What does the size w of the distribution box represent

In a variable-width box plot, the size "w" represents the width of the box, which is proportional to the size of the data group being plotted.

In a standard box plot, the box typically represents the interquartile range (IQR), showing the middle 50% of the data, with a line marking the median and whiskers extending to the minimum and maximum values (excluding outliers) . However, in a variable-width box plot, the width of the box itself conveys additional information: it reflects the number of observations in that group .

How the Width is Determined

A common convention is to make the box width proportional to the square root of the sample size. This means that larger groups have wider boxes, while smaller groups have narrower boxes, allowing viewers to quickly assess the relative sizes of different groups in a dataset .

Purpose and Interpretation

• Visual comparison: When comparing multiple groups, the width "w" helps indicate which groups have more data points, providing context for the reliability of the median and spread.
• Statistical insight: Wider boxes suggest larger sample sizes, which generally lead to more stable estimates of the median and interquartile range.
• Consistency: The width should be applied consistently across all boxes in the plot to ensure accurate visual comparison . In summary, the size "w" in a distribution box is not about the data values themselves but about the size of the group or sample, enhancing the interpretability of box plots when comparing multiple datasets.