### Video Transcript

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

Let’s begin by learning about this
term resolution. We may have heard this term before
when talking about images. Here, the resolution of the image
on the left is low, while that of the image on the right is high. The resolution of an image has to
do with how finally that image is divided up.

This term has the same meaning when
we’re talking about measurement tools. These two rulers are marked out in
different units, one in centimeters and the other in millimeters. They both measure the same maximum
length but they have different resolutions. We define resolution as the
fineness to which a measuring instrument can be read. By that standard, this ruler marked
out in millimeters has a higher resolution than this one marked out in
centimeters.

We can see the difference
resolution makes when we go to measure the length of some object. On our centimeter ruler, this
object goes out to this point on that ruler. Because of the way the ruler is
marked, we’ll report the length of this object to the nearest centimeter. Therefore, we have to pick which
centimeter marking the length is closest to. Even though this length is nearly
midway between one and two centimeters, we see it is a bit closer to two. And therefore, we report the length
of our object measured using this ruler as two centimeters.

On our millimeter ruler, a
measurement of the length of the same object turns out a bit differently. To the nearest marking on this
ruler, the length of our object is 16 millimeters. We could report that as the object
length. Or because one centimeter equals 10
millimeters, we could write it as 1.6 centimeters. One difference between these two
measured lengths is that this one, 1.6 centimeters, is more precise. We were able to make this more
precise measurement of length thanks to our higher-resolution ruler.

Notice though that the actual
object length isn’t exactly either one of these two measured lengths. We see that clearly in our
centimeter-marked ruler. It’s also the case though in our
ruler marked to the nearest millimeter. Our reported measured lengths are
approximations of the actual length of this object limited by the resolution of our
measurement tools. We say then that there is some
uncertainty in each one of these measurements. Uncertainty is defined as the
interval over which the true value of a measurement is likely to fall.

To better understand this, let’s
consider again our measurement on our centimeter ruler. Because the length of our object
was closer to two centimeters than to any other integer, we reported a measured
length of two centimeters. But notice that this means our
object could be anywhere longer than 1.5 centimeters and anywhere shorter than 2.5
centimeters if our ruler extended that far. And we would still record the same
value of two centimeters. This span then from 1.5 up to 2.5
centimeters is the interval over which the true value of our measurement is likely
to fall.

There’s even a fairly common way of
indicating that interval in our measured result. For a ruler marked out like this,
we would say that the length of this object is two plus or minus 0.5
centimeters. The actual length of this object
then could be two plus 0.5 centimeters or 2.5 centimeters up here, or the length
could be as small as two minus 0.5 centimeters or 1.5 centimeters. That’s here on our ruler. In words then, we would say that
the length of our object is two centimeters plus or minus 0.5 centimeters. 0.5 centimeters is the uncertainty
of this measurement.

We can go through this same process
using our other ruler. We’ve seen that to the nearest
millimeter, the length of our object is 16 millimeters, or 1.6 centimeters. This means though that our object
could be as short as 15 and a half millimeters or as long as 16 and a half. The uncertainty here then is 0.5
millimeters, which is equal to 0.05 centimeters. We would report our object’s length
then as 1.6 plus or minus 0.05 centimeters. This means the object could be as
long as 1.6 plus 0.05, or 1.65 centimeters, or it could be as short as 1.6 minus
0.05, or 1.55 centimeters. In this case then, the uncertainty
of our measurement is 0.05 centimeters. That tells us the interval over
which the true measurement of this object is likely to fall.

Notice that the uncertainty of our
measurement using the millimeter ruler is smaller than the uncertainty of our
measurement using the centimeter ruler. This means that our measurement
using the millimeter ruler is more precise. In general, a more precise
measurement is one that has a smaller uncertainty associated with it. Something interesting about
uncertainty is that all measurements have it to some extent.

Even if we were to measure the
length of our object using a ruler marked out, say, to the nearest micrometer, that
measurement would still have some uncertainty associated with it. We would still need to report the
length of our object to the nearest micrometer, and that could involve a measurement
uncertainty, or error, of as much as one-half of a micrometer. Uncertainty in measurements is
inescapable. The best we can do is make
uncertainties as small as possible.

