### Video Transcript

In this video, our topic is
combining uncertainties. In this sketch, we see a person
checking in for their flight at the airport. But in response to the gate agent’s
questions, this person doesn’t know where they’re going. They don’t know when they’re
leaving. And they don’t have their ID with
them. All these factors combine to create
a greater overall uncertainty about who this person is and where they’re going
to.

As we’ll see, something similar
takes place when we combine measured values. And in this lesson, we’ll learn
rules for doing that. As we get started, we can recall
that all measured values have some uncertainty about them. This is because any device we use
to make a measurement will not be infinitely precise. For example, let’s say that we
measure the height of this person here. We might use a tape measure to do
that. And even if we were very careful in
the measurement process, lining up the top of the person’s head with a tape
measure. Still, if we looked closely at the
tape measure, we would see that the markings on it limit the precision to which we
can report this person’s height.

Let’s say that each one of these
hash marks on our ruler indicated a difference in length of one centimeter. In looking at our dotted line
representing the height of the top of the person’s head, we could go one decimal
place beyond that precision to estimate this person’s height to the nearest tenth of
a centimeter. Say that when we do this, we come
up with a value of 165.4 centimeters for this person’s height. But this number is a bit
uncertain. The true height of the person could
be a little bit higher or a little bit lower than this value. Specifically, it could be one-tenth
of a centimeter above or one-tenth of a centimeter below the value we’ve
reported.

It’s important to include this
overall uncertainty in our reported value. We would say this person’s measured
height is 165.4 plus or minus 0.1 centimeters. We see then that uncertainties in
measurements come from limits in the precision of our measurement devices. Since no measurement device is
infinitely precise, there will always be some uncertainty in a measured value. Now, let’s say that using the same
ruler, we measure the height of a second person, and that this person’s height comes
out to be 148.6 plus or minus 0.1 centimeters. At this point, we have these two
measured values and they’re separate. They separately indicate the height
of these respective people.

But what if we wanted to combine
these heights? What if, through an impressive feat
of balance and strength, the second person was able to stand on the head of the
first? In that case, to get the combined
height of these two people, we’ll need to combine these two measured values. Now, if these were simply height
values with no uncertainties, that would be straightforward enough. But let’s think for a moment about
how we do this, including the uncertainties in our measured values.

We know that the measured height of
the first person has an uncertainty of one-tenth of a centimeter. This means that their true hight
could be a tenth of a centimeter higher or lower than this reported value. And then the same thing is true for
the height of the second person. Their height also could be a tenth
of a centimeter higher or lower than 148.6. Knowing this, we can write out
maximum and minimum possible height values for these two people’s heights. Now, if we were to add together
these two maximum possible heights, then we would effectively be adding the heights
of these two people along with 0.1 centimeters of additional height for each
one.

Or on the other hand, if we were to
consider adding together the minimum possible heights of these two people, then that
value would be equal to the reported heights minus 0.1 centimeters again for each
person. What we’re seeing then is that, in
combining the heights of these two people, not only do we add together the measured
values of their heights, but we also add together the uncertainties in those
heights. So then if we take the combined
heights of these two people and say we call that height 𝑇 sub 𝐻, that total height
will be equal to the height of the first person plus the height of the second
person. And we find that total height by
adding together the measured values of the two people as well as those measured
value uncertainties.

Doing this gives us a result of
314.0 plus or minus 0.2 centimetres. What we’re seeing here is a
specific example that can be expanded into a general rule. We can clear a bit of space and
then write out that general rule using words. We can say that when adding or
subtracting measured values, the uncertainties in the values are added together. Then, here’s how we can write this
mathematically. And we need to be a bit careful to
understand this notation.

If we have a value — we’ll call it
𝑣 one, and that’s equal to 𝑎 plus or minus the uncertainty in 𝑎 — and then if we
have a second measured value — and that’s equal to 𝑏 plus or minus the uncertainty
in 𝑏. Then, if we were either to add 𝑣
one and 𝑣 two together or subtract 𝑣 two from 𝑣 one, that would lead to a result
of 𝑎 plus or minus 𝑏, depending on whether we’re adding or subtracting between 𝑣
one and 𝑣 two. And then that value has an
uncertainty plus or minus of the uncertainty of 𝑎 plus the uncertainty of 𝑏. And by the way, this rule of adding
uncertainties together when we add or subtract measured values applies even if we
have more than two measured values.

For example, if we had three or
even four people standing on one another’s heads, to calculate the total height, we
would add together the heights of each person plus the uncertainties in each of
those heights. Now, what we’ve covered so far is
adding and subtracting measured values. But we know we can also multiply
and divide.

