1
$\begingroup$

What is the merit of $\chi^2$ analysis in the laboratory experiments? Why is so much emphasis put on it?

$\endgroup$
3
$\begingroup$

To test a model of the form $y=f_a(x)$ where $a$ parameterises a function in a family, finding best-fit $a$ from experimental data, we don't try to solve simultaneous equations $y_i = f_a (x_i)$ by assuming every data point is exact. Measurement error, random noise etc. means that's physically unrealistic, and you'd almost certainly find the equations are inconsistent anyway. Instead the data point $(x_i, y_i)$ satisfies $y_i = f_a (x_i)+\epsilon_i$.

In theory, one can find some collective measure of how large the error terms $\epsilon_i$ are, to assess (i) how to minimise it with the choice of $a$ and (ii) whether the result is good enough to believe the model. There are theoretical reasons to expect the $\epsilon_i$ to have mean-$0$ Normal distributions, and if we think we know the standard deviation is $\sigma$ then for $n$ data points we have $\sigma^{-2}\sum_i\epsilon_i^2\sim\chi_n^2$. The great thing about knowing this quadratic function's distribution is we can now express the empirical result's "weirdness" as a $p$-value.

Why do we use this function? Because it's easy. You can't use $\sum_i\epsilon_i$, because negative errors need to penalised rather than rewarded. Nor do we use $\sum_i|\epsilon_i|$, because working out the distribution of that is a lot harder. What's more, for many popular problems it's easy to compute which $a$ minimises $\sum_i\epsilon_i^2$. For example, any $f_a$ of the form $\sum_j a_j f_j(x)$ reduces this problem to matrix inversion.

$\endgroup$
  • $\begingroup$ "To test a model of the form $y=f_a(x)$ where $a$ parameterises a function in a family" By $y=f(x)$ do you mean a prediction of some model? Some kind of correlation between two physical observables $x$ and $y$? But what does $a$ stand here for? Can you give a simple physical example? $\endgroup$ – mithusengupta123 Sep 20 '18 at 18:15
  • $\begingroup$ @mithusengrupta123 The values of $x,\,y$ are empirical. We try to estimate a parameter $a$ on which $f$ depends, by minimising the errors. $\endgroup$ – J.G. Sep 20 '18 at 18:35

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.