The uncertainty we’ve considered so
far is called measurement uncertainty. As we’ve seen, it’s caused by the
limits of resolution of our measurement tools. As it turns out though, this isn’t
the only type of uncertainty. Say that we put a dish onto a scale
for measuring mass. Into the dish, we put a powder
that’s very highly absorptive. Imagine further that we make a
measurement of the mass of this powder and find that it’s 571.3 grams. If we wait 24 hours and then come
back and make another measurement, we might find a very slightly different result
than before. This difference could be caused by
the powder on our scale absorbing water from the atmosphere.

Later, perhaps on an especially dry
day, we might come back, make a mass measurement, and find this result. Over time, the mass of this powder
that we’re measuring changes ever so slightly. This leads to an uncertainty in the
powder’s mass, but it’s not a measurement uncertainty. Instead, this is called random
uncertainty. We can quantify the random
uncertainty of a measurement by taking the maximum measured value, in this case that
would be 571.4 grams, and subtracting from it the minimum measured value, in this
case 571.1 grams. Taking that difference and then
dividing by two gives us the random uncertainty of this measurement.

We can see that the only way to
discover a random uncertainty is to make more than one measurement. When measured values of what is
purportedly the same quantity differ, we call the resulting uncertainty in that
value random. Notice that all three of these
measurements were made using a scale that could measure to the nearest one-tenth of
a gram. We can say then that the
measurement uncertainty of the scale, or its absolute uncertainty, is 0.05
grams. When we go to report these measured
values then, we could use this absolute uncertainty written here. Or we could report this uncertainty
as a percent. Written as an equation, percent
uncertainty equals absolute uncertainty divided by given measured value multiplied
by 100 percent.

So for example, say that we had a
measured value of 100 plus or minus five centimeters. Here, five centimeters is the
absolute uncertainty of this measurement. Therefore, to find the percent
uncertainty involved in this measurement, we would take the absolute uncertainty,
that’s five centimeters, divide it by the measured value of 100 centimeters, and
then multiply that fraction by 100 percent. In this fraction, we see the units
of centimeters cancel out, and five divided by 100 is 0.05. So the percent uncertainty of this
measurement is five percent. Whether we use absolute uncertainty
or percent uncertainty, either one is a valid way of indicating uncertainty in a
measurement.

Getting back to our three measured
values, let’s say that when we made this measurement here that the readout on our
scale looked like this. We see that there’s a leading zero
and then 571.4 grams. Even though the zero is being
displayed on the scale, there isn’t any meaning to that digit. It doesn’t give us any additional
information about the mass being measured. We say then that it is not a
significant digit or not a significant figure. Significant figures are related to
measurement precision. The more figures there are in a
measurement that are significant, the more precise that measurement is.

For example, say that we had a mass
scale that could only measure to the nearest 10 grams. If we measure the mass of our
powder on that scale, we would get a result of 570 grams. This measurement has three
significant figures while our other measurements have four. These other measurements then are
more precise determinations of the mass of the powder. Any time we have a measured
numerical value, we’ll want to keep significant figures and their rules in mind. Partly, this is because, as we’ve
seen, significant figures indicate measurement precision.

We can think back to our
measurement of our object using our centimeter- and millimeter-marked rulers. Using the centimeter ruler, we
measured our object’s length to be two centimeters. This measurement has just one
significant figure. This one, on the other hand, has
two. It gives us a more precise idea of
the true length of our object. Any measurement of any quantity is
likely to yield a numerical result. We might measure, say, an object’s
mass or a gas’s pressure or the speed of a particle and so on. Here, we’re thinking about any type
of measurement. A measured value always has at
least one significant figure.

