### Video Transcript

For this video, our topic is
measurement uncertainty and resolution.

Whenever a measurement is made, for
example, in a sporting event, like we see on screen here, these topics of resolution
and uncertainty come into play. In the case of this track event,
apparently the measurements of the runners’ times are fairly uncertain. And because of that, we can’t say
for certain just who finished first, second, third, and so on. The first step to developing
instruments that are less uncertain is understanding what these terms resolution and
uncertainty refer to. Let’s start there, and we’ll begin
by defining this term “resolution.”

When we hear this term resolution,
we may think of it in terms of pictures or images. A picture with low resolution may
look very blocky. Edges in the image don’t look
smooth, but rather look very chunky or made of blocks. In contrast, a high-resolution
image has smooth-looking edges. When looking at it, our eye isn’t
able to detect the squares or rectangles we see in the low-resolution version. Indeed, thinking of resolution this
way does help us understand the definition of this term. Resolution is the fineness or the
specificity to which an instrument can be read. Here’s an example of that.

Say that we have two different
rulers. One ruler, the one on top, is
marked out with centimeter units. So here at the start is zero
centimeters. Then this is one centimeter, then
two centimeters. While the other ruler, the second
one, is marked out in units of millimeters. We can see the centimeter markings
on this one as well. But in between those markings, we
have these small hash marks which indicate millimeters. Let’s say further that using these
two rulers, we wanted to measure the length of some object.

Using the top ruler, the one marked
out to centimeters, we might look at the end of our object and say that it looks
like it’s closer to two centimeters than one centimeter, in which case the reading
for the length of this object would be recorded as two centimeters. But then, if we measure the same
object using our millimeter-marked-out ruler, we see the object’s length is one
centimeter plus one, two, three, four, five, six millimeters. Using our second ruler then, the
length of the object that we would record could be 1.6 centimeters.

From this example, we can see that
the second ruler — the one marked out in millimeters — has a greater degree of
fineness to which it can be read. That is, this ruler — the one
marked in millimeters — more finely marks out units of length than the one marked
out in centimeters. We would say then that this second
ruler enables more highly resolved measurements than the first one. And note that when we talk about an
instrument by which we make measurements, that could be measuring any physical
quantity. It might be measuring an object’s
mass or an amount of time elapsed. It could be measuring electric
current or gravitational attraction. Any measurement instrument will
have some degree of resolution, the fineness, to which the instrument can be
read.

Now this idea of resolution in
measurements leads us into measurement uncertainty. Let’s again consider the process of
measuring the length of this object with our two different rulers. In particular, let’s remember what
we did when we measured out the length of this object on our centimeter ruler. In determining the object’s length
from that ruler, we basically divided the distance between one and two centimeters
in half. That halfway point is roughly
here. And then we said, because our
object goes beyond that halfway point, we’ll report that it has a length of two
centimeters. That’s the whole number or integer
value of centimeters that its length is nearest to. But notice how much uncertainty in
this measured value that process creates.

What we’ve said is that if the
length of our object falls anywhere between this halfway point and two centimeters,
then we’ll record its length as though it were two centimeters. But we see that, really, the
object’s actual length could be as short as one and a half centimeters. Or imagine that it was even longer
than two centimeters. Then the objects length could go up
to 2.5 centimeters, and we would still record its length this way. Here’s what we’re finding then. The length of our object could fall
anywhere between 1.5 centimeters and 2.5 centimeters. And we would record its length as
two centimeters. We can see then just how uncertain
this measurement value of two centimeters is. It could be off by as much as one
half of a centimeter in either direction.

At this point, we can define just
what this term “uncertainty” means when it comes to making measurements. Uncertainty is the interval over
which a measured value is likely to fall. In the case of our object whose
length is being measured on the centimeter ruler, we can see that that interval is
anywhere from 1.5 centimeters up to 2.5 centimeters. That’s the uncertainty of this
measurement. Now we’ve been looking mostly at
the centimeter-marked ruler. But let’s look at the
millimeter-marked ruler. And we’ll see that this reading,
too, has an uncertainty in it. When we measure the length of our
object on this millimeter ruler, that length fell almost exactly at 1.6
centimeters. That made it easier to write down
1.6 centimeters as our measured length. But notice that we would’ve
recorded the same length even if our object had been a bit longer or a bit
shorter. Basically, so long as the measured
length of our object fell within one-half of one millimeter of this point on our
ruler, then we would’ve recorded its length as 1.6 centimeters or 16
millimeters.

