### Video Transcript

In this video, we’re talking about measurement accuracy and precision. As we’ll see, these are two terms that come up often when we make measurements of physical quantities. And even though sometimes these two terms are treated as though they mean the same thing, in fact, they have very particular and very distinct definitions. Since accuracy and precision both come up in the context of making measurements, let’s think for a moment about just what a measurement is.

We can define a measurement as a collection of numerical data that describes a physical quantity. Examples of this would include things like using a ruler to find the length of a pencil or using a scale to find the mass of some object or using a stopwatch to count how long it takes a runner to run around a track. In each one of these examples, we would collect some numerical data that describes a physical quantity. Now we know that the intention of a measurement is to come up with a value which matches the true value of the quantity we’re trying to measure.

For example, let’s say that the actual true length of our pencil is 12 centimeters. When we go to measure the pencil’s length, we would hope to come up with a value that very closely matches that true length. But for a number of possible reasons, this doesn’t always happen. For example, maybe we don’t have the bottom of the pencil and the bottom of our ruler perfectly aligned. Or maybe the pencil and the ruler aren’t perfectly parallel with one another. Or it’s even possible to have everything about our measurements set up correctly. But we simply read off the incorrect value from our ruler. These are all possible err resources that would lead to a measurement that doesn’t match the true value of our pencil’s length.

Let’s say that when we measure the length of this pencil, the value we read off of the ruler is 11 centimeters. And let’s further imagine that using this particular ruler, the best we can do in reading off values is to find a measurement to the nearest centimeter. So we make one measurement, which comes out to 11 centimeters. Now, if we don’t know the true value of the length of our pencil, we may very well think that this is it, that it’s 11 centimeters long. But aware that it’s possible to make a mistake in a measurement process, we decide to redo the experiment. We’re going to take a second measurement. Say that when we do, we find a result not of 11 but of 12 centimeters. And then say we repeat this measurement process one more time and we find a result of 13 centimeters.

Based on these results, if we were to give one value for what we thought the length of the pencil was, a common and useful strategy is to average all of the measurements we’ve collected. The average of 11 and 12 and 13 is 12. So our average measurement of these three is 12 centimeters. Now, given the inside information that the true length of this pencil is indeed 12 centimeters, we would know that our answer is an accurate answer. And when we say that an answer is accurate, we mean that it closely matches the true value we wanted to measure.

We can write that out this way. We can say that accuracy is an indication of how close measurements are to the true value of what is being measured. Our average value for the pencil length of 12 centimeters exactly matches its true value and therefore is a completely accurate result. On the other hand, if we had measured the average length of the pencil to be, say, four centimeters, then that would be less accurate than this result we found here. Now, when we’re talking about measurement accuracy, there are a couple of different ways to think of it.

One way is to think of the accuracy of individual measurements we make. So, for example, the measurement accuracy of this 11-centimeter reading or this 12-centimeter reading or this 13-centimeter reading individually. But another approach, the way that we went about doing it, was to find the average value of a set of measurements and then compare that average to the true value. Either way, whether we’re working with an individual measurement or an average measured value, we’re still able to talk about the accuracy of those measurements. That’s because accuracy involves comparing those measurements against the true value of what we’re trying to measure. The more closely these two values agree, the more accurate the measurement was. So that’s the meaning of accuracy. Now let’s consider a second set of measurements which would help us understand the term precision.

Say that once more we make three independent measurements of the length of our pencil. The first measurement comes out to be four centimeters. The second comes out to be 28 centimeters. And the third measurement is four centimeters. Now, if we were to once more consider these three values as a set of data and find the average of them. The average of four and 28 and four is 12. So once again, we report a pencil length of 12 centimeters. Now, even though these two final results are the same, we can see that there’s something different about the way we got them. The first set of measured values were all tightly clustered around 12 centimeters, while the second set was very spread out compared to that. It’s precisely that spread within a set of measured values that the term precision refers to.

