Error propagation formulas + error propagation examples with addition, multiplication, exponents.

[@otabek6270]

I have been spending hours trying to understand this concept and just came across two of your videos. Thanks a lot for uploading these!!!

[@ZaksLab]

you're welcome! -- Zak

[@mawesmo]

Even though I already sent in my assignment, now I know what should be done. Will redo it anyways. Thank you!!!!

[@ZaksLab]

you're welcome! z

[@bscutajar]

I don't understand the result we get from this. For example, in the second example at 5:20, if we take the lower values of both velocity and time we get a distance of 1.38, which is outside the predicted error range, so clearly wrong...

[@ZaksLab]

The propagation of error formula is relating a standard deviation on each measured variable to the standard deviation of the quantity derived from the individual measured variables. For example, given the standard deviation on length and the standard deviation on width, the propagation of errors formula gives you the standard deviation for the area. This is not as simple as just taking the lowest possible combined measurement and the highest possible combined measurement in order to create error bars on the derived quantity: that will overestimate the uncertainty on the derived quantity, because we are ignoring the fact that it is extremely unlikely (and increasingly unlikely as the number of variables increases) to end up with multiple measurements simultaneously taking their extreme low or high values. You can think of your approach as the most conservative possible approach, giving you larger error bars than necessary. The standard deviation gives you a more meaningful interpretation of uncertainty. Instead of "the mean value lies between this lowest and highest possible value" it's more like "there is a 68% chance the mean value lies within one standard deviation of this measurement, a 95% chance that the mean value lies within two standard deviations of this measurement", etc. All that said, I haven't been in touch with the guts of the theory for 20+ years, and I'm sure I could offer more insight into the propagation of error formula, it's range of validity and the geometric intuition behind it, the ways it is typically used/misused in publications, etc., if I refreshed my understanding of the derivation of the formula. That's on my to-do list, but not for a while! z

[@ummaymahfuzashapla6772]

Very helpful. thank you so much

[@charliegordon1982]

what would you do when you have seemingly mixed rules, like you measured the height, inner and outer diameter of a hollow cylinder and you need to calculate volume?

[@ZaksLab]

I suppose you could just do it step-by-step: compute the uncertainties on the inner and outer circle areas, then uncertainty on the area (difference of outer and inner circles), then uncertainty in volume. This seems way too cumbersome, so I doubt that's what's done in practice! I actually recently picked up a couple books on statistical methods in experimental science to really nail down the real answers for some of these more complicated questions, but I haven't had time to get into it yet! -- Zak

[@Ornitholestes1]

Given I have a number of measurements x1,x2,…,xn, I get that to get the uncertainty of the mean related to these uncertainties I’ll have to calculate the uncertainty of the sum, and then divide by n, i.e. sqrt(deltax1^2, deltax2^2,…deltaxn^2)/n. However I am interested in the entire uncertainty of the mean, including the statistical uncertainty (i.e. standard deviation/sqrt(n)) and not just the measurement uncertainty. My understanding is that here I need to again apply the sum formula to add them both together, i.e. sqrt((standard error of mean)^2+(propagated measurement error of mean)^2), am I getting this correctly?

[@robichag]

is there a reason you dont simply divide the diameter by 2 to get the radius instead of subsituting with d/2. wouldnt that be equivalent mathematically?

[@ZaksLab]

It's best to be able to phrase your entire uncertainty calculation in terms of the quantities you directly measure with an instrument. So you have the diameter and the uncertainty in the diameter (an uncertainty stamped on a set of calipers for example) as things you directly measured in the lab. You could halve your diameter measurement as long as you halve your uncertainty in diameter (the factors of 1/2 would cancel in the propagation of errors formula), but it's not an ideal way to present the work.

