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This video is about propagating uncertainties.

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We will learn how to propagate uncertainties when we add and subtract, multiply and divide, or raise a quantity to a power.

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Let's start with addition and subtraction.

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This is the formula for propagating uncertainties when we add or subtract.

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Let's see two examples.

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In the first example, we are given two quantities A and B, and we have to find the absolute uncertainty in A plus B.

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Let's see how to use the formula in the blue box.

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This formula tells us that when we add or subtract A and B and get Y as the result,

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then that Y, so the uncertainty in the result, is equal to delta A plus delta B.

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In other words, the absolute uncertainty in A plus the absolute uncertainty in B.

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Let's apply this formula.

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The uncertainty in A is plus minus 0.3, and the uncertainty in B is plus minus 1.2.

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To find the absolute uncertainty in A plus B, we have to add these two values.

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0.3 plus 1.2 is equal to 1.5, and this is the absolute uncertainty in A plus B.

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Using this, we can write that A plus B is equal to 9.3, which is the sum of 3.2 and 6.1,

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plus minus the uncertainty that we just found, which is 1.5.

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In the second example, we have two dogs, and we have to find the absolute uncertainty in the difference between their masses.

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The absolute uncertainty in the mass of the purple dog is plus minus 2 kilograms,

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and the absolute uncertainty in the mass of the green dog is plus minus 1 kilogram.

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Take care when finding the absolute uncertainty in the difference, because even though we are subtracting Mg from Mp,

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if you look at the formula in the blue box, you can see that to find the absolute uncertainty in the difference,

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we have to add the absolute uncertainty in Mp and in Mg.

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So we get delta Mp plus delta Mg is equal to 2 plus 1, which is 3 kilograms.

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This is the absolute uncertainty that we are looking for in this question.

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Using this result, we can express Mp minus Mg, and we get 21 minus 18, which is 3,

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plus minus the absolute uncertainty in the difference that we calculated, which is also 3 kilograms.

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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.

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Here, for example, we got 3 plus minus 3 kilograms for the difference, so the fractional uncertainty is 3 over 3, which is 1,

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and the percentage uncertainty is 100%.

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Next, let's see how to propagate uncertainties when we multiply or divide.

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Here is the formula that we will use in our two examples.

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In the first example, we are given two quantities A and B, and we have to find the absolute uncertainty in A times B.

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This process will have three steps.

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We will find the fractional or percentage uncertainty in A and in B.

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The fractional uncertainty in A is that I over A, which is 1 over 10.

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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.

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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%,

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why 1 over 5 would be 20%.

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Next, we will use the formula in the blue box to find the fractional or percentage uncertainty in A B.

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What the formula in the box tells us is that when we multiply two quantities, for example, A and B,

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in order to find the fractional uncertainty in the product, we have to add the fractional uncertainties in the two quantities.

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So the fractional uncertainty in A B is equal to the fractional uncertainty in A plus the fractional uncertainty in B.

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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.

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Finally, let's calculate the absolute uncertainty in A B.

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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.

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Next, we have to multiply this by the fractional uncertainty, which is 3 over 10, and this gives us 22.5.

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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.

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In the second example, we have two quantities, x and y, and we have to find the absolute uncertainty in 3x over 5y.

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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.

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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%.

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That y over y is 1 over 5, and this is equal to 20%.

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Next, we will apply the formula to find the fractional percentage uncertainty in 3x over 5y, and here you might be wondering,

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how the 3 and the 5 in front of the x and the y respectively will affect the uncertainty.

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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.

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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,

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we just add the fractional uncertainty in x and the fractional uncertainty in y.

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So we get that x over x plus that y over y, which is equal to 25%.

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Our final step is to calculate the absolute uncertainty in 3x over 5y.

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We will start by finding 3x over 5y, which is 3 times 200 over 5 times 5, and this is 24.

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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.

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This gives us 640 absolute uncertainty.

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Finally, let's see how to propagate uncertainties when we raise a quantity to a power.

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We are going to use this formula.

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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,

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we have to multiply the fractional uncertainty in the original quantity by the power.

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Let's look at two examples.

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In the first example, we have to calculate the absolute uncertainty in a cubed.

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Just like for multiplication and division, we will take 3 steps.

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First, we will find the fractional or percentage uncertainty in a.

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This is data over a, so 0.2 over 2, which is equal to 1 over 10.

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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.

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The vertical lines are absolute value signs, to ensure that the fractional uncertainty is a positive value.

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3x1 over 10 is 3 over 10.

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In the final step, we will find the absolute uncertainty in a cubed.

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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.

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This gives us 2.4.

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In the second example, we have quantity x, and we have to find the absolute uncertainty in the square root of x.

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Let's start by finding the fractional or percentage uncertainty in x.

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In this example, I will work with percentages.

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Data x over x is equal to 1 over 25, which is 4%.

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In step 2, we will find the percentage uncertainty in square root of x.

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Recall from math that we can write roots as powers.

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Square root of x can be written as x to the power of 1 health.

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So based on formula in the blue box, for the percentage uncertainty, we get the power which is 1 health,

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multiplied by the percentage uncertainty in x, and this is 4%.

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This gives us 2%.

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Now we are ready to find the absolute uncertainty.

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To do this, we will multiply the square root of x, so the square root of 25, by the percentage uncertainty,

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so 2%, which is 0.02.

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This gives us 0.1 for the absolute uncertainty in the square root of x.

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Let's see what we have covered in this video.

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We have worked with 3 formulae when propagating uncertainties, one for addition and subtraction,

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another for multiplication and division, and the third when raising the quantity to a power.

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We can also use this last formula when working with roots, such as square roots or cube roots.

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And through our examples we learned that when we are looking for the absolute uncertainty in the result of a multiplication division,

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or when raising a quantity to a power, we have to take 3 steps during the process.

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This concludes our discussion about propagating uncertainties.

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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.