The term precision means how close two or more measurements are to each other. Side by side then, we can see how precision and accuracy are different. When we talk about precision, we’re talking about two or more measurements of some value and comparing them to one another. If the values are close together, we say that they are precise. But then, if the values are very different from one another, we would say that that data set is not precise. So comparing the first set of measurements we made with the second set, we can say that the first set is more precise. Because the difference between these three measured values is smaller than the difference between these three measured values.

Interestingly, the precision of a measured value has nothing to do with the actual, true, value that we want to measure. As far as the precision of our measurements made in trying to find the true length of our pencil, we don’t even need to know that true length to know whether our results are precise or not. This is very different from what we found for accuracy, where accuracy necessarily compares the true value with the measured value. Along with this difference between accuracy and precision, there’s one other that we should mention.

It’s completely reasonable to talk about the accuracy of a single measurement we would make. That’s because we’re comparing that measured value with the true value of the quantity of interest. But when it comes to precision, it doesn’t make sense to talk about the precision of a single measurement. Why is that? It’s because precision necessarily involves a comparison between multiple measured values, at least two of them. Considering once more this second set of data, if we were asked, what is the precision of this measurement here of four centimeters, there’s no way we could give a sensible answer. That value has no precision except in comparison to the other measured values. It’s only when we consider two or more measurements together that we’re able to talk about the precision of data.

So summing all this up, to know the precision of a measured value, say our final value of 12 centimeters for the length of the pencil, we need to have two or more measured values we can compare with one another. And we don’t need to know the true value of the quantity we’re trying to measure. To find the accuracy of a result though, only one measured value is needed. But we do need to know the actual true value of the quantity we’re trying to measure. Along with our understanding of these two terms, we can now notice something else about our measurement results.

We saw that our first set of measurements taken together were more precise than our second set of measurements. Yet despite that difference, the accuracy of our final result in both cases was the same. This means it’s possible for us to arrive at an accurate final answer using an imprecise set of measurements. And what’s more, the reverse is also a possibility. We could have a very precise set of measured values. Say we made three measurements of the pencil’s length and we found the same values each time. So when we average them together, we get a result of 17 centimeters. We could say that this measurement set is very precise, as precise as it possibly could be, since the values are the same. And yet our answer is not accurate.

Here’s what we’re finding then. It’s possible to be precise but not accurate. And it’s possible to be imprecise and accurate. It comes down to the differences between these two terms, accuracy and precision. Let’s test our understanding of these terms through a couple of examples.

Which of the following statements most correctly describes the meaning of the precision of measurements? A) A precise measurement is more accurate than an accurate measurement. B) The more precise the measurement of a quantity is, the closer the measured value is to the actual value of the measured quantity. C) A precise measurement is made using a correct measurement method. D) The more precise the measurement of a quantity is, the smaller the predictable change that can be made between the measured value and other measured values of the same quantity.

Okay, we can see that the answer to this question is all about the specific meaning of the word precision when it comes to making measurements. We have these four different candidates for the most correct description of the precision of measurements. So let’s go through them one by one and evaluate each in turn.

Option A says that a precise measurement is more accurate than an accurate measurement. Now, one thing this option gets right is that it acknowledges that there’s a difference between a precise measurement and an accurate measurement. They don’t mean the same thing. But what it doesn’t get right is in claiming that a precise measurement is more accurate than an accurate one. Because precision and accuracy are two different terms with two different meanings, a precise measurement will not be more accurate than an accurate one. We can cross off option A then.

Option B says that the more precise the measurement of a quantity is, the closer the measured value is to the actual value of the measured quantity. So option B is saying that we have these two values: a measured one and the actual correct one for some quantity. And it says that the closer these two values are, the more precise the measurement is. But this statement is confusing the terms for precision and accuracy. If we replace this word “precise” with the word “accurate,” then this description would be correct. The closeness of a measured value to an actual true value is the meaning of an accurate measurement. But because precision and accuracy don’t mean the same thing, Option B is not a correct description of a precise measurement.

Moving on to option C, this says that a precise measurement is made using a correct measurement method. Well, it’s certainly more likely that using a correct measurement method will lead to a precise measurement. But that’s not always the case. It’s possible, for example, to have a correct measurement method. But the way we carry out that method has errors in it. And those errors, which would influence the measurements we would make, might lead to imprecise measurements. Overall then, option C is not looking like a great choice. But because it’s not explicitly incorrect, let’s table it for now, keep it in mind, and then move on to our last choice, option D.

