Video: Measurement Uncertainty and Resolution

In this lesson, we will learn how to define resolution based, and random measurement uncertainties, and show how they effect the values of measurements.


Video Transcript

For this video, our topic is measurement uncertainty and resolution.

Whenever a measurement is made, for example, in a sporting event, like we see on screen here, these topics of resolution and uncertainty come into play. In the case of this track event, apparently the measurements of the runners’ times are fairly uncertain. And because of that, we can’t say for certain just who finished first, second, third, and so on. The first step to developing instruments that are less uncertain is understanding what these terms resolution and uncertainty refer to. Let’s start there, and we’ll begin by defining this term “resolution.”

When we hear this term resolution, we may think of it in terms of pictures or images. A picture with low resolution may look very blocky. Edges in the image don’t look smooth, but rather look very chunky or made of blocks. In contrast, a high-resolution image has smooth-looking edges. When looking at it, our eye isn’t able to detect the squares or rectangles we see in the low-resolution version. Indeed, thinking of resolution this way does help us understand the definition of this term. Resolution is the fineness or the specificity to which an instrument can be read. Here’s an example of that.

Say that we have two different rulers. One ruler, the one on top, is marked out with centimeter units. So here at the start is zero centimeters. Then this is one centimeter, then two centimeters. While the other ruler, the second one, is marked out in units of millimeters. We can see the centimeter markings on this one as well. But in between those markings, we have these small hash marks which indicate millimeters. Let’s say further that using these two rulers, we wanted to measure the length of some object.

Using the top ruler, the one marked out to centimeters, we might look at the end of our object and say that it looks like it’s closer to two centimeters than one centimeter, in which case the reading for the length of this object would be recorded as two centimeters. But then, if we measure the same object using our millimeter-marked-out ruler, we see the object’s length is one centimeter plus one, two, three, four, five, six millimeters. Using our second ruler then, the length of the object that we would record could be 1.6 centimeters.

From this example, we can see that the second ruler — the one marked out in millimeters — has a greater degree of fineness to which it can be read. That is, this ruler — the one marked in millimeters — more finely marks out units of length than the one marked out in centimeters. We would say then that this second ruler enables more highly resolved measurements than the first one. And note that when we talk about an instrument by which we make measurements, that could be measuring any physical quantity. It might be measuring an object’s mass or an amount of time elapsed. It could be measuring electric current or gravitational attraction. Any measurement instrument will have some degree of resolution, the fineness, to which the instrument can be read.

Now this idea of resolution in measurements leads us into measurement uncertainty. Let’s again consider the process of measuring the length of this object with our two different rulers. In particular, let’s remember what we did when we measured out the length of this object on our centimeter ruler. In determining the object’s length from that ruler, we basically divided the distance between one and two centimeters in half. That halfway point is roughly here. And then we said, because our object goes beyond that halfway point, we’ll report that it has a length of two centimeters. That’s the whole number or integer value of centimeters that its length is nearest to. But notice how much uncertainty in this measured value that process creates.

What we’ve said is that if the length of our object falls anywhere between this halfway point and two centimeters, then we’ll record its length as though it were two centimeters. But we see that, really, the object’s actual length could be as short as one and a half centimeters. Or imagine that it was even longer than two centimeters. Then the objects length could go up to 2.5 centimeters, and we would still record its length this way. Here’s what we’re finding then. The length of our object could fall anywhere between 1.5 centimeters and 2.5 centimeters. And we would record its length as two centimeters. We can see then just how uncertain this measurement value of two centimeters is. It could be off by as much as one half of a centimeter in either direction.

At this point, we can define just what this term “uncertainty” means when it comes to making measurements. Uncertainty is the interval over which a measured value is likely to fall. In the case of our object whose length is being measured on the centimeter ruler, we can see that that interval is anywhere from 1.5 centimeters up to 2.5 centimeters. That’s the uncertainty of this measurement. Now we’ve been looking mostly at the centimeter-marked ruler. But let’s look at the millimeter-marked ruler. And we’ll see that this reading, too, has an uncertainty in it. When we measure the length of our object on this millimeter ruler, that length fell almost exactly at 1.6 centimeters. That made it easier to write down 1.6 centimeters as our measured length. But notice that we would’ve recorded the same length even if our object had been a bit longer or a bit shorter. Basically, so long as the measured length of our object fell within one-half of one millimeter of this point on our ruler, then we would’ve recorded its length as 1.6 centimeters or 16 millimeters.

