Another Error Analysis Question - how to go about it?

[Ryaners]

Hi folks, apologies if this is in the wrong forum. I'm a new undergrad physics student and I'm having trouble figuring out how to 'do' error analysis. (I had a quick look through some similar posts here & am still none the wiser.) I'll use my recent lab experiment on Resonance in a Tube as an example; feel free to skip the background given below if it's not necessary.

Background: The aim of the experiment was to find the speed of sound in air, using a function generator & loudspeaker attached to a hollow tube with plunger. We took several measurements at 1kHz, 500Hz and 3kHz and worked out the wavelengths of the standing wave formed in the tube. We noted the measurement error for each measurement we took. We averaged the values we got at each of those frequencies and then determined a speed v for each frequency, and averaged those to get v = 344m/s, using v = fλ.

Now I'm at the 'error analysis' part and a bit stuck. There are instructions in our handbook on how to find the average value, standard deviation, standard deviation of the average value and use Gauss' error law. Do I need to use all of these techniques? Also, to use Gauss' error law, I need to have a value Δf and Δλ and I'm not sure how to find those. Are they a combination of the measurement inaccuracy (eg ±0.1cm when using a meter stick, like we were) with the standard deviation of the average measurement? Or just the standard deviation alone?

Here is the form of Gauss' error law I'm fairly sure I'm supposed to use:
ΔV/V = √[(Δf/f)²+(Δλ/λ)²]

Thanks a million in advance for any help, this is something I've been having trouble with for weeks & the helproom in my college hasn't been able to see me about it yet.

 

[BvU]

You have my sympathy: it can be quite bewildering in the beginning.
Common sense is your best tool. Next comes knowledge about statistical methods.
Some things I remember from long ago:

• work with actual measurements as long as possible. Conversions using physical constants can be OK (unless your experiment is like determining relations among them, like m/e or $\hbar$). Conversions using a coefficient that you also determine are a nono. Reason: measurements that are to be converted hopefully have independent errors; conversion has an error too, so converted measurements no longer have independent errors and statistics don't work any more.
• along the same lines: make plots and do regression with actual measurements if possible. (e.g. logarithms, or squares, or whatever already complicate the error analysis)
• don't assume 0,0 is an actual measurement with infinite weight: first check if the relationship indeed meets expectations, then you can always re-analyse using 0,0.
• final result accuracies often depend on the range you used for your independent variables. Vary them as much as you can. You can always leave out the areas when undesired effects (e.g. non-linearity) disturb the assumed relationship -- but then you have a good reason to do so for big values.
• along the same lines: if you can reverse polarities, never hesitate to do so.
• realize you are often measuring differerences: even a length reading is a subtraction
• If you can, estimate if an error actually contributes to the final error. In your case I can imagine $\Delta f$ can be made so small that you can ignore it.
• don't average if you can make a plot and find e.g. V as a slope. Who knows how linear $\ v = \lambda f \$ really is and whether it holds up at low f ! Or worse: that your standing waves don't really precisely match the physical pipe length !

(ignore the imperative form; as the pirate says: "We treat them more as guidelines")

If you post your steps in the analysis, it'll be asier for potential helpers to comment and perhaps help a bit more..
After all you do have a set of instructions already at hand...

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worked out the wavelengths of the standing wave formed in the tube

There's several ways to do that: dragging a microphone all the way through the tube and record the amplitude is something else than applying a simple $L_{\rm \, pipe} = \lambda/4 (2n + 1)$ ...

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[Ryaners]

Thanks so much for the pointers! Sorry I didn't respond to your comment until now, it's been a hectic week. Thankfully I was finally able to get help on error analysis in a tutorial a few days after I posted. Your general advice for minimizing error is very welcome though, I'm just starting to get comfortable with plotting in Python at the moment & these tips are good excuses to practice and try different approaches. Thanks :)