This video is about propagating uncertainties.
We will learn how to propagate uncertainties when we add and subtract, multiply and divide, or raise a quantity to a power.
Let's start with addition and subtraction.
This is the formula for propagating uncertainties when we add or subtract.
Let's see two examples.
In the first example, we are given two quantities A and B, and we have to find the absolute uncertainty in A plus B.
Let's see how to use the formula in the blue box.
This formula tells us that when we add or subtract A and B and get Y as the result,
then that Y, so the uncertainty in the result, is equal to delta A plus delta B.
In other words, the absolute uncertainty in A plus the absolute uncertainty in B.
Let's apply this formula.
The uncertainty in A is plus minus 0.3, and the uncertainty in B is plus minus 1.2.
To find the absolute uncertainty in A plus B, we have to add these two values.
0.3 plus 1.2 is equal to 1.5, and this is the absolute uncertainty in A plus B.
Using this, we can write that A plus B is equal to 9.3, which is the sum of 3.2 and 6.1,
plus minus the uncertainty that we just found, which is 1.5.
In the second example, we have two dogs, and we have to find the absolute uncertainty in the difference between their masses.
The absolute uncertainty in the mass of the purple dog is plus minus 2 kilograms,
and the absolute uncertainty in the mass of the green dog is plus minus 1 kilogram.
Take care when finding the absolute uncertainty in the difference, because even though we are subtracting Mg from Mp,
if you look at the formula in the blue box, you can see that to find the absolute uncertainty in the difference,
we have to add the absolute uncertainty in Mp and in Mg.
So we get delta Mp plus delta Mg is equal to 2 plus 1, which is 3 kilograms.
This is the absolute uncertainty that we are looking for in this question.
Using this result, we can express Mp minus Mg, and we get 21 minus 18, which is 3,
plus minus the absolute uncertainty in the difference that we calculated, which is also 3 kilograms.
An important note here is that when you are subtracting quantities, the fractional or percentage uncertainty in the result might end up being very large.
Here, for example, we got 3 plus minus 3 kilograms for the difference, so the fractional uncertainty is 3 over 3, which is 1,
and the percentage uncertainty is 100%.
Next, let's see how to propagate uncertainties when we multiply or divide.
Here is the formula that we will use in our two examples.
In the first example, we are given two quantities A and B, and we have to find the absolute uncertainty in A times B.
This process will have three steps.
We will find the fractional or percentage uncertainty in A and in B.
The fractional uncertainty in A is that I over A, which is 1 over 10.
The fractional uncertainty in B is that I B over B, which is 1.5 over 7.5, and when we simplify we get 1 over 5.
Note that as I have shown in the title of this step, we can also work with percentages here, and 1 over 10 would be 10%,
why 1 over 5 would be 20%.
Next, we will use the formula in the blue box to find the fractional or percentage uncertainty in A B.
What the formula in the box tells us is that when we multiply two quantities, for example, A and B,
in order to find the fractional uncertainty in the product, we have to add the fractional uncertainties in the two quantities.
So the fractional uncertainty in A B is equal to the fractional uncertainty in A plus the fractional uncertainty in B.
This gives us 1 over 10 plus 1 over 5, and after finding the common denominator, which is 10, and adding, we get 3 over 10.
Finally, let's calculate the absolute uncertainty in A B.
To do this, first we have to calculate A B, and since A is 10 and B is 7.5, A B will be 75.
Next, we have to multiply this by the fractional uncertainty, which is 3 over 10, and this gives us 22.5.
However, since we will record the product A B as 75, which is rounded to the nearest whole number, we will also round the absolute uncertainty in A B to the nearest whole number, and this is 23.
In the second example, we have two quantities, x and y, and we have to find the absolute uncertainty in 3x over 5y.
We will follow the same steps as an example 1, and we will start by looking for the fractional or percentage uncertainty in x and in y.
So we get that x over x is equal to 10 over 200, and for a change, let's break in percentages here, so we get 5%.
That y over y is 1 over 5, and this is equal to 20%.
Next, we will apply the formula to find the fractional percentage uncertainty in 3x over 5y, and here you might be wondering,
how the 3 and the 5 in front of the x and the y respectively will affect the uncertainty.
Since 3 and 5 are not variables, but numbers, their absolute uncertainty is 0, so we can simply ignore them when calculating the fractional uncertainty.
So we will just focus on the fact that x is divided by y, which means that based on the formula in the blue box, similarly to multiplication,
we just add the fractional uncertainty in x and the fractional uncertainty in y.
So we get that x over x plus that y over y, which is equal to 25%.
Our final step is to calculate the absolute uncertainty in 3x over 5y.
We will start by finding 3x over 5y, which is 3 times 200 over 5 times 5, and this is 24.
To find the absolute uncertainty in 3x over 5y, we will multiply the percentage uncertainty, so 25% in other words 0.25, by 3x over 5y, so 24.
This gives us 640 absolute uncertainty.
Finally, let's see how to propagate uncertainties when we raise a quantity to a power.
We are going to use this formula.
What we see here is that when we raise a quantity to a power, and we want to calculate the fractional uncertainty in the result,
we have to multiply the fractional uncertainty in the original quantity by the power.
Let's look at two examples.
In the first example, we have to calculate the absolute uncertainty in a cubed.
Just like for multiplication and division, we will take 3 steps.
First, we will find the fractional or percentage uncertainty in a.
This is data over a, so 0.2 over 2, which is equal to 1 over 10.
In step 2, we will look for the fractional uncertainty in a cubed, and the formula shows us that we have to multiply the power by the fractional uncertainty in a.
The vertical lines are absolute value signs, to ensure that the fractional uncertainty is a positive value.
3x1 over 10 is 3 over 10.
In the final step, we will find the absolute uncertainty in a cubed.
To do this, we will multiply a cubed, so 2 to the power of 3, by the fractional uncertainty in a cubed, so by 3 over 10.
This gives us 2.4.
In the second example, we have quantity x, and we have to find the absolute uncertainty in the square root of x.
Let's start by finding the fractional or percentage uncertainty in x.
In this example, I will work with percentages.
Data x over x is equal to 1 over 25, which is 4%.
In step 2, we will find the percentage uncertainty in square root of x.
Recall from math that we can write roots as powers.
Square root of x can be written as x to the power of 1 health.
So based on formula in the blue box, for the percentage uncertainty, we get the power which is 1 health,
multiplied by the percentage uncertainty in x, and this is 4%.
This gives us 2%.
Now we are ready to find the absolute uncertainty.
To do this, we will multiply the square root of x, so the square root of 25, by the percentage uncertainty,
so 2%, which is 0.02.
This gives us 0.1 for the absolute uncertainty in the square root of x.
Let's see what we have covered in this video.
We have worked with 3 formulae when propagating uncertainties, one for addition and subtraction,
another for multiplication and division, and the third when raising the quantity to a power.
We can also use this last formula when working with roots, such as square roots or cube roots.
And through our examples we learned that when we are looking for the absolute uncertainty in the result of a multiplication division,
or when raising a quantity to a power, we have to take 3 steps during the process.
This concludes our discussion about propagating uncertainties.
In the next video we will learn about error bars, and talk about how to find the uncertainty in the gradient and the intercept of a best fit line.