So we see, even in this case, there’s a bit of uncertainty involved because our object could have been a bit longer than 1.6 centimeters or a bit shorter and we would’ve recorded this length as 1.6 anyway. So when we talk about the interval over which a measured value is likely to fall, in the case of our millimeter ruler, that interval is a half millimeter beyond 16 millimeters and one-half millimeter below it. In other words, we have a total interval or measurement uncertainty of one millimeter. Now what we’re seeing here, in terms of the uncertainty of the measurements we’re making of this length with our two rulers, would be true regardless of how finally resolved our measurement stick is. There is always a resolution limit to a measurement instrument. And therefore, every measurement ever made has some amount of uncertainty in it. In other words, because instrument resolution is never perfect, measurement uncertainty always exists, even if that uncertainty is very small.

Now if we get back to the two recorded lengths of our object, two centimeters for our centimeter-marked ruler and 1.6 centimeters for our millimeter-marked ruler, just by looking at these two numbers, even if we didn’t know where they came from. We can see that there’s more useful information in this value of 1.6 centimeters than in the value of two centimeters. The way we talk about that is we say that there are more significant figures in this number — 1.6 centimeters — than there are in this number. And in order to talk a bit about significant figures, let’s clear some space on screen.

Considering significant figures in general, here is how we can talk about them. Significant figures, also sometimes called significant digits, are digits that carry meaning contributing to measurement resolution. Think once more about our two measured lengths of our object: two centimeters with our one ruler and 1.6 centimeters with the other. If we were to count up the number of significant figures in each of these numbers, the two-centimeter figure has one significant figure and the 1.6-centimeter number has two. And this greater number of significant figures here shows us that the measurement method we used to get that result was more highly resolved or had better resolution than the method we used to get this result. And we can remember that’s true from thinking of our two rulers, marked down in centimeters and millimeters, respectively.

And we could imagine this trend continuing. Say that we had a third ruler that was marked out to tenths of millimeters. We could say then that the resolution of our first ruler was in centimeters. The resolution of the second one was in millimeters. And we’re now imagining a third ruler with a resolution of tenths of millimeters. If we were to measure the length of our object with this third, more highly resolved ruler, the answer we get might be something like 1.62 centimeters. This value then would have one, two, three significant figures in it, with the greater number of significant figures indicating a more highly resolved measurement instrument. We can see then that the better the resolution of our instrument is, the more significant figures will have in our measured value and therefore the less uncertain that measured value will be.

Now there are a few rules for working with significant figures that are worthwhile to know. The first rule has to do simply with what is and what is not a significant figure. And we can abbreviate that s.f. In general, we can think of any number as a significant figure unless it’s a leading zero. For example, in the number 014, the zero is a leading zero and therefore not significant. And we also don’t consider significant trailing zeros when they’re used specifically as placeholders. For example, say we measured a distance in kilometers to the nearest kilometer and it turned out to be a distance of five kilometers. If we wrote that out in units of meters, then we would write 5000 meters. That’s five kilometers. But we can see that these three zeros are placeholders. We don’t actually know the value of this distance that precisely. So then, these zeros are not significant. In this number then, there’s just one significant figure — that’s the five.

Now, sometimes in calculations, we’ll be working with values that have different numbers of significant figures. For example, let’s say that we measured some object to travel a distance of 14 meters in a time of 1.36 seconds. We can see that this first value, the 14 meters, has two significant figures. But the time has three. Now say we wanted to combine these numbers to calculate the speed of this object: the distance it travels divided by the time it took to travel that distance. In doing this, we would divide the distance traveled by the time elapsed. But then when we have to write down our final answer, we need to make a choice. How many significant figures do we use in that answer? This second rule tells us how to figure that out.

This rule says that when we combine values with different numbers of significant figures in them, in our answer, we use the least number from that set of values. So in this example, we have one value with three significant figures and one with two. So, following this rule, we would report our answer to two significant figures. We would write our answer as 10 meters per second, where in this case, the zero is not a place holder, but is in fact significant.

Now just one last rule with working with significant figures. And that is this: when we think of the significant figures in a value and the number of decimal places that value has, in general, these two numbers are not the same. Let’s consider this random number, 37.456. When we count the number of decimal places, we see there are one, two, three in this number. But because all of the digits in the number are not zero, that means all of them — all five — are significant. So then, there are five significant digits in this number, but only three decimal places.