For example, let’s say that we had
an empty room and we wanted to measure the volume of this room. So we make measurements of the
width, the height, and the depth of the room. And say that we come up with these
values. In each case, we have an
uncertainty of one-tenth of a meter based on the way we’ve measured these
distances. Now, we know that to find the
volume of the room we’ll need to multiply the height and width and depth
together. But then, as we do that, is finding
the total uncertainty in this volume as simple as multiplying together the
uncertainties for each dimension? As it turns out, no. If that were true, then our
uncertainty in volume would be 0.1 meters times itself three times. This is 0.1 meters quantity cubed,
which comes out to 0.001 cubic meters.

But this is much too small an
uncertainty. We can’t justify a number this
precise using the measured dimensions of the room. So this approach is not correct for
calculating the uncertainty in our volume. We can’t simply multiply the
individual uncertainties together. Instead, before we multiply the
room dimensions together, we take one extra step to modify the uncertainties in our
measured values. What we do is we convert them to
percents. That is, we calculate what percent
of 3.0 meters 0.1 meters is and, likewise, we calculate what percent of 4.7 0.1 is
and similarly with our last dimension.

When we do this, we find that 0.1
is 1.8 percent of 5.5, it’s 2.1 percent of 4.7, and it’s 3.3 percent of 3.0. Note that we haven’t made any
numerical changes to these measured values; we’re just expressing them in a
different equivalent way. But now that we’ve done that, we’re
ready to calculate the volume of this room by multiplying together 3.0 meters with
4.7 meters with 5.5 meters, but then adding together these percentage uncertainties
for each measured value. So here’s how that works. The volume of our room is 3.0
meters times 4.7 meters times 5.5 meters plus or minus 3.3 percent plus 2.1 percent
plus 1.8 percent. And note that we’re not multiplying
these percentages together, but rather we’re adding them.

Keeping two significant figures in
our calculations, all this comes out to 78 cubic meters plus or minus 7.2 percent of
that volume. Now, if we wanted to report our
final answer with an uncertainty written in cubic meters, rather than as a percent,
we could do that by calculating 7.2 percent of 78 cubic meters. When we do that, we get a final
result of 78 plus or minus 5.6 meters cubed. That’s the properly calculated
volume of this room including the uncertainties in each dimension. Just like before, we’re looking at
a specific example of a general rule. So let’s write out that rule that
applies to multiplying and dividing measured values with their uncertainties.

Whenever we multiply or divide
measured values, their percent uncertainties are added together to yield the total
uncertainty. We can write this out
mathematically like this. Say that we have three values, 𝑎
with its own uncertainty, 𝑏 which has its own uncertainty, and 𝑐. If 𝑐 is equal to the product of 𝑎
and 𝑏, then the uncertainty in 𝑐, that product, divided by 𝑐 is equal to the
uncertainty in 𝑎 divided by 𝑎 plus the uncertainty in 𝑏 divided by 𝑏. Now, this equation may seem a bit
confusing because were we talking about percents and adding those together. But actually, this equation is
closer to showing us percentages than we might realize.

If we were to multiply both sides
of the equation by 100 percent, then that would make each one of the terms in this
equation a percentage uncertainty. First, the percent uncertainty of
𝑐, then of 𝑎, and then of 𝑏. But since this factor of 100
percent is common to all the terms, we can cancel it out. And that leaves us with this
simplified expression. It’s worth noting that even though
we’ve said this expression applies for multiplying two values together to yield the
third, it’s also accurate if we were to do a division instead. For example, if we were to divide
𝑎 by 𝑏, that wouldn’t change anything about this equation for calculating the
total resulting uncertainty.

We’ve now talked about calculating
total uncertainty for adding, subtracting, multiplying, and dividing. There’s one special case, though,
that we can take a moment to look at. That special case is when we’re
raising a number to some integer power. We can see an example of this if we
imagine that all the dimensions of our room are the same. They’re all 3.0 meters, which means
that all the room dimensions have the same uncertainty, 3.3 percent. Now, according to this rule and to
our experience, we could compute the total uncertainty in the volume of this room by
adding together 3.3 percent with 3.3 percent with 3.3 percent which gives us 9.9
percent.

