How to express the agreement between experiment and theoretical observations?

[Arman777]

Let us suppose I have a value measured from experiment and given by
$$V_{\text{exp}} \pm \sigma_{V_{\text{exp}}}$$ and a theoretical value given as
$$V_{\text{the}} \pm \sigma_{V_{\text{the}}}$$

Is there a statistical way to measure how well $V_{\text{the}}$ matches with the $V_{\text{exp}}$.

In other words, what is the right way to tell that $V_{\text{the}}$ is a valid theory (or not) for the given experimental result?

It seems to be that we should take the difference,

$$ (V_{\text{exp}} \pm \sigma_{V_{\text{exp}}})- (V_{\text{the}} \pm \sigma_{V_{\text{the}}})$$

and that is $$(V_{\text{exp}} - V_{\text{the}}) \pm \sqrt{\sigma_{V_{\text{exp}}}^2 + \sigma_{V_{\text{the}}}^2} \equiv \Delta V \pm \sigma_{\Delta V}$$

If $$\Delta V - \sigma_{\Delta V} < 0 < \Delta V + \sigma_{\Delta V}$$ we say that the theory is valid I guess. But is a there a measure of how valid...like at which $\sigma$ ?

I guess it is $$\frac{\sigma_{\Delta V}}{\Delta V}$$, but I am not sure. Any help would be appreciated.

 

[FactChecker]

Given a sample, it is common to determine "confidence intervals" at particular levels (90%, 95%, 97.5%, 99%, etc.) for the distribution mean from the sample mean and variance. If your theoretical mean value is in a particular confidence interval (say 95%), then it should not be rejected on the basis of that sample. You can say that the true mean is in that interval with 95% confidence.

 

[Office_Shredder]

What does it mean for your theoretical prediction to have noise?

 

[FactChecker]

You might consider the Chi-squared goodness of fit test that compares a theoretical distribution to a sample set to give you the probability that the sample might have come from that theoretical distribution. But the requirements for that test are more than the sample mean and variance.

 

[Hornbein]

I'd like to know how these numbers were arrived at. But for now let's assume that we aren't concerned about sample sizes, bias, and so forth. You are doing fine until you give a test. Instead...

Chose the level of confidence you desire. This means how sure you want to be that the difference isn't due to chance. Divide delta V by sigma delta V. (I forget the name of this standardized value. p-score?) Look it up in a table and see whether or not it is past your chosen confidence level. If the score exceeds your chosen confidence level then you can declare your belief that the experiment and theory do not match. If not, you can say the experiment has not been shown to be inconsistent with the theory. (You've "failed to reject the null hypothesis.")

There is a subtlety about it being a one-sided or two-sided test that I'm going to disregard.

 

[Hornbein]

I'd like to know how these numbers were arrived at. But for now let's assume that we aren't concerned about sample sizes, bias, and so forth. You are doing fine until you give a test. Instead...

Chose the level of confidence you desire. This means how sure you want to be that the difference isn't due to chance. Divide delta V by sigma delta V. (I forget the name of this standardized value. p-score?) Look it up in a table and see whether or not it is past your chosen confidence level. If the score exceeds your chosen confidence level then you can declare your belief that the experiment and theory do not match. If not, you can say the experiment has not been shown to be inconsistent with the theory. (You've "failed to reject the null hypothesis.")

There is a subtlety about it being a one-sided or two-sided test that I'm going to disregard.

It's the z-score.

 

[Arman777]

It's the z-score.

I have learned at undergrad but know I have completely forget about it..