Often times though, there may be
one or more figures in a measured value that are not significant. The key for locating insignificant
figures is to look for any leading or trailing zeros in a number. When a number is preceded by zeros,
when it has leading zeros, like this one does here, those zeros are always
insignificant. When a number has trailing zeros,
like this one has a trailing zero, that may or may not be insignificant. If a trailing zero follows the
decimal point as this one here does, then that figure is actually a significant
figure. This digit meaningfully indicates
the precision of this measurement.

On the other hand, say that we had
a measurement device that could measure lengths to the nearest 10 meters. If we measured some object’s length
to be 30 meters using that device, then this trailing zero here is not
significant. We would say that this measured
value only has one significant figure. However, if we were measuring
object length using a device that could measure to the nearest one meter, then in
that instance the zero would be significant. It depends on the resolution of our
measurement device. In general, for a measured value,
the only figures that may be insignificant are leading and trailing zeros. Leading zeros are always
insignificant and trailing zeros may be. Other than in these cases, all the
figures in a measured value are significant.

We would say then that this
measurement here has one, two, three, four significant figures. This measurement here has one, two
significant figures because we don’t count leading zeros. This third measurement has one,
two, three, four, five, six, seven, and eight significant figures. As we’ve seen, this trailing zero
is significant because it follows a decimal point. It’s really only in cases that have
no decimal point, like this one here, where whether or not trailing zeros are
significant is less clear.

Once we know how many significant
figures are in measured values, it’s important to know how to combine those values
mathematically, say, by adding or subtracting or multiplying or dividing them. By way of example, say that we
wanted to add these two numbers here. As we’ve seen, the top number has
one significant figure and the bottom number has two. The rule for combining values that
have different numbers of significant figures is to count the number of significant
figures in the number with the fewest significant figures, in this case, that’s one
significant figure here, and report our answer with that many significant
figures. So then if we add two to 1.6, we
get 3.6.

But notice that this answer has
one, two significant figures in it. We want to give our final answer
with just one significant figure. What we would do then is round this
result so it’s to the nearest one significant figure. In this way, we would say that two
centimeters plus 1.6 centimeters is four centimeters. This rule for combining values with
different numbers of significant figures is important because it prevents us from
giving our final answer with more precision than is justified. This rule forces us to follow the
limits in precision of our actual measurements. Let’s see this in action now using
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?

We have here a situation where
a runner travels a distance, we’ll call it 𝑑, of 115 meters in a time, we’ll
call it 𝑡, of 12 seconds. The runner’s average speed 𝑣
is given by the distance traveled divided by the time taken to travel that
distance. When we calculate the speed,
though, we need to be careful to take into account the difference in significant
figures in our distance and time. The distance of 115 meters has
one, two, three significant figures. We know this because we’re told
the distance is measured to the nearest meter, meaning that each whole meter is
significant. Similarly, the time is measured
to the nearest second, which means that this time of 12 seconds has one, two
significant figures.

Whenever we combine values that
have different numbers of significant figures like these two do here, our final
answer keeps only the smallest number of significant figures of any of the
values involved. In this case, that smallest
number is two, the number of significant figures in our time 𝑡. When we calculate this
fraction, the exact answer we get is 9.583 repeating meters per second. But we recall that we’ll only
keep one, two significant figures in this final answer. All nonzero digits are
significant. So that means this is a
significant figure, and so is this. To round to two significant
figures then, we’ll look at this digit, which we see is greater than or equal to
five. And that means we will round
up. To two significant figures, the
runner’s average speed is 9.6 meters per second.

Let’s now review what we’ve learned
in this lesson. In this video, we learned that the
resolution of a measurement instrument is the fineness to which the instrument can
be read. An instrument with higher
resolution is said to deliver more precise measurements. We learned further that absolute
uncertainty is the interval over which a measured value is likely to fall. We learned also that there’s
something called percent uncertainty related to absolute uncertainty by this
equation.

And finally, we’ve learned about
significant figures — that these are digits that carry meaning contributing to
measurement precision. We learned what figures are and are
not significant in a measured value. And we also learned how to combine
values with different numbers of significant figures in carrying out
calculations. This is a summary of measurement
uncertainty and resolution.