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(x^{n})/(x^{n})
= 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 x_{0}. **If x is an angle, you have
to express it in radians for this rule to work.**

dF(x_{0}) = |dF/dx|_{x=x0}
dx_{0}

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:

P_{x} (total x-momentum) = p_{1} + p_{2}cosA_{2}

Your measurements:

- d
_{1}= 1.5 ± 0.1 cm (the distance between two points on the first puck's track) - d
_{2}= 3.4 ± 0.1 cm - t = 0.167 ± (1% of 0.167) seconds (the time interval between two sparks)
- m
_{1}= 512 ± 3 g (the mass of puck 1) - m
_{2}= 512 ± 3 g - A
_{2}= -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:

P_{x} = m_{1}d_{1}/t_{1}+ (m_{2}d_{2}/t_{2})cosA_{2}

The uncertainty on P_{x} 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[m_{1}d_{1}/t].
(This is the uncertainty on p_{1x}.) We use rule 2, since we're
only using multiplication and division:

d(m_{1}d_{1}/t) / (m_{1}d_{1}/t)
= [ (dm_{1}/m_{1})^{2}
+ (dd_{1}/d_{1})^{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 (dm_{1}/m_{1})^{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 p_{1x} is equal to the fractional uncertainty on d_{1}
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 p_{1x}.

**Step 2:** Find d[m_{2}d_{2}/t].
(This is the uncertainty on p_{2}.) This can be found in exactly
the same way--just plug in different numbers.

**Step 3:** Find d(cosA_{2}). Use
rule 3: the derivative of cosA is -sinA, so the uncertainty on cos(A_{2})
is:

d(cos(A_{2})) = |-sin(A_{2})|
dA_{2}.

Note that this is an **absolute** uncertainty, while our previous
results were **fractional** uncertainties. To convert it, we divide
d(cos(A_{2})) by cos(A_{2}).

What did you get? I got an uncertainty of 0.006 for dcos(A_{2}),
or 0.6%.

**Step 4:** Find d[p_{2}cosA_{2}].
(This is the uncertainty on p_{2x}). Use rule 2 again, using the
fractional uncertainties on p_{2} and cos(A_{2}).

__Approximation:__ decide whether cosA_{2}/cosA_{2}
is much smaller than dp_{2}/p_{2}.
If is is, then the fractional uncertainty on p_{2x} is almost
the same as that on p_{2}.

**Step 5:** Find d[p_{1} + p_{2}cosA_{2}].
(This is the uncertainty on P_{x}!) Use rule 1, but convert the
fractional uncertainties back to absolute uncertainties first:

d[p_{1} + p_{2}cosA_{2}]
= [(dp_{1})^{2} + (dp_{2x})^{2}]^{1/2}

**Your Measurement is:** P_{x} ± dP_{x},
or P_{x} ± 100(dP_{x}/P_{x})%.
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.