Video: Measurement Error

In this video, we will learn how to define and calculate the absolute and relative errors of measured values.

10:31

Video Transcript

In this video, we’re talking about measurement error. Errors in measurement can happen for many different reasons, for example, in this case, a tape measure with improper markings. But our focus in this lesson is going to be on describing and quantifying measurement error. When it comes to an error in a measured physical quantity, we may already have an intuitive sense of what this means. There is some physical quantity, say, the mass of this block, that has a true or an accurate value, in this case, five kilograms. If we then go to measure that quantity and come up with a different number than the true value, then we’re witnessing an example of measurement error.

An important thing to see here is that for measurement error to exist, there has to be some right standard with which we compare a measured value. There’s a name for this; it’s called the accepted value of a quantity. And this is simply the value of some physical quantity when it’s measured accurately; that is, it’s not subject to measurement error. But this might raise a question, how is it that we know that some measured physical quantity hasn’t been altered by measurement error to some extent? When the value of a quantity is very important to know precisely, for example, if the quantity we were talking of was some universal constant like the gravitational constant or the charge of an electron. In cases like these, the accepted value of some physical quantity comes out of many different experiments performed to find that value.

This way, any measurement errors that are made, say, in an individual experiment, can be identified and rooted out. At the end of what may be quite a lot of work then, we have an accepted value for some physical quantity. This value is well tested and well confirmed over a broad range of experiments. So, as we mentioned, the accepted value is our standard. It’s what we make measurements with respect to. And when we make such a measurement, we would hope that our result would agree with that accepted value. If it doesn’t, then some kind of measurement error has taken place.

Whatever the causes of those errors may be, there are a couple of different ways of quantifying these errors. One is to talk in terms of what’s called absolute error. This is defined as the absolute value of the accepted value of some physical quantity minus the measured value. We can see then that if there’s no difference between these two values, then our absolute error is zero. But if there is a difference, as there is in the case of measuring this mass, then we can use this relationship to calculate a number which is the absolute error of our measurement.

In the case of our mass measurement, we saw that the accepted value of this block’s mass is five kilograms. And so we take that value and subtract from it the measured value indicated by our scale. And if we keep just one significant figure in our answer, then our absolute error is one kilogram. And this is just the absolute difference between our accepted value and our measured value.

Now, sometimes we want to know more than just the difference between our accepted and measured values. To see why this might be so, imagine we’ve been tasked to build a gigantic boat. By design, this boat is meant to have a mass of one million kilograms. Let’s say, though, that when we actually get done constructing this boat, we find it has a mass of one million and one kilograms. Now, if we say that one million kilograms is the accepted value of this quantity and that our measured value is one million and one kilograms. Then we could say that the absolute error of this whole boat-construction process is one kilogram. On the scale of construction we’re talking about for a massive boat like this, this absolute error might be acceptably small.

But what if we were looking instead to test the accuracy of a scale that measures much smaller masses? In a case like that, the same exact absolute error might be unacceptably large. In order to show the difference, so to speak, between an absolute error of one kilogram in each of these two different cases, we could rely on something called relative error. And the relative error of a measurement is given by taking the absolute error of that measurement and dividing it by the accepted value. So in the case of our scale measuring the mass of this block, we would have an absolute error of one kilogram divided by an accepted value of five kilograms. And this would equal 0.2. We could say that this is the relative error of our scale in indicating the mass of this five-kilogram block.

But then, how about for our gigantic boat? Here, just like before, our absolute error was one kilogram, but our accepted value is now one million kilograms. This gives us a relative error of 10 to the negative sixth or one part in one million. So, now we’re starting to see the real difference between these identical absolute errors. Relative error shows us that a one-kilogram absolute error in measuring a five-kilogram mass is quite significant. But a one-kilogram absolute error in measuring a very large one-million-kilogram mass makes very little difference.

And then, there’s one way this idea of relative error is extended a step further. We do this by calculating what’s called a percent relative error. And this is simply the relative error of a measured value multiplied by 100 percent. So, recall that our relative error for the mass measured by our scale was 0.2. If we multiply 0.2 by 100 percent, we get 20 percent. That’s the percent relative error. And then, if we take the relative error of our boat’s mass and multiply that by 100 percent, we get 0.0001 percent. And, once again, we see a notable difference between these two values, whereas the absolute error of these two measurements was the same, one kilogram. Now that we know a bit about these different types of measurement errors, let’s get some practice with these ideas through an example.

