Finding uncertainty for index of refraction

[bubothedog]

Homework Statement

In an experiment to find the index of refraction, n, of a block of glass, the angle of incidence, θi , and angle of refraction, θr , were measured a number of times as follows:

θi 10◦ 20◦ 30◦ 40◦ 50◦

θr 7.0 ◦ 13.5 ◦ 20.0 ◦ 25.5 ◦ 31.0 ◦

calculate the index of refraction n with its uncertainty. Are your answer consistent with the manufacturer’s claim that n = 1.50? Explain.

Homework Equations

n = sinθi/sinθr

The Attempt at a Solution

I managed to find some answers that are similar to this question and how it approach the question is to calculate the individual index n, which in this case 5 of them and then calculate the uncertainty. But the problem is that the question I refer to, provides the uncertainty for both θi and θr which makes it easy to compare with the n=1.5 that the question wants to compare.

So how do I find the uncertainty for each of the index of refraction without any uncertainty given for θi and θr for my question? Sorry, I was given an assignment on this and I have never learned this before so any help is appreciated.

 

[Quantum Defect]

Homework Statement

In an experiment to find the index of refraction, n, of a block of glass, the angle of incidence, θi , and angle of refraction, θr , were measured a number of times as follows:

θi 10◦ 20◦ 30◦ 40◦ 50◦

θr 7.0 ◦ 13.5 ◦ 20.0 ◦ 25.5 ◦ 31.0 ◦

calculate the index of refraction n with its uncertainty. Are your answer consistent with the manufacturer’s claim that n = 1.50? Explain.

Homework Equations

n = sinθi/sinθr

The Attempt at a Solution

I managed to find some answers that are similar to this question and how it approach the question is to calculate the individual index n, which in this case 5 of them and then calculate the uncertainty. But the problem is that the question I refer to, provides the uncertainty for both θi and θr which makes it easy to compare with the n=1.5 that the question wants to compare.

So how do I find the uncertainty for each of the index of refraction without any uncertainty given for θi and θr for my question? Sorry, I was given an assignment on this and I have never learned this before so any help is appreciated.

Based upon the presentation of the data (I assume that you did not measure these numbers) -- it looks like the uncertainties in the measurements are about +/- 0.5 degrees. You could probably assume similar errors on the theta_i numbers as well. Any introductory book on error analysis will tell you how to propagate the errors in the thetas through to errors in n.

 

[bubothedog]

Hi, thanks for the reply. But how do I approach this question?

If I use the mean, std.dev method and then using the std.dev to find the std.error. Will the answer be correct? I tried using this method and I got an answer of 1.5+- 0.5 but I am not sure if this method is the right way since the answer I found doesn't use this method. Also, I don't think the above angles are considered repeated measurements? Correct me if i am wrong.

But from the answer I found, it finds the individual n which in this case 5 of them and its uncertainty and then explain the relation with the n=1.5. The problem with the answer I found is that the question actually provides the uncertainty for the i and r angles as mentioned above so I am clueless on how to find the uncertainty for them.

Can you show me an example?

 

[Quantum Defect]

Hi, thanks for the reply. But how do I approach this question?

If I use the mean, std.dev method and then using the std.dev to find the std.error. Will the answer be correct? I tried using this method and I got an answer of 1.5+- 0.5 but I am not sure if this method is the right way since the answer I found doesn't use this method. Also, I don't think the above angles are considered repeated measurements? Correct me if i am wrong.

But from the answer I found, it finds the individual n which in this case 5 of them and its uncertainty and then explain the relation with the n=1.5. The problem with the answer I found is that the question actually provides the uncertainty for the i and r angles as mentioned above so I am clueless on how to find the uncertainty for them.

Can you show me an example?

Look for a good book on error analysis. Bevington, "Data Reduction and Error Analysis for the Physical Sciences" is the book I used to learn this. Taylor's book "An Introduction to Error Analysis" is also good. A good library will have either/both of these. Basically, you want to propagate the error in the angular measurements through to an error in sin(theta) through to an error in the index (assumes errors are "normal"). You will then have a bunch of measurements for n, with varying uncertainties -- not all of the measurements of n will have the same percent error I think. What this basically means is that some of the measurements are going to be more trustworthy than others. You can take an average over all of the measurements that takes this trust-worthiness into account -- you are basically calculating a weighted average, where the weights are lower when the relative error is higher. You will also be able to calculate an uncertainty of this average value. The books referenced above will tell you exactly how to do this, and I bet the same equations are on the intertubes, somewhere.

