The chi-square statistic (chi being the Greek letter) is a test of statistical significance of a set of values or as a test for independence between two categorical variables in a table. You add up the difference between each value and the value expected under the null hypothesis, squared, and divide that by the expected value. Adding up all the terms yields the chi-square statistic, denoted by chi-squared. Chi-square is always >= zero. An inordinately high (or possibly an inordinately small1) value indicates statistical significance. Just how significant requires obtaining looking up or computing a P-value from a chi-square curve.

Example: A company accused of sex discrimination employs 630 men and 470 women. According to the null hypothesis, this difference is due to random variation, and the expected number of men and women would be 550. Chi-square would thus be (630 - 550)2/550 + (470-550)2/550 = 23.3 Looking this up on a chi-square curve with one degree of freedom gives a P-value between 1 and 2 percent. This provides strong evidence to reject the null hypothesis--we can say with better than 98% confidence that something is causing an imbalance of men (note that we haven't shown that they are biased against women, only that there really are too many men in the company).

1dogboy points out that only an inordinately high value indicates significance. This is generally true but it depends on what we are looking at. Sometimes an inordinately low number can indicate that there is too little chance error. If I have reported data for coin tosses, for instance, and there are supposedly 50,006 heads and 49,994, chi-square will be about .001. This is so low, that the p-value is about .99. This would be excellent evidence that the data was fabricated or massaged! This is precisely the technique used to show that Gregor Mendel certainly fudged his pea-plant experiments in genetics to fit his (correct) theory. But dogboy's well-written nitpicking is generally valid, though I think we were pretty much driving at the same thing.