## Chi-Squared Test

Let the probabilities of various classes in a distribution be , , ..., . The expected frequency

is a measure of the deviation of a sample from expectation. Karl Pearson proved that the limiting distribution of is (Kenney and Keeping 1951, pp. 114-116).

where is Pearson's Function. 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).

References

Cuzzi, J. The Subsurface Nature of Mercury and Mars from Thermal Microwave Emission. Ph.D. Thesis. Pasadena, CA: California Institute of Technology, 1972.

Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, 1951.