Chi-Square Calculator


The chi-square statistic for an experiment with k possible outcomes, performed n times, in which Y1, Y2,... Yk are the number of experiments which resulted in each possible outcome, with probabilities of each outcome p1, p2,... pk is:

X2 will be larger to the extent that the observed results diverge from those expected by chance. The probability Q that a X2 value calculated for an experiment with d degrees of freedom (where d=k-1, one less the number of possible outcomes) is due to chance is:

Where Gamma is the generalisation of the factorial function to real and complex arguments:

Unfortunately, there is no closed form solution for Q, so it must be evaluated numerically. This page allows you to calculate the probability of chance occurrence of a given X2 for an experiment with d degrees of freedom, or to calculate X2 given d and the probability Q. All calculations are performed with six decimal places of accuracy; the maximum X2 accepted is thus 99999.

Note that the probability calculated from the X2 is an approximation which is valid only for large values of n, and is therefore only meaningful when calculated from a large number of independent experiments.

In order to use this page, your browser must support JavaScript. The text field below indicates whether JavaScript is available; if not, consider switching to a browser which implements it.


Calculate probability from X2 and d

One of the most common chi-square calculations is determining, given the measured X2 value for a set of experiments with a degree of freedom d, the probability of the result being due to chance. Enter the X2 and d values in the boxes below, press the Calculate button, and the probability will appear in the Q box.

Given X2= and d=

The chance probability, Q, is:


Calculate X2 from probability Q and d

To determine the chi-square value indicating a probability Q of non-chance occurrence for an experiment with d degrees of freedom, enter Q and d in the boxes below and press Calculate.
Given probability Q= and d=

The X2 value is:

Mathematical Details

Table of Contents

Run Experiments

RetroPsychoKinesis Project Home


by John Walker
kelvin@fourmilab.ch