This is a basic problem in science. You have a bunch of measurements made with varying levels of precision. How can you use all of the data to come up with a best estimate. In the simple case where you have one really, really precise measurement, and others not so precise, the average value will be closest to the most precise value and the error will look like the best precision. If you have a bunch of values with similar errors, the average will look similar to the unweighted average, and the uncertainty will look close to the uncertainty of the mean.

 

[haruspex]

I tried using this method

Then please post your attempt, don't just describe it.

 

[bubothedog]

Then please post your attempt, don't just describe it.

I find the mean for both i and r. i=30, r=19.4 by using all the 5 values of i(10+20+30+40+50)/5 and similar for r.

I then proceed to find the std.dev and std.error using the formulas. My std.error i=7.07, r=4.246. All in degrees.

With that I find the best estimated n using the formula of n=sini/sinr=1.505. To find the uncertainty I used the formula for the division propagation error to get a value of 0.4842.

Therefore, rounding off I get a final answer of 1.5+- 0.5.

But the problem is, I am not too sure if using this method is correct? Because it feels like I should find the individual values for n for each pair of i and r and compare the value with that of n=1.50 from the manufacturer. But I can't seem to find a way to calculate the uncertainty if I try to find the individual values instead. Any tips? Is the mean method I used any correct? The question would have been clearer if the uncertainty were provided for the i and r angles.

 

[haruspex]

I find the mean for both i and r. i=30, r=19.4 by using all the 5 values of i(10+20+30+40+50)/5 and similar for r.

No, that's not the way at all. That has nothing to with uncertainties in the data or in the answer.
Having said that, I'm not entirely sure what you are expected to do here.

Some uncertainties in the data can be inferred from the precision given. It's a bit awkward here because the input angles are expressed with no decimal part (10, not 10.0), so it looks as though the input angles are only specified to the nearest degree. On the other hand, the output angles are all specified to one decimal place. Since there's no reason why they should be more accurately measured than the input angles, I would presume all angles are stated to the nearest 0.1 degree.
I.e., the given data should be taken as +/-0.05 degrees.
The error range that leads to in the sin function is nonlinear, of course - not sure how to handle that.

A completely different approach is to look at the variance in the calculated index. Making the somewhat crude assumption that the variance is independent of the input angle ("homoscedasticity"), you can estimate the variance for the result. Use the $\hat{\sigma}^2 = \frac 1{N-1} \Sigma(X_i-\bar X)^2$ form. Then see how many standard deviations from the mean to the manufacturer's value.

There may be a way to combine these approaches, but that's too advanced for me.

 

[bubothedog]

Anyway, I got another question. If i am given a set of numbers of repeated values. For example I am to find the velocity and given a equation of Dist/Time. I have 10 repeated timings let's say 1-10s. The distance is given to be 5+-0.5m. How do I find the uncertainty for the velocity?

No, that's not the way at all. That has nothing to with uncertainties in the data or in the answer.
Having said that, I'm not entirely sure what you are expected to do here.

Some uncertainties in the data can be inferred from the precision given. It's a bit awkward here because the input angles are expressed with no decimal part (10, not 10.0), so it looks as though the input angles are only specified to the nearest degree. On the other hand, the output angles are all specified to one decimal place. Since there's no reason why they should be more accurately measured than the input angles, I would presume all angles are stated to the nearest 0.1 degree.
I.e., the given data should be taken as +/-0.05 degrees.
The error range that leads to in the sin function is nonlinear, of course - not sure how to handle that.

A completely different approach is to look at the variance in the calculated index. Making the somewhat crude assumption that the variance is independent of the input angle ("homoscedasticity"), you can estimate the variance for the result. Use the $\hat{\sigma}^2 = \frac 1{N-1} \Sigma(X_i-\bar X)^2$ form. Then see how many standard deviations from the mean to the manufacturer's value.

There may be a way to combine these approaches, but that's too advanced for me.

Hi,

Thanks for the reply. Those are exactly my questions. I managed to find the original question from the internet and the answer that was provided was to calculate the individual n for each of the 5 angles and then compare it with the manufacture value of n=1.50 and make an explanation.The difference between the original question and my question is that the uncertainty was not provided. The original question has an uncertainty level of +- 1 for each of the data provided in particular i and r.

But for my question, no uncertainty values were provided so I am quite confused on how to solve it since the question wants me to find the n value and its uncertainty.

So what I did was just to make use of the mean values of n, since std.error could be found using this way. But I am quite sure this is the wrong method since the original answer doesn't work this way.

So do you think the best way for this question is to assume a certain uncertainty level such as
+- 0.05 degrees as you mentioned earlier on and then try to solve for each of the n?