Non-parametric tests — common mistakes

For the Wilcoxon rank-sum test, the tables in MF19 give the critical region for the rank sum of the smaller sample. Always calculate the rank sum for the smaller sample to use the tables directly. If you accidentally sum the ranks for the larger sample ($R_L$), you can find the sum for the smaller sample ($R_S$) using the formula $R_S + R_L = \frac{1}{2}N(N+1)$, where $N$ is the total sample size.

When using a normal approximation for a discrete test statistic like $W$ or $R$, you must apply a continuity correction. If you are calculating $P(W \le k)$, you use $k+0.5$. If you are calculating $P(W \ge k)$, you use $k-0.5$. This adjustment accounts for approximating a discrete distribution with a continuous one.

The signed-rank test is for a single sample (or paired data, where you test the median of the differences). It tests if the population median is a specific value. The rank-sum test is for two independent samples, and it tests whether the two populations they come from have the same median.

The median is a more 'robust' measure of central tendency than the mean. This means it is less affected by extreme values (outliers) or skewed data. Since non-parametric tests are designed for situations where the data might not be normally distributed, using the median is a more reliable approach.

If two or more observations are identical, they are each assigned the average of the ranks they would have occupied. For example, if two values are tied for the 5th and 6th positions, they both receive a rank of (5+6)/2 = 5.5. The next value then receives rank 7. The formulae for the variance in the large sample approximations should be adjusted for ties, but this is not typically required in the 9231 syllabus unless the formula is given.