This situation came up in Rebecca’s class last Wednesday. Some students had gathered data that, looked something like this made up version:
This shows for each subject, their RTs on two conditions of the experiment (Green/Blue). If you throw this sucker into the paired t-test, you find that the difference between conditions is, in fact, not significant (t = -1.8994, df = 10, p > 0.08), even though the difference in means is large (-1.28), and every subject shows the correct trend. But the signed-rank test comes out (p < 0.003).
As Talia pointed out, there’s one subject, the first one, which has a much bigger effect size than the others, and the t-test takes this into account. And, if you remove this first subject, the T-test comes out significant (t = -5.8054, df = 9, p < 0.0005). This is because the t statistic takes cares about the variance in the differences (for each subject) between conditions. If you leave in the first subject, this variance is large and the t statistic decreases in magnitude.
So adding a data point with the correct trend can actually decrease your t statistic, and make this difference nonsignificant. Why is this counterintuitive? Because the first point is a much bigger effect than the others in the right direction! And removing it actually decreases the mean difference between conditions (to -0.62).
Beware!