Chi-Square Test for Goodness of Fit

Scientists often use the Chi-square (χ) test to determine the “goodness of fit” between theoretical and experimental data. In this test, we compare observed values with theoretical or expected values. Observed values are those that the researcher obtains empirically through direct observation. The theoretical or expected values are developed on the basis of an established theory or a working hypothesis. For example, we might expect that if we flip a coin 200 times, that we would tally 100 heads and 100 tails. In checking our hypothesis, we might find only 92 heads and 108 tails. Should we reject this coin as being fair? Should we just attribute the difference between expected and observed frequencies to random fluctuation? Consider a second example: let’s suppose that we have an unbiased, six-sided die. We roll this die 300 times and tally the number of times each side appears: