This lecture is about the definition and difference between absolute, fractional,
and percent uncertainty. Absolute uncertainty is uncertainty expressed as a number independent of
the original number. The symbol we use for absolute uncertainty is the symbol delta,
and then the letter variable for what we're measuring. Fractional uncertainty is uncertainty
expressed as a fraction of the original number, is delta and then the variable over the original
variable. I'll explain why that is in a moment. And percent uncertainty is uncertainty expressed
as a percentage of the original number, and you can see I have that symbol, an equation written
there I'll give you a visual to understand the difference between each of these. Let's say that
we have 4 plus or minus 1. What this would look like on the number line is we have the number 4,
but the actual value of this measurement could be as high as 4 minus 1 or 4 plus 1,
so the red line indicates the value of the measurement and the blue line indicates the value
of the uncertainty. I can understand how big that uncertainty is without referring to the 4. I can
understand that no matter how big the red number is the uncertainty itself is going to be plus or
minus 1. It'll go up 1 and down 1. So because that number is independent of the measurement number,
that uncertainty is expressed as an absolute uncertainty. I'm going to write the same number
and uncertainty but with fractional uncertainty. I need to understand what fraction the uncertainty
is of the number, and I can see here that 1/4 of my number is uncertain. 1/4 of my total
measurement I'm uncertain about. It's not that full range from 3 to 5, it's not 2 over 4. It's
half of the range because that's how we usually express uncertainty as half of the full range,
so 1/4 of my measurement is uncertain, so that fraction expressed as a decimal is 0.25,
so 0.25 times my measurement is uncertain. So this is how I would express that uncertainty with
fractional uncertainty. This is the uncertainty expressed as a fraction of my original number. And
percent uncertainty is uncertainty expressed as a percentage of the original number, and if 1/4 of
my measurement is uncertain, that means 25% of my measurement is uncertain. You can see that these
expressions all express the same uncertainty. They're all referring to that same red and blue
line. They're just different ways of writing that same uncertainty. So absolute, fractional,
and percent uncertainty are three different ways of recording the same fundamental thing, and you
can see now why the equations work in the way that they do. Because the fractional uncertainty is
the absolute uncertainty delta-A divided by the original number A (in my case it was 1 over 4)
and the percent uncertainty is that fractional uncertainty multiplied by a hundred percent,
25 percent is 0.25 times a hundred percent, these equations will always work to convert from one
uncertainty to another. In comparison to absolute uncertainty, we call fractional and percent
uncertainty relative uncertainty, because the value of the uncertainty is relative to the value
of the measurement, so with absolute uncertainty even if I don't know what the measurement is I
can still understand the exact size of the uncertainty. Here it's just plus or minus 1.
That's absolute on its own, so I can understand it independently of any other number, but for
fractional and percent uncertainty if I can't see the original measurement I don't actually know how
large that is, so if I'm dealing with 25 percent of 1 that's going to be a very different size of
uncertainty compared to 25 percent of a hundred, so the actual physical size of the uncertainty
is relative to the original measurement in a way that it's not with absolute uncertainty, so we say
that fractional and percent uncertainties are both examples of relative uncertainty for that
reason. These are instructions for converting between each type of uncertainty. When you're
going from absolute to fractional you divide the absolute uncertainty by the number. When you're
going from fractional to percent uncertainty you multiply the fractional uncertainty by 100%,
and in the other direction if we're going from a percent to a fractional uncertainty you divide by
a hundred percent, and from fractional to absolute uncertainty means you multiply the fraction by
the number. The unit of measurement is written in a different place for absolute, fractional,
and percent uncertainty. For absolute uncertainty it's written to the far right of the number,
because both of those numbers are measurements in that unit. They're both independent of each other
and absolute so they're both measuring amounts of that unit, whereas with fractional and percent
uncertainty we put the unit only next to the measurement value, not next to the uncertainty,
because that uncertainty is not actually measured in the same units as the original number. It's
some fraction or percentage of that original number. So we're not using the same units for
it. As an example, if we were measuring something in kilograms, this is how we would write the
absolute, fractional, and percent uncertainty with the unit in the correct place, and just looking at
the bottom right it wouldn't really make sense to say oh this is plus or minus 25% kilograms,
like that statement on its own doesn't really make sense, so we say instead that it's 4 kilograms
plus or minus 25% of that value. The placement of the unit can tell us whether the uncertainty
is absolute or fractional. You can definitely have decimal uncertainties that are absolute. So a way
of understanding whether those uncertainties are absolute or fractional is to look at where the
unit is in the measurement. If it's an absolute uncertainty, the unit occurs after the number and
uncertainty, but if it's a fractional uncertainty the unit is only written after the number,
not after the uncertainty. There are rules for decimal places in absolute uncertainty,
but not really rules for fractional and percent uncertainty. If the absolute uncertainty goes
to more decimal places than the number the uncertainty is, round it to the same number
of decimal places as that number, and if the number goes to more decimal places than the
absolute uncertainty, the number is rounded to the same number of decimal places as the uncertainty.
