## Degrees of Freedom: A Correction to Chi Square For Physical Hypotheses

**Authors:** John Michael Williams

In common practice, degrees of freedom (df) may be corrected for the number of
theoretical free parameters as though parameters were the same as data categories.
However, a free physical parameter generally is not equivalent to a data category in
terms of goodness of the fit.
Here we use synthetic, nonrandom data to show the effect of choice of
categorization and df on goodness of fit. We then explain the origin of the df
problem and show how to avoid it in a three-step process:
First, the theoretical curve is fit to the data to remove its
variance, leaving what, under the null hypothesis, should be
structureless residuals.
Second, the residuals are fit by a set of orthogonal polynomials up
to the degree, should it occur, at which significant variance was
removed.
Third, the number of nonsignificant polynomial terms in the
original + orthogonal set become the df in a standard chi square
test.
This process reduces a general df problem to one of polynomial df and allows
goodness of a fit to be determined by data categorization and significance level
alone. An example is given of an evaluation of physical data on neutrino
oscillation.

**Comments:** 47 Pages.

**Download:** **PDF**

### Submission history

[v1] 16 Aug 2010

**Unique-IP document downloads:** 2004 times

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