We can see, though, that this is
mathematically equivalent to taking that percent uncertainty, 3.3 percent, and
multiplying it by three. And three has a special
significance when we’re talking about calculating the volume of some object. And that’s because if we have a
cubic object, like we do in this example, then the object’s volume is equal to the
length of each side cubed. Now, it turns out that this number
here and this number here, being the same, is not a coincidence. Once more, this is a specific
example of a general rule in calculating uncertainties when we’re raising a number
to an integer power. We can write that out this way.

We can say that when raising a
measured value to an integer power, the total uncertainty equals that power times
the percent uncertainty of the measure value. That’s a mouthful. But if we write it in symbols, it
just means that if we’re calculating the integer power of some value, we’ll call
that value 𝑥, and the integer power 𝑏. Then the uncertainty in the result
of that calculation, the uncertainty in 𝑦, can be found by using this equation. And we see that this equation
involves multiplying the fractional uncertainty in 𝑥 by that power 𝑏.

As we noted, this rule is really a
specific application of the rule we discovered earlier for multiplying and
dividing. When we are multiplying together
identical measured values, either approach will give the right answer. It’s just that one of them, for
example, has us adding the percent uncertainty together as many times as it appears,
while the other approach, the one we’ve just learned, takes that percentage
uncertainty and multiplies it by the number of times that uncertainty is there. Either way, we come out with the
same result in the end. Now that we’ve learned various ways
of combining uncertainties, let’s get some practice with these ideas through an
example.

Two resistors have a resistances of
20 plus or minus 0.1 ohms and 80 plus or minus 0.2 ohms. If the two resistors were placed in
series, what would the uncertainty of the two resistors together be?

Okay, so in this example, we have
two resistors. We’ll say that this is our first
and then this is our second. And we’re told these two resistors
are placed in series with one another. For both, we’re given the value of
their resistances. And we see that those resistances
include an uncertainty, 0.1 ohms in one case and 0.2 ohms in the other. We can recall that when resistors
are placed in series, like they are here, their resistances add together for a total
combined resistance value. If these resistance values were
stated without uncertainties, it would be straightforward enough to add 20 ohms to
80 ohms to get a total resistance of 100 ohms. But in this case, we do have these
uncertainties that we’ll need to consider as well.

The way to do this is to recall our
rule for combining uncertainties, specifically when we’re adding two values
together. Let’s say that we have one
value. We’ll call it 𝑣 sub one. And this is equal to 𝑎 plus or
minus the uncertainty in 𝑎. And similarly, we have a second
value, 𝑣 two, which is equal to 𝑏 plus or minus the uncertainty in 𝑏. Now, if we were to add 𝑣 one to 𝑣
two, then the way we would do that is we would add 𝑎 and 𝑏 and then, with the
uncertainties, add those together as well. We can apply this rule to our
particular scenario of adding the values of these two resistors.

If we were to solve for the total
resistance, we can call it capital 𝑅, of these two resistors together, then by our
rule, that would be equal to 20 plus 80 plus or minus 0.1 plus 0.2 ohms or, in other
words, 100 plus or minus 0.3 ohms. Now, it’s not the overall
resistance 𝑅 that we want to solve for, but rather the total uncertainty of these
two resistors together. In solving for 𝑅, though, we have
found that total uncertainty. We see that it’s equal to the sum
of the individual uncertainties. That total is 0.3 ohms.

Let’s summarize now what we’ve
learned about combining uncertainties. We first learned that when measured
values, and all measured values have uncertainties, are added together or subtracted
from one another, then their uncertainties add. Mathematically, if 𝑐 is equal to
𝑎 plus or minus 𝑏, where 𝑎 and 𝑏 are measured values with uncertainties, then
the uncertainty in that sum or difference is equal to the uncertainty in 𝑎 plus the
uncertainty in 𝑏. This was the first rule we saw for
combining uncertainties.

The second rule we discovered was
that when measured values are multiplied or divided, their percent uncertainties add
together. Using symbols, if 𝑐 is equal to 𝑎
times 𝑏 or 𝑐 is equal to 𝑎 divided by 𝑏, where 𝑎 and 𝑏 are measured values,
then the uncertainty in 𝑐 divided by 𝑐 is equal to the uncertainty in 𝑎 divided
by 𝑎 plus the uncertainty in 𝑏 divided by 𝑏. And recall that these fractional
uncertainties can quickly be turned into percent uncertainties. We do that by multiplying both
sides of the equation by 100 percent.

Lastly, we learned a specific
application of this multiplication rule. When a measured value, we’ll call
it 𝑎, is raised to an integer power, we’ll call that 𝑛, then the fractional
uncertainty of the result is equal to 𝑛 times the fractional uncertainty of 𝑎. This is a summary of combining
uncertainties.