This option says that the more precise the measurement of a quantity is, the smaller the predictable change that can be made between the measured value and other measured values of the same quantity. Now here’s what this description is saying. Let’s say that we have some true value for a quantity. And we’ll call that value 𝑉. This could be the length of an object or its mass. But the point is it’s the accurate representation of that quantity. In order to discover what that correct quantity is, we make a series of measurements. We can say that the result of our first measurement is 𝑀 one. And then we make another measurement 𝑀 two, another one 𝑀 three, and so on. We can make any number of these measurements attempting to figure out the true value of the quantity we’re interested in.

Now, statement D is not talking about this true value 𝑉. What it’s comparing is the different measured values of this quantity one to another. And it’s saying that the smaller the difference between these measured values, the more precise our measurement is. And this is a good description of measurement precision. It compares measured values to one another and says the closer they are to one another, the more precise our measurements are. Since option D is an explicitly correct description of the precision of measurements, we choose this as our answer. Let’s look now at one more example, this one helping us understand what measurement accuracy means.

Which of the following statements most correctly describes the meaning of accuracy of measurements?

Okay, let’s go through these statements one by one. A) An accurate measurement has a value that is the same value when a quantity is repeatedly measured. Now to set this up a bit, let’s say that this quantity that we’re measuring has some true accurate value. We’ll call that value 𝑇. And what we’re doing is we’re making measurements of this quantity with the hope that those measurements reveal to us 𝑇, that true value. So we make a series of measurements. And their results we can call 𝑀 one, 𝑀 two, 𝑀 three, and so forth. Now this option A is saying that if 𝑀 one, 𝑀 two, 𝑀 three, and the other measured values we have all have the same value, then that’s what an accurate measurement is. But notice that all these measurements could be the same thing and yet be very different from our true value 𝑇. So just because our measured values agree with one another, just because they have the same value doesn’t mean our measurement is accurate. So option A is off our list.

Moving on to option B, the more accurate the measurement of a quantity is, the smaller the predictable change that can be made between the measured value and other measured values of the same quantity. Similar to option A, option B is comparing these measured values one to another, and it’s not comparing them to the true value 𝑇. Option B is saying that the closer 𝑀 one, 𝑀 two, 𝑀 three, and so on are together, the more accurate the measurement is. But again, this leaves out a comparison with the true value of this quantity. So option B can’t be our choice either.

Option C says that an accurate measurement is made using a correct measurement method. It’s true that using a correct measurement method makes it more likely that our measurement will be accurate, but it doesn’t make it inevitable. It’s possible to have a perfectly correct measurement method. But to execute that method incorrectly, leading to measurement values which are not accurate. That is, do not match up closely with the true value of the quantity we’re measuring. So option C doesn’t seem like it will be our choice either.

On then, to our last hope, option D. This says the more accurate the measurement of a quantity is, the closer the measured value is to the actual value of the measured quantity. In Option D, for the first time, we’re talking about making a comparison between the measured quantities and the true value of the quantity we’re measuring. Option D says that the smaller that gap is, the smaller the difference between the true value and a measured value, the more accurate that measured value is. And this is true. Accuracy refers to a comparison between the true value of some quantity and the measured value of that quantity. This then is the statement that most correctly describes the meaning of accuracy of measurements. Let’s take a moment now to summarize what we’ve learned in this lesson on measurement accuracy and precision.

We’ve seen that accuracy and precision are two terms we use in describing measurements. And that each one of these terms has a distinct meaning. We learned that the closer a measured value or average measured value is to the true value of the quantity being measured, the more accurate that measured value is. And then the closer a set of two or more measured values are to one another, the more precise the measurement is. So accuracy then involves comparing measured values with the true or actual value of the quantity being measured. While precision involves a comparison among measured values, without any reference to the true or actual value of the quantity of interest. So accuracy in measurement and precision in measurement are both useful terms. And each one has its own distinct meaing.