# Error Propagation

### Error Propagation (adopted from Ben Mathiesen)

Error propagation is a way of combining two or more random errors together to get a third. It is only used in situations where you don't have the luxury or ability to measure the same thing several times and thereby estimate the random error on your final result directly. These equations assume that the errors are Gaussian in nature.

It can be used when you need to measure more than one quantity to get at your final result. For example, in the "Impact and Momentum" experiment you need to measure distances, times, and masses but you really want to calculate energy and momenta. The equations introduced here let you propagate the uncertainties on your data through the calculation and come up with an uncertainty on your results.

Error propagation can also be used to combine several independent sources of random error on the same measurement. For example, you could have a known random error associated with your equipment and find that thing you're studying is physically fluctuating to some degree.

Having said that, error propagation can be easy and fun! OK, not really, but this discussion should get you through it with a minimum of blood, sweat, and tears. There are three basic rules that you need to know. Let's say you're trying to measure something which you can't observe directly, but depends on two other quantities which you can measure: x and y. The absolute uncertainties on these measurements are ± dx and ± dy, respectively. The relative uncertainties on your data are just a different way of expressing the same concept: ± 100(dx/x)%, for example.

Sometimes you will need to use the relative uncertainties in these equations, and sometimes you will need to use the absolute uncertainties. Be careful to keep track as you switch back and forth! It helps to write down the units or percent sign as you go through the process.

RULE 1: If you add or subtract x and y, the absolute uncertainty on x+y or x-y is obtained by adding the absolute uncertainties dx and dy in quadrature:

d(x-y) = d(x+y) = [(dx)2 + (dy)2]1/2

RULE 2: If you multiply or divide x and y, the fractional uncertainty of x times y or x/y is obtained by adding the fractional uncertainties dx/x and dy/y in quadrature.

d(xy)/xy = d(x/y)/(x/y) = [(dx/x)2 + (dy/y)2]1/2

RULE 3: The fractional uncertainty on x to a power n is just n times the fractional uncertainty of x:

d(xn)/(xn) = n(dx/x)

In fact, this is true for any power of x, not just positive integer powers! Rules 1 and 2 depend on the assumption that the uncertainties on x and y have nothing to do with each other and may cancel each other out partially. This is clearly not true if we multiply a number by itself, which is why rule 3 can't be derived from rule 2.

RULE 4: The absolute uncertainty on an arbitrary function of x is obtained by taking the derivative of that function, evaluating it at the value of your measurement, taking the absolute value, and multiplying the result by the absolute uncertainty on x. For clarity, we'll call the value of the measurement x0. If x is an angle, you have to express it in radians for this rule to work.

dF(x0) = |dF/dx|x=x0 dx0

That's it! All you need to know is up there, and the error on any arbitrarily complicated function of your measurements can be evaluated using those rules just by going through it one step at a time. Here's a simple example which uses all three rules:

You want to measure the total momentum of two air table pucks before impact, along the x-direction. If you choose the x-axis to be along the track of puck #1, the formula is:

Px (total x-momentum) = p1 + p2cosA2

• d1 = 1.5 ± 0.1 cm (the distance between two points on the first puck's track)
• d2 = 3.4 ± 0.1 cm
• t = 0.167 ± (1% of 0.167) seconds (the time interval between two sparks)
• m1 = 512 ± 3 g (the mass of puck 1)
• m2 = 512 ± 3 g
• A2 = -20¡ ± 1¡ (the angle that the second puck makes with the x-axis)

The way you use these measurements in this formula is as follows:

Px = m1d1/t1+ (m2d2/t2)cosA2

The uncertainty on Px is really horrible and complicated, right? Right. I'll walk you through how to do it exactly, at the same time showing you that you really don't have to do all that work. The key thing to remember is this: your uncertainty only needs to be accurate to one decimal place! Therefore, when you're adding up uncertainties you only have to keep the largest; any uncertainties an order of magnitude smaller that the biggest can be safely dropped. (When adding or subtracting, you should look at the magnitudes of the absolute uncertainties when deciding whether or not to drop a term; otherwise look at the magnitude of the fractional uncertainties.)

Step 1: Find d[m1d1/t]. (This is the uncertainty on p1x.) We use rule 2, since we're only using multiplication and division:

d(m1d1/t) / (m1d1/t) = [ (dm1/m1)2 + (dd1/d1)2 + (dt/t)2]1/2

Approximating: the fractional uncertainty on the spark timer is about 1%, and the fractional uncertainty on the mass is even less than that. The fractional uncertainty on the distance, however, is more like 7%. In this case, you can definitely neglect the (dm1/m1)2 term, and may be able to neglect the (dt/t)2 term as well. If you can neglect both, then the fractional uncertainty on p1x is equal to the fractional uncertainty on d1 to a very good approximation. To convince yourself that this is reasonable, try it yourself both ways and see how much of a difference it makes.

What did you get? I got an uncertainty of 6.8% on p1x.

Step 2: Find d[m2d2/t]. (This is the uncertainty on p2.) This can be found in exactly the same way--just plug in different numbers.

Step 3: Find d(cosA2). Use rule 3: the derivative of cosA is -sinA, so the uncertainty on cos(A2) is:

d(cos(A2)) = |-sin(A2)| dA2.

Note that this is an absolute uncertainty, while our previous results were fractional uncertainties. To convert it, we divide d(cos(A2)) by cos(A2).

What did you get? I got an uncertainty of 0.006 for dcos(A2), or 0.6%.

Step 4: Find d[p2cosA2]. (This is the uncertainty on p2x). Use rule 2 again, using the fractional uncertainties on p2 and cos(A2).

Approximation: decide whether cosA2/cosA2 is much smaller than dp2/p2. If is is, then the fractional uncertainty on p2x is almost the same as that on p2.

Step 5: Find d[p1 + p2cosA2]. (This is the uncertainty on Px!) Use rule 1, but convert the fractional uncertainties back to absolute uncertainties first:

d[p1 + p2cosA2] = [(dp1)2 + (dp2x)2]1/2

Your Measurement is: Px ± dPx, or Px ± 100(dPx/Px)%. If you used the approximations, this calculation should have been relatively painless. I got an answer of -5200 ± 400 g cm sec-1 for the total x-momentum. If you didn't get something similar, check your work again.