[@Abdulwahabyekini]

How to quote measurement in physics

[@sathiyavathymurugan7674]

4.18 how did you calculate the delta d/d... I'm not getting that

[@ZaksLab]

we started with delta-d/d= (a big square root of the sum of squares of the relative uncertainties of the two measured quantities). So I calculated the big square root and got delta-d/d=.0398. This is the relative uncertainty in d, meaning we can look at this number and say "d has about a 4% uncertainty"! To find the absolute uncertainty in d, we multiply both sides by d and obtain delta-d=d*.0398, and now we have to sub in the d we found earlier (1.4616m): so delta-d=1.4616*.0398=.0582. A popular convention is to round this to 1 significant digit as .06m. Finally, we take the value of d and round to that same decimal place, since writing digits in decimal places past the uncertainty doesn't make sense: d=1.46+/-.06m

[@sathiyavathymurugan7674]

Thank you so much for your reply nd explanation ..👍🏻✨

[@anthonygarcia7810]

Amazing video, thank you!!

[@ZaksLab]

thanks! -- Zak

[@willjackson5885]

What if you wanted to propagate error for averages? Using the first example, would the average mass of M and m be "1.54 +- delta z" or "1.54 +- (delta z)/2"?

[@ZaksLab]

My best attempt at the correct answer here is that if you averaged over n trials with each trial having an uncertainty of 0.1 for M, then the uncertainty in the average goes down by a factor of sqrt(n) if I remember my stats correctly. Then this would be your uncertainty in the average, and you could propagate that as usual. I'm super rusty on stats though! -- Zak

[@Haneet]

Thank you so much, this helped a lot!

[@ZaksLab]

you're welcome! -- Zak

[@alaricboikanyo]

Good video I wanna understand this, you are doing an experiment then you collect data 5 times and those are x values, How will you find the uncertainty of "d" value if "d" is found by saying (d= 1/x). Mind you there are 5 d values as you collected the data for the experiment five times. Lastly will the uncertainty of d be the same or vary?

[@ZaksLab]

There are two different things going on here: statistical uncertainty for the data set of "x" values, then error propagation. The uncertainty in x would be found by using the variation between the measurements, so the crude way to do that would be to just say the uncertainty is equal to the standard deviation of the measurements. The more precise way would be to use a t-distribution to construct a confidence interval with a desired level of confidence (usu. 95%), where the margin of error on that CI could be taken as the uncertainty in your measurement of x. Applying the propagation of errors formula, and saying y=1/x, you get del-y/y=sqrt((-del-x/x)^2), but that's just del-x/x, then you get del-y by multiplying both sides by y to get del-y=y*del-x/x, in other words both variables get the same relative uncertainty (percent uncertainty). All this amounts to a reasonable way to handle the data, but if I was publishing a paper that required this uncertainty calculation, I would spend some serious research time in a statistical methods for laboratory sciences book etc. and check against norms in my discipline and specific to the journal before publishing. -z

[@alaricboikanyo]

@@ZaksLab You are the best. Finding information on uncertainty on internet is not easy. You can get stuck only on uncertainty when doing lab reports 😭 Anyways, Thanks. Your help is much appreciated 🙏.

[@sushantdalvi1042]

Thanks you zak's.

[@nicolelegge7839]

Great video! I was wondering what the process would be to find the propagration of error for the equation x= (a-b)/c? thanks!

[@muthurpe]

Did you happen to find this? I am wondering the same, for X = (B-A)/A

[@marabunya]

Amazing.

[@ZaksLab]

Thanks! z

[@ggxsky4811]

Thanx

[@ZaksLab]

you're welcome! z

[@jenm1]

I'm a little confused by why the 2 came out due to d^2 in the error propagation.

[@ZaksLab]

I didn't mean for this to be a *derivation* of the propagation of errors formula, but the exponent comes out in front of that term due to a partial derivative. In terms of application in intro physics, you just have to remember the exponent will result in that same number popping out in front of the corresponding term in the formula. I do intend to make a video on the derivation of the propagation of errors formula eventually though! z

[@jenm1]

@@ZaksLab Okay, I get it now! Thank you, I'll watch it again with that in mind. It still helped in my lab today :)

[@PatienceMtchire]

❤

[@lisajose9219]

❤🎉🎉❤

[@sixgunslobes621]

i am having a hard time proving the formulars

[@ZaksLab]

on my to-do list for a while, but I haven't had a chance to make a video on it. In lower div. physics it's traditional to just give the formulas out in lab without proof, but that's not very satisfying. z