The Sign Test and Median-Based Inference

Testing whether a strategy's daily returns have a median of zero doesn't require knowing anything about the shape of the return distribution — no normality, no symmetry, nothing beyond each day's return being drawn independently. The sign test exploits this by throwing away almost all the information in each observation and keeping only its sign relative to the hypothesized median: does each return fall above or below zero? Under the null hypothesis that the true median is zero, each observation is above or below with equal probability, exactly like a fair coin flip, so the count of "above" observations out of $n$ total follows a simple binomial(n, 0.5) distribution. A result far from the expected roughly 50/50 split — say 80 positive days out of 100 — is strong evidence the true median isn't zero.

Because it uses only signs and not magnitudes, the sign test is far less powerful than a test that uses the actual values (like a t-test, when its assumptions hold), meaning it needs more data to detect a real effect of a given size. Its advantage is robustness: it works even when returns are wildly skewed, fat-tailed, or contain extreme outliers that would distort a mean-based test, because a single huge outlier still only counts as one "above" or "below" sign, not an outsized value that drags the whole test off course.