So the lowest decimal place of both the number and the uncertainty always have to match, and
you always rounded up the value that goes to the lowest decimal place. There are no clear rules for
recording significant figures or decimal places in a fractional and percent uncertainty. The amount
of significant figures your record will usually be told to you in the problem, and if you're not
told about them it's okay to write the number of sig figs that you would like, because the problem
isn't being clear enough on its own. The rest of the video will just be example problems where I
convert between different types of uncertainty using the rules that I introduced earlier. I'm
going to start by converting each absolute uncertainty to fractional uncertainty, so
to do that I divide each absolute uncertainty by its original number and rewrite it as a decimal,
and I can now rewrite these values with fractional uncertainties, so the numbers stay the same,
the unit is moved next to the number, and I'm now writing in the fractional instead of the absolute
uncertainties. Going to the right to convert each fractional uncertainty to absolute uncertainty,
I multiply the fraction by the original number, and when I do that I get the absolute uncertainty
for each number, and you'll notice that I've rounded the uncertainties correctly to go to the
same decimal place as the original numbers, and as a policy we usually round absolute uncertainty to
just one sig fig, so the number of places in the uncertainty you must match the number of decimal
places in the original number. Going down and to the left I can convert each fractional uncertainty
to percent uncertainty. This is one of the easiest ones. You just have to multiply the fractional
uncertainty by a hundred percent. So this is how you would write those fractional uncertainties
as percent uncertainties, and then I can convert each percent uncertainty to fractional uncertainty
by just dividing by a hundred percent. That's the rule for conversion there. I can now take 2 steps
and convert each absolute uncertainty to percent uncertainty, so to do that I'll convert from
absolute to fractional and then from fractional to percent, so I divide the absolute uncertainty by
the original number and then multiply by a hundred percent, and this is what I get for those values,
and again I move the unit to the correct spot in each number. Finally I'll convert each percent
uncertainty to absolute uncertainty, so to do that I can divide by a hundred percent and then
multiply by the original number, and when I do that these are the absolute uncertainties
that I get. And notice again that I'm not changing the actual value of the uncertainty,
I'm just writing it in different ways in each problem. So in each problem those answers are
the same values as the others, it's just that I'm changing how I express them between absolute
fractional and percent uncertainties, but the underlying value of each number is the same.
I'll close with some rounding problems where I just round each number to the correct decimal
place and then round each uncertainty to the correct decimal place. So all I need to do is
round each number to the same decimal place as its uncertainty. When I do that this is what I
get for each number, and on the right because the uncertainties go to lower decimal places
than the original numbers I have to round those up to the lowest decimal point of each number,
and when I do that this is what I get for each uncertainty so that's what you have to know about
the definition of absolute fractional and percent uncertainty and how to convert between them.