So we see, even in this case,
there’s a bit of uncertainty involved because our object could have been a bit
longer than 1.6 centimeters or a bit shorter and we would’ve recorded this length as
1.6 anyway. So when we talk about the interval
over which a measured value is likely to fall, in the case of our millimeter ruler,
that interval is a half millimeter beyond 16 millimeters and one-half millimeter
below it. In other words, we have a total
interval or measurement uncertainty of one millimeter. Now what we’re seeing here, in
terms of the uncertainty of the measurements we’re making of this length with our
two rulers, would be true regardless of how finally resolved our measurement stick
is. There is always a resolution limit
to a measurement instrument. And therefore, every measurement
ever made has some amount of uncertainty in it. In other words, because instrument
resolution is never perfect, measurement uncertainty always exists, even if that
uncertainty is very small.

Now if we get back to the two
recorded lengths of our object, two centimeters for our centimeter-marked ruler and
1.6 centimeters for our millimeter-marked ruler, just by looking at these two
numbers, even if we didn’t know where they came from. We can see that there’s more useful
information in this value of 1.6 centimeters than in the value of two
centimeters. The way we talk about that is we
say that there are more significant figures in this number — 1.6 centimeters — than
there are in this number. And in order to talk a bit about
significant figures, let’s clear some space on screen.

Considering significant figures in
general, here is how we can talk about them. Significant figures, also sometimes
called significant digits, are digits that carry meaning contributing to measurement
resolution. Think once more about our two
measured lengths of our object: two centimeters with our one ruler and 1.6
centimeters with the other. If we were to count up the number
of significant figures in each of these numbers, the two-centimeter figure has one
significant figure and the 1.6-centimeter number has two. And this greater number of
significant figures here shows us that the measurement method we used to get that
result was more highly resolved or had better resolution than the method we used to
get this result. And we can remember that’s true
from thinking of our two rulers, marked down in centimeters and millimeters,
respectively.

And we could imagine this trend
continuing. Say that we had a third ruler that
was marked out to tenths of millimeters. We could say then that the
resolution of our first ruler was in centimeters. The resolution of the second one
was in millimeters. And we’re now imagining a third
ruler with a resolution of tenths of millimeters. If we were to measure the length of
our object with this third, more highly resolved ruler, the answer we get might be
something like 1.62 centimeters. This value then would have one,
two, three significant figures in it, with the greater number of significant figures
indicating a more highly resolved measurement instrument. We can see then that the better the
resolution of our instrument is, the more significant figures will have in our
measured value and therefore the less uncertain that measured value will be.

Now there are a few rules for
working with significant figures that are worthwhile to know. The first rule has to do simply
with what is and what is not a significant figure. And we can abbreviate that s.f. In general, we can think of any
number as a significant figure unless it’s a leading zero. For example, in the number 014, the
zero is a leading zero and therefore not significant. And we also don’t consider
significant trailing zeros when they’re used specifically as placeholders. For example, say we measured a
distance in kilometers to the nearest kilometer and it turned out to be a distance
of five kilometers. If we wrote that out in units of
meters, then we would write 5000 meters. That’s five kilometers. But we can see that these three
zeros are placeholders. We don’t actually know the value of
this distance that precisely. So then, these zeros are not
significant. In this number then, there’s just
one significant figure — that’s the five.

Now, sometimes in calculations,
we’ll be working with values that have different numbers of significant figures. For example, let’s say that we
measured some object to travel a distance of 14 meters in a time of 1.36
seconds. We can see that this first value,
the 14 meters, has two significant figures. But the time has three. Now say we wanted to combine these
numbers to calculate the speed of this object: the distance it travels divided by
the time it took to travel that distance. In doing this, we would divide the
distance traveled by the time elapsed. But then when we have to write down
our final answer, we need to make a choice. How many significant figures do we
use in that answer? This second rule tells us how to
figure that out.

This rule says that when we combine
values with different numbers of significant figures in them, in our answer, we use
the least number from that set of values. So in this example, we have one
value with three significant figures and one with two. So, following this rule, we would
report our answer to two significant figures. We would write our answer as 10
meters per second, where in this case, the zero is not a place holder, but is in
fact significant.

