Error of Product: Calculating Speed of Sound Formula

[Opus_723]

I'm teaching one of the physics labs for non-science majors at my school this year, and I ran across a formula in the first lab that's confusing me.

They are using an oscilloscope to measure the wavelength of sound at a given frequency in order to determine the speed of sound. The lab manual then asks them to find the error in their calculation, using the mean of several measurements as the true $\lambda$, and this formula:

$\Delta$v = v$\sqrt{(\frac{\Delta\lambda}{\lambda})^{2}+(\frac{Δf}{f})^{2}}$

I'm not sure where this formula comes from, but I get a very different formula for the error in a product when I multiply out ($\lambda$+Δ$\lambda$)(f+Δf) and divide by $\lambda$f to get the relative error (dropping high-order terms, of course):

$\Delta$v = v($\frac{Δ\lambda}{\lambda}$+$\frac{Δf}{f}$)

And trying a few examples on my calculator, this latter formula seems to give better results. Has anyone seen that first formula above? I'm trying to figure out if I'm just being dense and it should give me better results, or if someone made a mistake printing this lab. The only thing I can think of is that someone squared both sides of the equation I got and then dropped the middle term for some reason, even though it's the same order as the others.

 

[DrGreg]

Your own formula would make sense if you wanted to interpret Δλ, Δf and Δv as maximum possible errors. But a more useful interpretation would be to think of them as most likely (or average) errors, or as, say, 90% confidence intervals for errors. It's quite unlikely both errors will be simultaneously large (assuming they're statistically independent), but more likely one will be larger and the other smaller.

See the Wikipedia article Propagation of uncertainty for the maths.

 

[szynkasz]

The first formula is for random errors, the other for systematical ones.

 

[Opus_723]

Thank you. That makes sense. I haven't studied any statistics yet, so I just tried the simplest thing I could think of.

 

[mfb]

The first formula is for random errors, the other for systematical ones.

Systematic uncertainties can be uncorrelated, too. And statistic errors can be correlated (not very likely in this example, however).

The first formula is for uncorrelated errors. The second assumes maximal correlation - you can use it as conservative approach if you are unsure about the correlation, but you will usually overestimate the uncertainty with it.