Let the probabilities of various classes in a distribution be 
, 
, ..., 
, with observed frequencies 
, 
, ..., 
. The quantity
 |
(1)
|
is therefore a measure of the deviation of a sample from expectation, where 
is the sample
size. Karl Pearson proved that the limiting distribution of 
is a chi-squared
distribution (Kenney and Keeping 1951, pp. 114-116).
The probability that the distribution assumes a value of 
greater than the measured value 
is then given by
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
There are some subtleties involved in using the 
test to fit curves (Kenney and Keeping 1951, pp. 118-119).
When fitting a one-parameter solution using 
, the best-fit parameter value can be found by calculating
at three points, plotting against
the parameter values of these points, then finding the minimum of a parabola
fit through the points (Cuzzi 1972, pp. 162-168).
