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Error analysis

I am struggling with error analysis with a practical!!!

Please help:

I am plotting a straight line graph, between two variables I measured, each variable with its own absolute errors.

One of the variables is length. Say I estimate the error to be +/- 0.1 cm. Can take an average of all the lengths, <l>, and find an average percentage error for length by +/- 0.01/<l>; or do I have to find the percentage error of all the values?

I draw a best-fit line for the graph and calculate the gradient of this graph.

What is the error for this gradient then? The sum of average percentage errors of the two variables? Since you divide one variable by the other to find the gradient. Or is more complicated than that...

And in general, what are important procedures, things to note, for error analyses.

Thank you very much in advance. :smile:

Reply 1

ApeXaviour

celeritas

I am struggling with error analysis with a practical!!!

Please help:

I am plotting a straight line graph, between two variables I measured, each variable with its own absolute errors.

Ok, it's been a while since I did labs that required error analysis but I'll give it a shot.

One of the variables is length. Say I estimate the error to be +/- 0.1 cm. Can take an average of all the lengths, <l>, and find an average percentage error for length by +/- 0.01/<l>; or do I have to find the percentage error of all the values?

The measurement of length is usually a statistical error. ie if you make a set of measurements the error is the same for each. This should make the errorbars on your graph pretty simple in one axis.

There is no need to get the percentage error for each one unless this length is not one of the quantities you are graphing. ie you have to perform an operation (multiply, divide, differentiate etc) on it with another variable. Say for example you were graphing average velocities vs something else (for some reason!). You would have to add the % error for each measurement of time taken for journey and distance covered in that time. If you've taken a lot of readings this can obviously be tedious to do with a calculator. So you're better off using something like MS Excel:

Say you have Columns A,B,C and D. Time, time error, distance and distance error. And you want columns E and F to be your velocity and velocity error respectively. You merely highlight grid E1 (top square in E column) and type into it =C1/A1 (ie distance over time). Then you grab little bottom right corner notch of the E1 square and drag it down for the whole column of E. Excel will calculate the velocity for each reading for you!
Ok that's easy, what about the error? Well it's the same, highlight grid F1 and type =((B1/A1)+(D1/C1))*E1 (the first bit works out your combined percentage error and then you multiply {* means multiply} that by your velocity to get your velocity error ok?) then drag it down for the whole column of F. Excel does all the grunt work for you! If you have more than 4 or 5 readings this saves you ONE HECK of a lot of time.

I draw a best-fit line for the graph and calculate the gradient of this graph.
What is the error for this gradient then? The sum of average percentage errors of the two variables? Since you divide one variable by the other to find the gradient. Or is more complicated than that...

The gradient as in the slope? Technically you should divide difference between the x variables and the y variables. Remember your school maths m=(y2-y1)/(x2-x1)? But to complicate things even more, none of the points on the graph (that your variables represent) have to even touch the best fit line! Really the above y2-y1 formula only applies to any two different points on your best fit line.
So are you just drawing a best fit graph by hand? In this case I cannot see how you would find the error of the gradient, or an accurate gradient for that matter.
How we worked it out when I was in 2nd year Uni physics labs was to use Excel's regression tool. It would work out the slope and error in slope of a straight line graph plotted from points (with errorbars). It would give an answer. But the error wasn't the "true error" as it only took into account the variation of points from the best fit line and not the errorbars of those points. This was accepted at that level once it was stated that this was not the true error but merely an approximation. Which, at the end of the day is all error is supposed to be, an approximation. Hence the "one significant figure" you always hear demonstrators going on about.

And in general, what are important procedures, things to note, for error analyses.

Em.. lets see. Personally the best advice I think is to learn to use Excel or something like it. It saves a whole lot of time on tedious repetitive calculations, it even plots all sorts of graphs, fits lines/curves, does error bars and finds slope etc too.

I know it feels wrong often but always keep your errors to one significant figure. Like you cannot have 5.624 +/- 0.016 that just doesn't make sense! It's too accurate a number for a description of how inaccurate you are :biggrin:

It should be 5.62 +/- 0.02

Watch out for decay rate errors if you're doing geiger counter/radiation experiments. They're a little tricky and not quite the same as you're used to.

Oh there's a formula too, for getting the error of things that aren't simply multiply, divide, add etc.:

Ok say a = f(x) (ie the value a equals some function of x)

Then the error of a is given by:
Δa = Δx * d[f(x)]/dx
I'd write the above out, it's difficult to see properly when typed.

For example say a = 3x^2
then Δa = Δx * 6x

So if x +/- Δx = 3.0 +/- 0.5, then a = 3 * (3^2) = 27
and Δa = 0.5 * 6 * 3 = 9

This formula always seems to work, try it for a few simple things like multiplication to test it if you don't have confidence in it. Write it down in the back of your lab book or something (I still have it in mine). You rarely need it but man is it useful for those times when you're like "wtf? how the hell do I work out the error for this!? Bastards".

Thank you very much in advance. :smile:

Hope at least some of this was some help. Good thing at times to refresh the brain in these utterly necessary if not altogether fascinating aspects of our studies :smile:

Best of luck