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Essentially, Pearson's formula translates qualitative data from a set of observations into a single number. Probability tables with corresponding numbers, with variances built in for different levels of significance and different degrees of freedom (the number of available data points used for the estimation/prediction of other data, the calculation of which in Chi Square analysis is provided for by another straightforward equation), provide the probability of dependence for any given Chi Square statistic.
The most simple example of a Chi Square test uses two populations and one variable of examination with a binary ("yes/no") set of possibilities. One example used is examining the high school graduation rate of students in a special program vs. The graduation rate of a control group of students not involved in the program (Lane 2010). If a grid is constructed to fill in data points, there would be two rows -- one for each population -- and two columns -- one recording the number of students who graduated per population, the other recording the number of students who did not (Lane 2010). Using Pearson's formula to develop the Chi Square statistic, the columns and the rows would each be added separately, yielding four different numbers. These numbers multiplied together become the denominator in the fraction (or decimal) that is the Chi Square statistic. The four original...
The resulting number -- dividing the denominator by the numerator -- is the Chi Square statistic (HWS 2010).
In order to utilize a Chi Square table to see the probability of dependence associated with the statistic, the degrees of freedom must be known. A simple way to derive this is to subtract one from the number of rows and one from the number of columns, then multiplying these two together. In a 2 x 2 grid, there is one degree of freedom. The formula becomes somewhat more complex with larger data sets, but the basics stay the same; the Chi Square analysis can be used to measure populations with many differences in just the same way.
References
HSW. (2010). "The Chi Square Statsitic." Hobart and William Smith College. Accessed 26 February 2010. http://math.hws.edu/javamath/ryan/ChiSquare.html
Lane, D. (2010). "Introduction to the Chi Square Test of Independence." Accessed 26 February 2010. http://davidmlane.com/hyperstat/B143466.html
Plackett, R.L. (2010). "Karl Pearson and the Chi Squared Test." International statistical review 51, pp. 59-72.
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