13.6 : Sign Test for Median of Single Population

In general, the sign test serves as a nonparametric method to test hypotheses about the median of a single population when the data does not follow a known distribution. This simplicity makes it particularly useful for small sample sizes or when the assumptions of parametric tests cannot be met. The process begins with identifying a null hypothesis, typically stating that the population median equals a specific value. The alternative hypothesis could be that the median is either not equal to, less than, or greater than the tested value, depending on the research question.

The sign test compares each data point to the proposed median under the null hypothesis and uses their differences to calculate the test statistics and obtain conclusions. With this, data points greater than the hypothesized median are labeled with positive signs, and the smaller ones are marked with negative signs. The test then focuses on the count of these signs, ignoring any data points that exactly match the median, as they do not contribute evidence towards either hypothesis.

Unlike other sign tests, such as those for paired samples or matched pairs (often used in before-and-after studies), the single-population sign test does not compare two groups or conditions but focuses solely on assessing the central tendency within a single group relative to a fixed value. This test is useful for determining if a single sample median deviates from a standard, providing a robust alternative when sample sizes are small or distributions are skewed. The test then evaluates the balance between positive and negative signs, which reflects on the median's true position relative to the hypothesized value. A significant imbalance suggests the sample data contradicts the null hypothesis, indicating an alternative median value for the population.