Error Calculation for a Diffraction Grating's Performance

[zehkari]

Homework Statement

I need to calculate the error of an average value (N for diffraction grating).
My results were 4 different average angles. With which I calculated their uncertainty using std.
Using equation (1), I found the number of lines per meter (N) with a known wavelength (λ) and the correct diffraction order (p).
However, I now need an average of those 4 N values to find a mean N value.
What is my uncertainty on the mean N value, considering sin(θ) used in the equation is based off 4 results with their uncertainty?
I understand how to propagate error. I am just confused with how to take the 4 propagated errors based off the uncertainty of the angles to find an error on the mean N.

Homework Equations

(1) sin(θ)=pNλ

The Attempt at a Solution

My only idea would be adding all 4 propagation errors on N to give the total error for the average N?

Any help would be great, many thanks!

 

[gneill]

Suppose that your N values are $N_1, N_2,...N_4$ with uncertainties $\Delta_1,\Delta_2,...,\Delta_4$. I'd think that the uncertainties should sum in quadrature (square root of sum of squares) and be divided by the number of samples. So:

$$\Delta_{avg} = \frac{1}{4}\sqrt{\sum_{i=0}^4{\Delta_i^2}}$$

 

[Dr Transport]

what is your uncertainty in angle? You have a measurement equation, that determines your uncertainty.

 

[zehkari]

Suppose that your N values are $N_1, N_2,...N_4$ with uncertainties $\Delta_1,\Delta_2,...,\Delta_4$. I'd think that the uncertainties should sum in quadrature (square root of sum of squares) and be divided by the number of samples. So:

$$\Delta_{avg} = \frac{1}{4}\sqrt{\sum_{i=0}^4{\Delta_i^2}}$$

Hey, thanks, do you know of the weighted standard diviation? Does that apply here?

 

[gneill]

Hey, thanks, do you know of the weighted standard diviation? Does that apply here?

I know of it. I'll state right away that I am not an expert in this area.

Having said that, I feel It might be applicable if the percent error in angular measurements is untowardly biasing the results (if Δθ is of fixed size then the fraction Δθ/θ becomes larger as θ gets smaller, even though you measure with the same accuracy).

 

[zehkari]

I know of it. I'll state right away that I am not an expert in this area.

Having said that, I feel It might be applicable if the percent error in angular measurements is untowardly biasing the results (if Δθ is of fixed size then the fraction Δθ/θ becomes larger as θ gets smaller, even though you measure with the same accuracy).

Yeah, the error in angle measurements is quite small. I think I will stick with sum of uncertainty in quadrature like you suggested and then talk it over with my lecturer. Thank you for your help.

 

[gneill]

You're very welcome.