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In this video, we’re talking about measurement error.
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Errors in measurement can happen for many different reasons, for example, in this case, a tape measure with improper markings.
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But our focus in this lesson is going to be on describing and quantifying measurement error.
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When it comes to an error in a measured physical quantity, we may already have an intuitive sense of what this means.
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There is some physical quantity, say, the mass of this block, that has a true or an accurate value, in this case, five kilograms.
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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.
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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.
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There’s a name for this; it’s called the accepted value of a quantity.
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And this is simply the value of some physical quantity when it’s measured accurately; that is, it’s not subject to measurement error.
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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?
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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.
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In cases like these, the accepted value of some physical quantity comes out of many different experiments performed to find that value.
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This way, any measurement errors that are made, say, in an individual experiment, can be identified and rooted out.
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At the end of what may be quite a lot of work then, we have an accepted value for some physical quantity.
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This value is well tested and well confirmed over a broad range of experiments.
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So, as we mentioned, the accepted value is our standard.
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It’s what we make measurements with respect to.
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And when we make such a measurement, we would hope that our result would agree with that accepted value.
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If it doesn’t, then some kind of measurement error has taken place.
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Whatever the causes of those errors may be, there are a couple of different ways of quantifying these errors.
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One is to talk in terms of what’s called absolute error.
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This is defined as the absolute value of the accepted value of some physical quantity minus the measured value.
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We can see then that if there’s no difference between these two values, then our absolute error is zero.
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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.
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In the case of our mass measurement, we saw that the accepted value of this block’s mass is five kilograms.
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And so we take that value and subtract from it the measured value indicated by our scale.
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And if we keep just one significant figure in our answer, then our absolute error is one kilogram.
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And this is just the absolute difference between our accepted value and our measured value.
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Now, sometimes we want to know more than just the difference between our accepted and measured values.
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To see why this might be so, imagine we’ve been tasked to build a gigantic boat.
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By design, this boat is meant to have a mass of one million kilograms.
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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.
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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.
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Then we could say that the absolute error of this whole boat-construction process is one kilogram.
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On the scale of construction we’re talking about for a massive boat like this, this absolute error might be acceptably small.
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But what if we were looking instead to test the accuracy of a scale that measures much smaller masses?
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In a case like that, the same exact absolute error might be unacceptably large.
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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.
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And the relative error of a measurement is given by taking the absolute error of that measurement and dividing it by the accepted value.
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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.
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And this would equal 0.2.
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We could say that this is the relative error of our scale in indicating the mass of this five-kilogram block.
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But then, how about for our gigantic boat?
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Here, just like before, our absolute error was one kilogram, but our accepted value is now one million kilograms.
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This gives us a relative error of 10 to the negative sixth or one part in one million.
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So, now we’re starting to see the real difference between these identical absolute errors.
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Relative error shows us that a one-kilogram absolute error in measuring a five-kilogram mass is quite significant.
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But a one-kilogram absolute error in measuring a very large one-million-kilogram mass makes very little difference.
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And then, there’s one way this idea of relative error is extended a step further.
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We do this by calculating what’s called a percent relative error.
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And this is simply the relative error of a measured value multiplied by 100 percent.
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So, recall that our relative error for the mass measured by our scale was 0.2.
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If we multiply 0.2 by 100 percent, we get 20 percent.
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That’s the percent relative error.
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And then, if we take the relative error of our boat’s mass and multiply that by 100 percent, we get 0.0001 percent.
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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.
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Now that we know a bit about these different types of measurement errors, let’s get some practice with these ideas through an example.
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In an experiment, the atmospheric pressure at sea level on Earth is measured to be 101,150 pascals.
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Find the absolute error in the measurement using an accepted value of 101,325 pascals.
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Okay, so in this experiment, there’s a measurement made of atmospheric pressure at sea level.
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We can refer to this measured value using a capital 𝑀, and we know it’s 101,150 pascals.
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We want to compare our measured value to an accepted value of atmospheric pressure at sea level given here.
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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.
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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.
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Basically, we’ll take our measured value, capital 𝑀, and we’ll subtract it from our accepted value for atmospheric pressure at sea level.
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And we’ve called that value capital 𝐴.
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And then, lastly, we’ll take the absolute value of this difference.
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We can now substitute in the values for 𝐴 and 𝑀.
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And when we do and then calculate this difference, we find it’s equal to 175 pascals.
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That’s the magnitude of the difference between our measured and accepted values.
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And, therefore, it’s our absolute error.
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Let’s look now at a second example exercise.
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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.
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Find the percent relative error in the measurement using an accepted value of 344 meters per second.
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Give your answer to one decimal place.
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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.
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We can call that measured speed 𝑠 sub m.
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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.
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Knowing these values, we want to calculate the percent relative error in our measurement.
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To help us figure this out, we can recall the equation for the percent relative error of a measured value.
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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.
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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.
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And that gives us this expression.
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And when we subtract 333 meters per second from 344, we get a value in our numerator of 11 meters per second.
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Notice now that these units, meters per second, cancel out.
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And when we calculate 11 divided by 344 multiplied by 100 percent to one decimal place, we get a result of 3.2 percent.
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This is the percent relative error in our measurement.
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Let’s look now at one last measurement-error example.
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In an experiment, the acceleration due to gravity at the surface of the Earth is measured to be 9.90 meters per second squared.
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Find the absolute error in the measurement using an accepted value of 9.81 meters per second squared.
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So we have here these two values indicating the acceleration due to gravity at Earth’s surface.
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One, the measured value, that we’ll call 𝑔 sub m, is 9.90 meters per second squared.
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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.
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In our comparison, we specifically want to solve for the absolute error of our measured value.
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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.
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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.
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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.
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And the absolute value of that difference is equal to 0.09 meters per second squared.
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This is the absolute error in our measured value.
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Let’s summarize now what we’ve learned about measurement error.
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In this lesson, we saw that measurement error occurs whenever a measured value of a physical quantity differs from that quantity’s accepted value.
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When such a difference exists, it’s possible to quantify it by calculating the absolute error of the measurement.
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This is accomplished by taking the absolute value of the difference between the measured value and the accepted value.
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Another way to quantify measurement error is by calculating what’s called relative error.
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This is equal to the absolute error of a measurement divided by the accepted value.
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And lastly, it’s possible to quantify measurement error using what’s called percent relative error.
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This is calculated by taking the relative error and multiplying it by 100 percent.
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This is a summary of measurement error.