In an experiment, the atmospheric pressure at sea level on Earth is measured to be 101,150 pascals. Find the absolute error in the measurement using an accepted value of 101,325 pascals.

Okay, so in this experiment, there’s a measurement made of atmospheric pressure at sea level. We can refer to this measured value using a capital 𝑀, and we know it’s 101,150 pascals. We want to compare our measured value to an accepted value of atmospheric pressure at sea level given here. And specifically, we want to calculate the absolute error in this measurement compared to our accepted value that we’ll represent using a capital 𝐴. To do this, we can recall that the absolute error of a measured value is equal to the absolute value of the measured value subtracted from the accepted value.

Basically, we’ll take our measured value, capital 𝑀, and we’ll subtract it from our accepted value for atmospheric pressure at sea level. And we’ve called that value capital 𝐴. And then, lastly, we’ll take the absolute value of this difference. We can now substitute in the values for 𝐴 and 𝑀. And when we do and then calculate this difference, we find it’s equal to 175 pascals. That’s the magnitude of the difference between our measured and accepted values. And, therefore, it’s our absolute error.

Let’s look now at a second example exercise.

In an experiment, the speed of sound waves on Earth at sea level at a temperature of 21 degrees Celsius is 333 meters per second. Find the percent relative error in the measurement using an accepted value of 344 meters per second. Give your answer to one decimal place.

So, in this scenario, we’re talking about making a measurement of the speed of sound waves where, under certain conditions, at sea level and at a particular temperature, we measure a sound wave speed of 333 meters per second. We can call that measured speed 𝑠 sub m. And we’re to compare it to an accepted speed of sound, we’ll call that 𝑠 sub a, of 344 meters per second at the same elevation and temperature. Knowing these values, we want to calculate the percent relative error in our measurement. To help us figure this out, we can recall the equation for the percent relative error of a measured value. It’s equal to the magnitude of the accepted value minus the measured value all divided by the accepted value and then multiplied by 100 percent.

We can apply this relationship to our scenario by substituting 𝑠 sub a for the accepted value and 𝑠 sub m for the measured value. And that gives us this expression. And when we subtract 333 meters per second from 344, we get a value in our numerator of 11 meters per second. Notice now that these units, meters per second, cancel out. And when we calculate 11 divided by 344 multiplied by 100 percent to one decimal place, we get a result of 3.2 percent. This is the percent relative error in our measurement.

Let’s look now at one last measurement-error example.

In an experiment, the acceleration due to gravity at the surface of the Earth is measured to be 9.90 meters per second squared. Find the absolute error in the measurement using an accepted value of 9.81 meters per second squared.

So we have here these two values indicating the acceleration due to gravity at Earth’s surface. One, the measured value, that we’ll call 𝑔 sub m, is 9.90 meters per second squared. We’re to compare this to the accepted value of gravity’s acceleration, we’ll call it 𝑔 sub a, of 9.81 meters per second squared. In our comparison, we specifically want to solve for the absolute error of our measured value.

To do this, we can recall that the absolute error of a measured value is equal to the difference between a measured value and an accepted value and then, if that number is negative, taking the absolute value of it. To apply this relationship, we’ll substitute 𝑔 sub a as our accepted value, and we’ll use 𝑔 sub m as our measured value. So, the absolute value of 𝑔 sub a minus 𝑔 sub m is equal to 9.81 meters per second squared minus 9.90 meters per second squared. And the absolute value of that difference is equal to 0.09 meters per second squared. This is the absolute error in our measured value.

Let’s summarize now what we’ve learned about measurement error. In this lesson, we saw that measurement error occurs whenever a measured value of a physical quantity differs from that quantity’s accepted value. When such a difference exists, it’s possible to quantify it by calculating the absolute error of the measurement. This is accomplished by taking the absolute value of the difference between the measured value and the accepted value.

Another way to quantify measurement error is by calculating what’s called relative error. This is equal to the absolute error of a measurement divided by the accepted value. And lastly, it’s possible to quantify measurement error using what’s called percent relative error. This is calculated by taking the relative error and multiplying it by 100 percent. This is a summary of measurement error.

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