This video is about absolute, fractional, and percentage uncertainty.
We will briefly return to absolute uncertainty,
the topic that we covered in more depth in the previous video,
and then we will see a few examples of calculating fractional and percentage uncertainty.
Due to the measuring tools that we use and the way we carry out measurements,
we never actually get an exact value when we measure something.
Instead, we get a range in which our measured value lies.
We use absolute uncertainty to show this range when we record our data values.
So when we say that the mass of an apple is 278.6 plus minus 0.1 gram,
where 0.1 gram is the absolute uncertainty,
we mean that we don't know exactly the mass of the apple,
but it lies somewhere between 278.5 and 278.7 grams.
Similarly, when we measure the temperature of the C and get 6.0 plus minus 0.5 degrees Celsius,
we mean that the water temperature is somewhere between 5.5 and 6.5 degrees Celsius,
and probably conclude that we should wear a wet suit when we go for a swim.
Finally, when the time taken by an object to fall from a table is 0.50 plus minus 0.25 seconds,
the estimated value of this time is somewhere between 0.25 and 0.75 seconds.
Next, let's discuss fractional and percentage uncertainty.
Let's say we record our measured value as x plus data x, where data x is the absolute uncertainty.
Then, the fractional uncertainty is expressed as the absolute uncertainty divided by the measured quantity,
in other words, data x over x. To find a percentage uncertainty,
we just have to convert our fractional uncertainty into a percentage, so multiply by 100%.
Let's see two examples. In the first example, we have a kitten whose mass is given to be 400
plus minus 80 grams, and we have to find the fractional and the percentage uncertainty in the
mess of this kitten. The measured mass is 400 grams, and the absolute uncertainty in the mass,
data m is 80 grams. Based on what we discussed above, the fractional uncertainty is data m over m.
This is equal to 80 over 400, and after simplifying, so canceling the zeros and dividing 8 by 40,
we get 1 over 5. This is the fractional uncertainty in the mess of the kitten.
To find the percentage uncertainty, we simply have to multiply the fractional uncertainty by 100%.
So we get 1 over 5 times 100%, and since 1 over 5 is 0.2, this is equal to 20%.
This is the percentage uncertainty in the mess of the kitten.
The uncertainty, in example, is given a bit differently. Sometimes, instead of the absolute uncertainty,
the percentage uncertainty is given together with the measured value.
So here we are given that the time it takes for a toy car to roll down a slope is 15 seconds
plus minus 10%. So we are given the percentage uncertainty, and we have to find the fractional
and the absolute uncertainty. Let's start with the fractional uncertainty in the time,
which is delta t over t. To find this, we simply have to convert 10% into a fraction or decimal.
So we get 10%, divided by 100%, which is equal to 0.1 or 1 over 10. This is the fractional uncertainty.
It is okay to leave the fractional uncertainty in a decimal form.
Finally, let's find the absolute uncertainty, which is delta t. To do this, we have to multiply the
measured value, so 15 seconds, by the fractional uncertainty, so 0.1. Another way to think about it
is that here we have to find 10% of 15 seconds. This is equal to 1.5 seconds, and this is the
absolute uncertainty here. Let's summarize what we have discussed in this video. We have seen that
measurement in sciences does not give us an exact value, so we use absolute uncertainty to
express the range where measured value lies. We also learned that when a quantity is given as x
plus minus delta x, the absolute uncertainty in this quantity is plus minus delta x. The fractional
uncertainty is delta x over x, and the percentage uncertainty is delta x over x multiplied by 100%.
This wraps up our discussion about absolute, fractional, and percentage uncertainty.
In the next video, we will learn about propagating uncertainties.