Calculating the Percentage Error in Measurement Uncertainties

[Nerd10]

Homework Statement

Let x=5.234+/-0.0005 and y=5.123+/-0.0005. Find the percentage error of the difference a=x-y when relative errors delta_x=delta_y=0.0001.

Homework Equations

I think a=0.111 because 5.234-5.123=0.111 and +/-0.0005 cancel each other out. But how do I find the answer from here?

The Attempt at a Solution

 

[Simon Bridge]

The plus-and-minuses don't cancel each other out.

Are x and y independent?
You must have done some theory of errors before now?

I prefer sigmas for standard errors:
$$\sigma_a^2=\sigma_x^2+\sigma_y^2\\ p_a=100\frac{\sigma_a}{a}$$

 

[Nerd10]

Thanks.

 

[Simon Bridge]

If x and y are measurements of properties X and Y, then we can represent uncertainties in the measurement process by writing: $$X=x\pm\sigma_x\qquad Y=y\pm\sigma_y$$.
When we do, we are basically saying that X and Y can be modeled as continuous random variables which are normally distributed with mean equal to their measured value and standard deviation equal to the uncertainty.

When you deal with them like that, then all that stuff you probably did in secondary school about hypothesis testing comes into play.

X and Y may have dependent of independent measurements.

If independent then:$$z=x+y\implies \sigma_z^2=\sigma_x^2+\sigma_y^2\\
z=xy\implies \left(\frac{\sigma_z}{z}\right)^2=\left(\frac{\sigma_x}{x}\right)^2+ \left( \frac{\sigma_y}{y}\right)^2$$note that ##x-y=x+(-y)## and ##x\div y = x\times (1/y)## so I don't need to provide specific rules for subtraction and division: the uncertainties part always adds.
BTW: If you think these rules look like Pythagoras - you are right.

If the measurements are dependent though - like we measure length y starting from where length x left off or we add/multiply a measurement to itself - then the rules get looser:$$z=x+y\implies \sigma_z=\sigma_x+\sigma_y\\ z=xy\implies \frac{\sigma_z}{z}=\frac{\sigma_x}{x}+\frac{\sigma_y}{y}$$

These are rules of thumb - from them, the others can be derived.
These rules come from a huge body of math that is often covered at senior undergrad and/or junior postgrad levels in science courses. Below that just remember the rules.