A Summary of Error Propagation Suppose you measure some quantities a b c with uncertainties a b c Now you want to calculate some other quantity Q which depends on a and b and so forth What is the uncertainty in Q The answer can get a little complicated but it should be no surprise that the uncertainties a b etc propagate to the uncertainty of Q Here are some rules which you will occasionally need all of them assume that the quantities a b etc have errors which are uncorrelated and random These rules can all be derived from the Gaussian equation for normally distributed errors but you are not expected to be able to derive them merely to be able to use them 1 Addition or Subtraction If Q is some combination of sums and di erences i e then Q a b c x y z Q cid 112 a 2 b 2 c 2 x 2 y 2 z 2 In words this means that the uncertainties add in quadrature that s the fancy math word for the square root of the sum of squares In particular if Q a b or a b then 3 Example suppose you measure the height H of a door and get 2 00 0 03 m This means that H 2 00 m and H 0 03 m The door has a knob which is a height h 0 88 0 04 m from the bottom of the door Then the distance from the doorknob to the top of the door is Q H h 1 12 m What is the uncertainty in Q Using equation 3 Q cid 112 a 2 b 2 Q cid 112 H 2 h 2 cid 112 0 03 m 2 0 04 m 2 cid 112 0 0009 m2 0 0016 m2 0 0025 m2 0 05 m 1 2 4 5 6 7 So Q 1 12 0 05 m You might reasonably wonder Why isn t Q just equal to a b After all if I add a a to b b the answer is de nitely at least a b a b and at most a b a b right The answer has to do with the probabalistic nature of the uncertainties a and b remember they represent 68 con dence intervals So 32 of the time the true value of a is outside of the range bounded by a a likewise for b But how often is the sum outside of the range bounded by a b a b A little bit less often than that In order for it to be higher than a b a b for instance you d need either both a and b to be on the high end of the expected range or one of them to be very high which is less likely than one being high and the other being low Remember that this formula assumes that the uncertainties in a and b are uncorrelated with each other So a b is actually a slight overestimate of the uncertainty in a b If you were to go through the math in detail you d arrive at the conclusion that the expected uncertainty is given by equation 3 rather than by the simpler expression a b 1 Physical Sciences 2 Harvard University Fall 2007 There is good news though The more complicated expression in equation 3 has a very nice feature it puts more weight on the larger uncertainty In particular when one of the uncertainties is signi cantly greater than the other the more certain quantity contributes essentially nothing to the uncertainty of the sum For instance if a 5 cm and b 1 cm then equation 3 gives a b cid 112 a 2 b 2 cid 112 5 cm 2 1 cm 2 cid 112 25 cm2 1 cm2 5 1 cm Since we generally round uncertainties to one signi cant gure anyway 5 1 isn t noticeably di erent from 5 So the 1 cm uncertainty in b didn t end up mattering in our nal answer As a general rule of thumb when you are adding two uncertain quantities and one uncertainty is more than twice as big as the other you can just use the larger uncertainty as the uncertainty of the sum and neglect the smaller uncertainty entirely However if you are adding more than two quantities together you probably shouldn t neglect the smaller uncertainties unless they are at most 1 3 as big as the largest uncertainty As a special case of this if you add a quantity with an uncertainty to an exact number the uncertainty in the sum is just equal to the uncertainty in the original uncertain quantity 2 Multiplication or Division cid 115 cid 18 a cid 19 2 cid 18 b cid 19 2 b a Q Q cid 18 x cid 19 2 x cid 18 y cid 19 2 y cid 18 z cid 19 2 z Q ab c xy z cid 19 2 cid 18 c c simplest to convert all of the uncertainties into percentages before applying the formula What this means is that the fractional uncertainties add in quadrature In practice it is usually Example a bird ies a distance d 120 3 m during a time t 20 0 1 2 s The average speed of the bird is v d t 6 m s What is the uncertainty of v 8 9 10 11 12 13 14 15 16 17 v v cid 115 cid 18 d cid 18 t cid 19 2 cid 19 2 cid 115 cid 18 3 m cid 18 1 2 s cid 19 2 cid 112 2 5 2 6 2 120 m d t 20 0 s cid 19 2 0 000625 0 0036 6 5 2 If then So Physical Sciences 2 Harvard University Fall 2007 v v 6 5 18 19 20 So the speed of the bird is v 6 0 0 4 m s Note that as we saw with addition the formula becomes much simpler if one of the fractional uncertainties is signi cantly larger than the other At the point when we noticed that t was 6 uncertain and d was only 2 5 uncertain we could have just used 6 for the nal uncertainty and gotten the same nal result 0 36 m s which also rounds to 0 4 6 m s 6 5 0 39 m s The special case of multiplication or division by an exact number is easy to handle since the exact number has 0 uncertainty the nal product or quotient has the same percent uncertainty as the original number For example if you measure the diameter of a sphere to be d 1 00 0 08 cm then the fractional uncertainty in d is 8 Now suppose you want to know the uncertainty in the radius The radius is just r d 2 0 50 cm Then the fractional uncertainty in r is also 8 8 of 0 50 is 0 04 so r 0 50 0 04 cm …