# On propagation of errors

I’m reading John R. Taylor’s textbook An Introduction to Error Analysis: The study of uncertainties in physical measurements. This is meant for people taking introductory physics lab classes, but it never hurts to revisit these things. I actually double-majored in math and chemistry in college. It’s fun watching the contortions that chemists go to to avoid math.

Anyway, measurements come with uncertainities: that is, they have the form $x \pm \delta_x$, where $x$ is our best estimate of the quantity and $\delta_x$ is an estimate of the uncertainty. (We can think of this as being roughly the standard deviation of the distribution from which $x$ is drawn.) In these intro lab classes one quickly learns some rules for manipulating these uncertainties. These can be thought of as defining arithmetic on intervals; however this isn’t the usual interval arithmetic but actually an abbreviation of arithmetic on probability distributions.

• $(x \pm \delta_x) + (y \pm \delta_y) = (x+y) \pm (\sqrt{\delta_x^2 + \delta_y^2})$ – that is, the variances attached to the measurements add. Similarly, for differences, $(x \pm \delta_x) - (y \pm \delta_y) = (x-y) \pm (\sqrt{\delta_x^2 + \delta_y^2})$. For example, $(30 \pm 4) + (20 \pm 3) = (50 \pm 5)$ and $(30 \pm 4) - (20 \pm 3) = (10 \pm 5)$. Note that the error for the sum and the difference are the same – but for the difference, the error is relatively much bigger.
• To find $(x \pm \delta x) \times (y \pm \delta_y)$, start by finding the fractional uncertainties $\delta_x/x$ and $\delta_y/y$. Then the squares of the fractional uncertainties add: the fractional uncertainty of the product is $\sqrt{(\delta_x/x)^2 + (\delta_y/y)^2}$. The same fractional uncertainty holds for quotients. For example, the fractional uncertainty in $30 \pm 4$ is $4/30 \approx 0.133$, and that in $20 \pm 3$ is $3/20 = 0.15$. So the fractional uncertainty in their product is $\sqrt{(0.133)^2 + (0.15)^2 = 0.201$. Thus we have for the product $600 \pm 120$ and for the quotient $1.5 \pm 0.3$.
• Perhaps one learns rules for dealing with powers, logarithms, and the like. These are all easily derived from the rule$f(x \pm \delta x) = f(x) \pm |f\prime(x) \delta x|$.

For example,
$(30 \pm 4)^2 = 30^2 \pm |(2)(30)(4)| = 900 \pm 240$ – in fact, when taking nth powers, the fractional uncertainty is raised to the $n$ power. Similarly,
$\log (30 \pm 4) = \log 30 \pm |(1/30) (4)| = 1.48 \pm 0.13.$
In this case, the fractional uncertainty becomes the absolute uncertainty in the logarithm. If we know a number to within ten percent, we know its log to within 0.1 unit.

But implicit in the rules for sums, differences, products, and quotients is the idea that the errors $\delta x, \delta y$ in the measurements of $x, y$ are independent! So these rules can’t be used if there’s correlation between the errors. More simply, they can’t be used if the quantity that you’re interested in is a function of many variables, some of which occur more than once. Consider for example the Atwood machine, as Taylor does in his problem 3.47. This consists of two objects, of masses $M$ and $m$ with $M > m$; the larger mass accelerates downward, with acceleration $a = g(M-m)/(M+m)$. Here $g$ is the acceleration due to gravity. We assume this is known exactly. So there may be correlation between the numerator and the denominator.

So what can we do? In this particular case it’s not hard to rewrite as
$a = g {1-(m/M) \over 1+(m/M)} = g f(m/M)$
where $f(z) = (1-z)/(1+z)$, and use the rules that I’ve already discussed. (But it may be hard to see that this is worth doing!) For example (I’m taking these numbers from Taylor) say $M = 100 \pm 1, m = 50 \pm 1$. Then the fractional uncertainty in the quotient $M/m$ is$\sqrt{(0.01)^2 + (0.02)^2} \approx 0.022$, and we get $M/m = 0.5 \pm 0.011$. Then $f^\prime(z) = -2/(1+z)^2$, so $f^\prime(1/2) = -8/9$, and thus we have $f(M/m) = f(1/2) \pm (0.011)(8/9) = (1/3) \pm 0.01$.

Alternatively, we think that $m/M$ is likely to lie in the interval $0.5 \pm 0.011 = [0.489, 0.511]$; then f(0.489) = 0.343 and f(0.511) = 0.324, so we figure that f(m/M) is likely to lie in the interval [0.324, 0.343].

But we are not guaranteed that our rewriting trick will always work. What else can we do? I’ll address that in a future post.