Finding uncertainty for varying error bars

[joelwong]

Homework Statement

I am currently doing an experiment on Compton scattering and have plotted a linear graph of 1/E' on (1-cos θ), where E' is the scattered gamma ray energy. My goal is to find the value of the gradient and y-intercept with their corresponding uncertainties. Instead of using the statistical uncertainty as given by excel's LINEST function, I want to use the physically obtained ones. i.e. Due to the limitations of the apparatus. This has resulted in varying error bars for each point, which makes it very difficult to find the lines with max/min gradient. Is there any function in excel to find the max/min gradient lines?

I use Mathematica too if it helps.

Homework Equations

The equation is:

1/E' = 1/E + 1/mc²(1-cos θ)

The gradient is 1/E and the y-intercept is 1/mc²

The Attempt at a Solution

I know I can probably do it the brute force way- adjusting the lines until it fits within the error bars. But I' ve encountered this a few times already and I'm wondering if there is a function to do it quickly.

 

[haruspex]

A crude way is to repeat data points according to the degree of certainty. Can you figure out the relationship?
But what you really want is a weighted regression fit. I don't know if Mathematica offers that. If not, you could write some spreadsheet formulas to do it.

And welcome to PF.

 

[joelwong]

Thanks haruspex for the reply.

I'm not too sure what you mean by repeat data points according to the degree of uncertainty.

The uncertainty for (1-cosθ) is simple, d(1-cosθ) = sinθ dθ, so the uncertainty increases with θ. I have taken results for 15°, 30°, 45°,...

The uncertainty relationship for E' is not as simple- it is based on the FWHM of each photopeak of the spectrum, so there is no clear relationship.

The spectrum is taken by using a NaI(Tl) detector over a period of time. Thus, multiple readings will not make any difference, since every time period is independent of the other. Thus, the data points I have obtained over 180s are total counts over that period of time. Taking multiples readings and adding them up will get me the same result.

Note: I decided to do it the brute force way for now, since I have to submit the report. But I still welcome any responses, since it's something good to learn. I'll see if I can craft some excel function to do the trick when I have the time.