How to verify how good my data is?

Question

I'm in an undergrad physics lab right now and I have taken some data. The theoretical curve should be proportional to $\cos^2(\theta)$. How can I quantify how close my data values are to this theoretical curve? Can I linearize it somehow and then let Excel do a least squares type line for me? Or is there some better way of quantifying how close my data is? Sorry I haven't taken any statistics and this physics lab doesn't explain anything to me. Thanks in advance for any answers. :)

Answer 1

Suppose one of your input values is $\theta$ and the experimental value for that $\theta$ is $\hat y.$ The theoretical output value is $y=\cos^2\theta$. The difference between $\hat y$ and $y$ measures how good the experimental data is for that particular $\theta$ value.

To measure how good the experimental data is overall, you could take the average of these differences over all data points. Suppose that your experimental data points are $(\theta_1,\hat y_1),(\theta_2,\hat y_2),\ldots,(\theta_n,\hat y_n).$ The corresponding points on the theoretical curve are $(\theta_1,y_1),(\theta_2,y_2),\ldots,(\theta_n,y_n),$ where $y_i=\cos^2\theta_i.$ The average of these differences is then $$\frac{1}{n}\sum_{i=1}^n|\hat y_i-y_i|$$ This is called the mean absolute error.

Another alternative is to take the square root of the sum of the squares of the errors $$\sqrt{\frac{1}{n}\sum_{i=1}^n(\hat y_i-y_i)^2}$$ This is called the root-mean-square error. The formula seems more magical, but is more commonly used in practice (as far as I can tell).

Both of these quantify how close the experimental data is to the theoretical curve. The smaller either of these values is, the better the experimental data is.

Answer 2

well, first off - I cannot fathom a collection of huge data (by experiments, analytical or empirical) that follows a $$cos^2(\theta)$$ distribution ! naturally most "good" data distributions/collections must follow a Normal distribution( preferrably 0 mean and standard deviation 1 kind of data) , for large amount of observations.

But I'll leave the validity of your 2nd statement , to you.

Coming to the closeness of data. For an undergraduate , generally, its very helpful to learn MATLAB.

Its a software, used for many purposes( which you might want to google). And it would be helpfull if you could learn basic statistics,mean , standard deviation ,variance, mean square values , statistical distributions etc. Also some concepts like cross correlation can help too.

But I would stress on you getting to learn to use MATLAB. There are a lot of resources online to help you learn it .

Pertaining to your question , you can get the "goodness" measure or any such parameter about your data and the theoretical curve you want it to follow by using - the curve fitting module/app of MATLAB/Simulink. In that you can check the closeness of your data to the your desired model by ensuring SSE (sum of squared residuals) is nearly 0 (it can never be absolute 0 because of random factors contributing to the deviation in the data)

Does that help ?