### 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.