How do you calculate the margin of error in Physics?

[daiyleena]

Homework Statement

Margain of error for force and Jung's module

Relevant Equations

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Hello! So, I have a horrendous lab work about deformations... I have done most of it but have not done the margin of error because I honestly don't know how to. My professor mentioned the relative margin error method, differentiation and substitution(?), but it was quite a bit ago and I don't remember how to use them. I tried figuring out my older works, but god even more confused because I don't get partial differentiation at all and we used to do that and I kinda just copied the formula and..yeah.

I apologise for the language, english is not my main one :D

I tried to do the error margin in the first graph by just using the atandard deviation but it looks hellishly wrong... and I honestly have no idea what else to use...

(if anyone is confused by the headers of the graphs, they mean: force depending on elongation and jung's module depending on force)

 

[WWGD]

Welcome to PF!

Maybe this
https://en.m.wikipedia.org/wiki/Propagation_of_uncertainty

Could help?

 

[WWGD]

There are worked examples in the link. Do they help?

 

[haruspex]

First, I'm not sure about your graphs. The translations of the titles are "elongation dependence on force" and "Jung's module dependence on applied force". (I take the second to mean Young's modulus.) But the first graph has force on the y axis and elongation on the x axis, which is the wrong way round according to the title.

Next, the regression line in the first graph does not look like a very good fit. I am guessing you forced it to go through (0,0), which is ok, but it looks to me that the slope should be a bit steeper. How did you obtain it?

You don’t say how you calculated the standard deviation. If you just took the s.d. of the y values that would certainly be wrong. Take the root mean square of the differences between the y values and where, according to the regression line, they should be. If your regression line is $y=ax+c$ then that's $\sqrt{\frac 1N\Sigma_1^N(y_i-ax_i-c)^2}$.

 

[WWGD]

First, I'm not sure about your graphs. The translations of the titles are "elongation dependence on force" and "Jung's module dependence on applied force". (I take the second to mean Young's modulus.) But the first graph has force on the y axis and elongation on the x axis, which is the wrong way round according to the title.

Next, the regression line in the first graph does not look like a very good fit. I am guessing you forced it to go through (0,0), which is ok, but it looks to me that the slope should be a bit steeper. How did you obtain it?

You don’t say how you calculated the standard deviation. If you just took the s.d. of the y values that would certainly be wrong. Take the root mean square of the differences between the y values and where, according to the regression line, they should be. If your regression line is $y=ax+c$ then that's $\sqrt{\frac 1N\Sigma_1^N(y_i-ax_i-c)^2}$.

I guess they're using the sample standard deviation.

 

[haruspex]

I guess they're using the sample standard deviation.

You mean divide by N-1 instead of N? Yes, that's technically more accurate, but with nine datapoints it won't make much difference (6%).

 

[mjc123]

There are other "errors" here. You seem to be taking the strain as (initial length)/(extension), which is the wrong way round. In row 2, the strain should be 1/125, not 125. Consequently, your modulus values are nonsensical.

 

[WWGD]

You mean divide by N-1 instead of N? Yes, that's technically more accurate, but with nine datapoints it won't make much difference (6%).

I meant I believe they use estimators for the samples because they don't know the population parameters.

 

[haruspex]

I meant I believe they use estimators for the samples because they don't know the population parameters.

I'm still not sure what you mean. Do you mean the a priori estimate of the error in taking each measurement?

 

[WWGD]

I'm still not sure what you mean. Do you mean the a priori estimate of the error in taking each measurement?

No, that the formulas for propagatioj of uncertainty are defined in terms of the actual, rather than damole, deviation of the random variables in question. In this problem, from what Ive seen, we don't have the actual parameters, so can't used the standard formulas I linked to in post #2.

 

[haruspex]

No, that the formulas for propagatioj of uncertainty are defined in terms of the actual, rather than damole, deviation of the random variables in question. In this problem, from what Ive seen, we don't have the actual parameters, so can't used the standard formulas I linked to in post #2.

That's what I meant by the a priori estimates, the errors to be expected in the individual measurements. Yes, these are propagated through formulas to arrive at the errors in plotted data, but these still constitute a priori estimates. The alternative approach is a posteriori, looking at the spread of results.
The references in post #1 to relative error and differentiation indicate a priori, but the mention of standard deviation implies a posteriori. Perhaps this is one of the confusions baffling @daiyleena.

@daiyleena, if you want to use the propagation formulas (a priori estimates) the first step is to estimate your measurement errors. When you measure an elongation, how large might your error be?

There are other "errors" here. You seem to be taking the strain as (initial length)/(extension), which is the wrong way round. In row 2, the strain should be 1/125, not 125. Consequently, your modulus values are nonsensical.

Which is why the graph rises quadratically instead of being roughly constant.