Now just one last rule with working
with significant figures. And that is this: when we think of
the significant figures in a value and the number of decimal places that value has,
in general, these two numbers are not the same. Let’s consider this random number,
37.456. When we count the number of decimal
places, we see there are one, two, three in this number. But because all of the digits in
the number are not zero, that means all of them — all five — are significant. So then, there are five significant
digits in this number, but only three decimal places.

Or consider a different number. Consider this one: 0.32000. We can see that there are one, two,
three, four, five decimal places in this number. When it comes to significant
figures, let’s say that these last three zeros, these trailing zeros, are
placeholder values that we don’t actually know the number to that precision. In that case, based on rule number
one of what is and is not a significant figure, we would say that these three zeros
are not significant and neither is this leading zero here. This number then just has two
significant figures: the three and the two. So in this case, while we only have
two significant figures, we have five decimal places. Once again, these two values are
not the same.

Related to this, let’s consider how
we might round these two different values. Let’s say we wanted to round this
first value to three significant figures and we wanted to round this second value to
one significant figure. The way we would go about doing
that is we would count off the number of significant figures we want to keep in our
answer. So in the case of this first value,
we would count one, two, three significant figures. And then we look at the next digit
in our number significant or not after this. If that digit is equal to five or
higher, then we round up our last significant figure. In this example, that digit is
equal to five and therefore we do round up. So to three significant figures, we
would report this value as 37.5.

Then for our second value that we
want to round to one significant figure, we count off the first significant
figure. This is the one we’ll keep. And then, we look at the next digit
following on whether it’s significant or not. Once more, if that digit is five or
higher, we round our three up. And if it’s lower than five, we
keep the three the same. Since two is less than five, we
won’t change the three. We won’t round it up to four. So to one significant figure, this
number rounds to 0.3. Let’s get a bit more practice now
with these ideas through 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?

Alright, so in this example, we
have a runner running at distance, which is measured to the nearest meter, to be 115
meters. Along with this distance, the time
it takes the runner to cover that distance is also measured. And that time — we can call it 𝑡 —
is measured to the nearest second to be 12 seconds. The question is what is the
runner’s average speed. And importantly, we want to report
this average speed to the appropriate number of significant figures. Along these lines, we can begin by
counting the number of significant figures in each of these two numbers, the time 𝑡
and the distance 𝑑. The measure time has one, two
significant figures and the measured distance has one, two, three of them.

At this point, let’s recall that
the average speed an object has — we’ll call it 𝑠 — is equal to the distance the
object travels divided by the time it takes to travel that distance. We can see then that our runners
average speed is going to equal the distance, 115 meters, divided by the time, 12
seconds. But the question is, how should we
report our answer? After all, our distance has three
significant figures, but the time just has two. Should our final answer be in terms
of three or two significant figures? To answer this, we can recall a
rule for combining numbers that have different numbers of significant figures in
them.

That rule goes like this: when
combining values with different numbers of significant figures, answer using the
smallest number of significant figures. Applying this rule to our
situation, we see that our two values have different numbers of significant figures:
three and two, respectively. This rule says that as we combine
them, like we are in calculating average running speed, that we should keep the
least number or the smallest number of significant figures any one of them
possesses. That smallest number is two. And this means we’ll round our
final answer to be given in terms of two significant figures.

Now, when we calculate this
fraction 115 meters divided by 12 seconds, we get this result of 9.583 repeating
meters per second. We know this isn’t our final answer
because we’ll round it. And in particular, we’ll round it
to two significant figures. To do that, let’s count off two
significant figures. Starting at the front of the value,
nine and five are the first two. And then we look at the next digit
in our number, eight. Since eight is greater than or
equal to five, that means we’ll go to our previous number, the five, and we’ll round
that up one. The five rounds up to a six. And our final answer to two
significant figures is 9.6 meters per second. That’s the average running speed
reported to two significant figures.

Let’s take a moment now to
summarize what we’ve learned about measurement uncertainty and resolution.

Starting off, we saw that
measurement resolution is the fineness to which a measuring instrument can be
read. We then learned that measurement
uncertainty is the interval over which a measured value is likely to fall. We also learned that because no
instrument has perfect resolution, every measured value has some degree, however
small, of uncertainty. Further, we learned that
significant figures — sometimes abbreviated s.f. — are digits that carry meaning
contributing to measurement resolution. Related to this, we learned a rule
for determining what is and what is not a significant figure, a rule for how to
combine different numbers of significant figures. We saw that in general the number
of significant figures in a value and the decimal places in that value are not the
same. And lastly, we saw how to round a
value to a particular number of significant figures.