123 Median Test based on Notched Boxplots
The Notched Boxplots have been discussed in Chapter 69. Hence, we only discuss the underlying assumptions.
123.1 Hypotheses
For two independent groups, the visual-notched approach is used as a quick screening tool for:
\[ \begin{cases}\text{H}_0: \text{the group medians are equal} \\\text{H}_A: \text{the group medians are different}\end{cases} \]
Unlike formal tests (e.g. Mann-Whitney), this is an approximate graphical rule.
123.2 Assumptions
The notches of the Notched Boxplot represent 95% confidence intervals for the Median (McGill, Tukey, and Larsen 1978). The lower and upper bounds are computed as follows
\[ M_x \pm 1.57 \times \frac{IQR}{\sqrt{n}} \]
where \(IQR\) is the Interquartile Range and \(n\) is the number of observations.
The formula for the confidence interval of the Median depends on the underlying assumption that the two data sets are:
- independent
- identically distributed random samples from two populations with unknown medians but with a normal distributional shape in the central position
In practice, this implies that the Notched Boxplot approach is more robust (i.e. less sensitive to outliers) than the traditional t-Test. On the other hand, it will (just like the t-Test) not provide reliable information when the distributions are extremely skewed.
Also note that the empirical density of the median (as obtained through bootstrapping) often turns out to be multimodal. This can make simple parametric approximations difficult, but exact sampling distributions for medians (order statistics) are well established.
123.3 Decision Rule (Graphical Approximation)
- If notches do not overlap, this is evidence that medians differ.
- If notches overlap, there is insufficient evidence (at this level of approximation) to claim different medians.
For reporting in academic work, use this as exploratory evidence and confirm with a formal inferential test.