The Art of Statistics: Learning from Data

by David Spiegelhalter

Often we make use of approximations that were developed by the pioneers of statistical inference. For example, around 1900 Karl Pearson developed a series of statistics for testing associations in cross-tabulations such as Table 10.1, out of which grew the classic chi-squared test of association.fn3 These test statistics involve calculating the expected number of events in each cell of the table, were the null hypothesis of no-association true, and then a chi-squared statistic measures the total discrepancy between the observed and expected counts. Table 10.2 shows the expected numbers in the cells of the table, assuming the null hypothesis: for example, the expected number of females with left arm on top is the total number of females (14), times the overall proportion of left-armers (22/54), which comes to 5.7. Female Male Total Left arm on top 5 (5.7) 17 (16.3) 22 Right arm on top 9 (8.3) 23 (23.7) 32 Total 14 40 54 Table 10.2 Observed and expected (in parentheses) counts of arm-crossing by gender: expected counts are calculated under the null hypothesis that arm-crossing is not associated with gender. It is clear from Table 10.2 that the observed and expected counts are fairly similar, reflecting that the data are just about what we would expect under the null hypothesis. The chi-squared statistic is an overall measure of the dissimilarity between the observed and expected counts (its formula is given in the Glossary), and has the value 0.02. The P-value correspondin...

Chi square for testing null hypothesis