Analyze statistical distributions efficiently today. Precision matters.
Statistical analysis often requires verifying whether experimental count data adheres to a theoretical expectation. When working with discrete events occurring independently at a constant average rate, Poisson statistics provide the ideal framework. The chi-square test evaluates the goodness of fit between observed frequencies and expected theoretical values.
The standard Pearson's chi-square statistic with Poisson uncertainties is calculated using the following mathematical formulation:
$$\chi^2 = \sum_{i=1}^{n} \frac{(O_i - E_i)^2}{\sigma_i^2}$$
Where $O_i$ represents the observed count in bin $i$, $E_i$ denotes the expected model value, and $\sigma_i$ corresponds to the standard uncertainty derived from the Poisson distribution ($\sigma_i = \sqrt{O_i}$). The reduced chi-square ($\chi^2_\nu$) is derived by dividing the chi-square sum by the degrees of freedom $\nu = n - p$, where $n$ is the number of observations and $p$ represents fitted parameters.
What does a high reduced chi-square value indicate?
A reduced chi-square significantly greater than 1 suggests that the model does not adequately describe the data, or that uncertainties have been underestimated.
Why use Poisson errors instead of standard normal errors?
Poisson errors are uniquely tailored for countable event data where the variance equals the expected mean, ensuring high estimation accuracy.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.