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When you carry out an experimental measurement, you generally acquire a series of data points ( ). The value of each will usually deviate from its expected value ( ), because of measurement noise or other nefarious reasons. Often, it is assumed that may be approximated as a Gaussian random variable (GRV) with mean and some standard deviation  - a.k.a. the measurement error.
When counting the number of times that independent events occur within some range, we expect a Poisson distribution of counts. In this case, the error in  counts is  . In other examples, that do not involve counting, the error will be different. Be aware that a Poisson distribution is not Gaussian. Nevertheless, for more than a few counts, a Poisson distribution is well-approximated by a Gausssian.
Least mean-squares fitting generally seeks to minimize ("chi-squared'')
 (1)
where , , and are the experimental value, the error, and the theoretical value, respectively, of data point  , and  is the total number of data points. Usually, is a function of the fitting parameters, which are varied to minimize  corresponding to the best fit.
Just as is a random variable, so is . When  can be considered a GRV, it turns out that  can be shown to have a probability density
 (2)
where  and  is the number of fitting parameters.
Thus, we can answer the question: what is the probability ( ), that exceeds the observed value ( )? Specifically, we have
 (3)
where  is the Euler gamma function and  is the incomplete gamma function, both of which Mathematica can calculate for you. The idea, of course, is that if it turns out that there is only a small probability of finding a value of greater than or less than the observed value ( ), why then you probably should be worried, either about whether your theoretical model is in fact appropriate, or the quality of your measurements. This assumes that you have appropriate errors. The most common reason for funky values of is incorrect errors.
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