is and, likewise, we calculate what percent of 4.7 0.1 is and similarly with our last dimension.

When we do this, we find that 0.1 is 1.8 percent of 5.5, it’s 2.1 percent of 4.7, and it’s 3.3 percent of 3.0. Note that we haven’t made any numerical changes to these measured values; we’re just expressing them in a different equivalent way. But now that we’ve done that, we’re ready to calculate the volume of this room by multiplying together 3.0 meters with 4.7 meters with 5.5 meters, but then adding together these percentage uncertainties for each measured value. So here’s how that works. The volume of our room is 3.0 meters times 4.7 meters times 5.5 meters plus or minus 3.3 percent plus 2.1 percent plus 1.8 percent. And note that we’re not multiplying these percentages together, but rather we’re adding them.

Keeping two significant figures in our calculations, all this comes out to 78 cubic meters plus or minus 7.2 percent of that volume. Now, if we wanted to report our final answer with an uncertainty written in cubic meters, rather than as a percent, we could do that by calculating 7.2 percent of 78 cubic meters. When we do that, we get a final result of 78 plus or minus 5.6 meters cubed. That’s the properly calculated volume of this room including the uncertainties in each dimension. Just like before, we’re looking at a specific example of a general rule. So let’s write out that rule that applies to multiplying and dividing measured values with their uncertainties.

Whenever we multiply or divide measured values, their percent uncertainties are added together to yield the total uncertainty. We can write this out mathematically like this. Say that we have three values, 𝑎 with its own uncertainty, 𝑏 which has its own uncertainty, and 𝑐. If 𝑐 is equal to the product of 𝑎 and 𝑏, then the uncertainty in 𝑐, that product, divided by 𝑐 is equal to the uncertainty in 𝑎 divided by 𝑎 plus the uncertainty in 𝑏 divided by 𝑏. Now, this equation may seem a bit confusing because were we talking about percents and adding those together. But actually, this equation is closer to showing us percentages than we might realize.

If we were to multiply both sides of the equation by 100 percent, then that would make each one of the terms in this equation a percentage uncertainty. First, the percent uncertainty of 𝑐, then of 𝑎, and then of 𝑏. But since this factor of 100 percent is common to all the terms, we can cancel it out. And that leaves us with this simplified expression. It’s worth noting that even though we’ve said this expression applies for multiplying two values together to yield the third, it’s also accurate if we were to do a division instead. For example, if we were to divide 𝑎 by 𝑏, that wouldn’t change anything about this equation for calculating the total resulting uncertainty.

We’ve now talked about calculating total uncertainty for adding, subtracting, multiplying, and dividing. There’s one special case, though, that we can take a moment to look at. That special case is when we’re raising a number to some integer power. We can see an example of this if we imagine that all the dimensions of our room are the same. They’re all 3.0 meters, which means that all the room dimensions have the same uncertainty, 3.3 percent. Now, according to this rule and to our experience, we could compute the total uncertainty in the volume of this room by adding together 3.3 percent with 3.3 percent with 3.3 percent which gives us 9.9 percent.

We can see, though, that this is mathematically equivalent to taking that percent uncertainty, 3.3 percent, and multiplying it by three. And three has a special significance when we’re talking about calculating the volume of some object. And that’s because if we have a cubic object, like we do in this example, then the object’s volume is equal to the length of each side cubed. Now, it turns out that this number here and this number here, being the same, is not a coincidence. Once more, this is a specific example of a general rule in calculating uncertainties when we’re raising a number to an integer power. We can write that out this way.

We can say that when raising a measured value to an integer power, the total uncertainty equals that power times the percent uncertainty of the measure value. That’s a mouthful. But if we write it in symbols, it just means that if we’re calculating the integer power of some value, we’ll call that value 𝑥, and the integer power 𝑏. Then the uncertainty in the result of that calculation, the uncertainty in 𝑦, can be found by using this equation. And we see that this equation involves multiplying the fractional uncertainty in 𝑥 by that power 𝑏.

As we noted, this rule is really a specific application of the rule we discovered earlier for multiplying and dividing. When we are multiplying together identical measured values, either approach will give the right answer. It’s just that one of them, for example, has us adding the percent uncertainty together as many times as it appears, while the other approach, the one we’ve just learned, takes that percentage uncertainty and multiplies it by the number of times that uncertainty is there. Either way, we come out with the same result in the end. Now that we’ve learned various ways of combining uncertainties, let’s get some practice with these ideas through an example.

Two resistors have a resistances of 20 plus or minus 0.01 [0.1] ohms and 80 plus or minus 0.02 [0.2] ohms. If the two resistors were placed in series, what would the uncertainty of the two resistors together be?

Okay, so in this example, we have two resistors. We’ll say that this is our first and then this is our second. And we’re told these two resistors are placed in series with one another. For both, we’re given the value of their resistances. And we see that those resistances include an uncertainty, 0.1 ohms in one case and 0.2 ohms in the other. We can recall that when resistors are placed in series, like they are here, their resistances add together for a total combined resistance value. If these resistance values were stated without uncertainties, it would be straightforward enough to add 20 ohms to 80 ohms to get a total resistance of 100 ohms. But in this case, we do have these uncertainties that we’ll need to consider as well.

The way to do this is to recall our rule for combining uncertainties, specifically when we’re adding two values together. Let’s say that we have one value. We’ll call it 𝑣 sub one. And this is equal to 𝑎 plus or minus the uncertainty in 𝑎. And similarly, we have a second value, 𝑣 two, which is equal to 𝑏 plus or minus the uncertainty in 𝑏. Now, if we were to add 𝑣 one to 𝑣 two, then the way we would do that is we would add 𝑎 and 𝑏 and then, with the uncertainties, add those together as well. We can apply this rule to our particular scenario of adding the values of these two resistors.

If we were to solve for the total resistance, we can call it capital 𝑅, of these two resistors together, then by our rule, that would be equal to 20 plus 80 plus or minus 0.1 plus 0.2 ohms or, in other words, 100 plus or minus 0.3 ohms. Now, it’s not the overall resistance 𝑅 that we want to solve for, but rather the total uncertainty of these two resistors together. In solving for 𝑅, though, we have found that total uncertainty. We see that it’s equal to the sum of the individual uncertainties. That total is 0.3 ohms.

Let’s summarize now what we’ve learned about combining uncertainties. We first learned that when measured values, and all measured values have uncertainties, are added together or subtracted from one another, then their uncertainties add. Mathematically, if 𝑐 is equal to 𝑎 plus or minus 𝑏, where 𝑎 and 𝑏 are measured values with uncertainties, then the uncertainty in that sum or difference is equal to the uncertainty in 𝑎 plus the uncertainty in 𝑏. This was the first rule we saw for combining uncertainties.

The second rule we discovered was that when measured values are multiplied or divided, their percent uncertainties add together. Using symbols, if 𝑐 is equal to 𝑎 times 𝑏 or 𝑐 is equal to 𝑎 divided by 𝑏, where 𝑎 and 𝑏 are measured values, then the uncertainty in 𝑐 divided by 𝑐 is equal to the uncertainty in 𝑎 divided by 𝑎 plus the uncertainty in 𝑏 divided by 𝑏. And recall that these fractional uncertainties can quickly be turned into percent uncertainties. We do that by multiplying both sides of the equation by 100 percent.

Lastly, we learned a specific application of this multiplication rule. When a measured value, we’ll call it 𝑎, is raised to an integer power, we’ll call that 𝑛, then the fractional uncertainty of the result is equal to 𝑛 times the fractional uncertainty of 𝑎. This is a summary of combining uncertainties.