Or consider a different number. Consider this one: 0.32000. We can see that there are one, two, three, four, five decimal places in this number. When it comes to significant figures, let’s say that these last three zeros, these trailing zeros, are placeholder values that we don’t actually know the number to that precision. In that case, based on rule number one of what is and is not a significant figure, we would say that these three zeros are not significant and neither is this leading zero here. This number then just has two significant figures: the three and the two. So in this case, while we only have two significant figures, we have five decimal places. Once again, these two values are not the same.

Related to this, let’s consider how we might round these two different values. Let’s say we wanted to round this first value to three significant figures and we wanted to round this second value to one significant figure. The way we would go about doing that is we would count off the number of significant figures we want to keep in our answer. So in the case of this first value, we would count one, two, three significant figures. And then we look at the next digit in our number significant or not after this. If that digit is equal to five or higher, then we round up our last significant figure. In this example, that digit is equal to five and therefore we do round up. So to three significant figures, we would report this value as 37.5.

Then for our second value that we want to round to one significant figure, we count off the first significant figure. This is the one we’ll keep. And then, we look at the next digit following on whether it’s significant or not. Once more, if that digit is five or higher, we round our three up. And if it’s lower than five, we keep the three the same. Since two is less than five, we won’t change the three. We won’t round it up to four. So to one significant figure, this number rounds to 0.3. Let’s get a bit more practice now with these ideas through an example.

A distance of 115 meters is measured to the nearest meter. The distance is run in a time of 12 seconds, measured to the nearest second. Rounding to an appropriate number of significant figures, what was the average running speed?

Alright, so in this example, we have a runner running at distance, which is measured to the nearest meter, to be 115 meters. Along with this distance, the time it takes the runner to cover that distance is also measured. And that time — we can call it 𝑡 — is measured to the nearest second to be 12 seconds. The question is what is the runner’s average speed. And importantly, we want to report this average speed to the appropriate number of significant figures. Along these lines, we can begin by counting the number of significant figures in each of these two numbers, the time 𝑡 and the distance 𝑑. The measure time has one, two significant figures and the measured distance has one, two, three of them.

At this point, let’s recall that the average speed an object has — we’ll call it 𝑠 — is equal to the distance the object travels divided by the time it takes to travel that distance. We can see then that our runners average speed is going to equal the distance, 115 meters, divided by the time, 12 seconds. But the question is, how should we report our answer? After all, our distance has three significant figures, but the time just has two. Should our final answer be in terms of three or two significant figures? To answer this, we can recall a rule for combining numbers that have different numbers of significant figures in them.

That rule goes like this: when combining values with different numbers of significant figures, answer using the smallest number of significant figures. Applying this rule to our situation, we see that our two values have different numbers of significant figures: three and two, respectively. This rule says that as we combine them, like we are in calculating average running speed, that we should keep the least number or the smallest number of significant figures any one of them possesses. That smallest number is two. And this means we’ll round our final answer to be given in terms of two significant figures.

Now, when we calculate this fraction 115 meters divided by 12 seconds, we get this result of 9.583 repeating meters per second. We know this isn’t our final answer because we’ll round it. And in particular, we’ll round it to two significant figures. To do that, let’s count off two significant figures. Starting at the front of the value, nine and five are the first two. And then we look at the next digit in our number, eight. Since eight is greater than or equal to five, that means we’ll go to our previous number, the five, and we’ll round that up one. The five rounds up to a six. And our final answer to two significant figures is 9.6 meters per second. That’s the average running speed reported to two significant figures.

Let’s take a moment now to summarize what we’ve learned about measurement uncertainty and resolution.

Starting off, we saw that measurement resolution is the fineness to which a measuring instrument can be read. We then learned that measurement uncertainty is the interval over which a measured value is likely to fall. We also learned that because no instrument has perfect resolution, every measured value has some degree, however small, of uncertainty. Further, we learned that significant figures — sometimes abbreviated s.f. — are digits that carry meaning contributing to measurement resolution. Related to this, we learned a rule for determining what is and what is not a significant figure, a rule for how to combine different numbers of significant figures. We saw that in general the number of significant figures in a value and the decimal places in that value are not the same. And lastly, we saw how to round a value to a particular number of significant figures.

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