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The bixplot: A variation on the boxplot suited for bimodal data

This paper introduces the bixplot, a new visualization tool that extends the traditional boxplot by incorporating a univariate clustering method to effectively detect and display bimodal or multimodal data structures, thereby facilitating the identification of meaningful subgroups and isolated points in univariate datasets.

Original authors: Camille M. Montalcini, Peter J. Rousseeuw

Published 2026-05-05

📖 5 min read🧠 Deep dive

CC0 1.0

Original authors: Camille M. Montalcini, Peter J. Rousseeuw

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to understand a crowd of people. You want to know: Are they all just one big, mixed-up group? Or are there distinct subgroups hiding in plain sight?

For decades, statisticians have used a tool called the boxplot to take a quick look at data. Think of a boxplot as a simple summary card. It tells you the "middle" of the crowd, how spread out they are, and if there are any weird outliers (people standing far away from the group). It's great, but it has a blind spot: it assumes everyone belongs to one single, smooth group. It's like looking at a crowd and only seeing one big blob, even if there are actually two distinct groups of people standing back-to-back.

This paper introduces a new tool called the bixplot (a clever mix of "boxplot" and "mix"). It's designed specifically to catch those hidden subgroups, especially when the data has two or more "peaks" (bimodal or multimodal).

Here is how the bixplot works, using simple analogies:

1. The "Sniff Test" (Checking for One or Many Groups)

Before drawing anything, the bixplot takes a deep "sniff" of the data using a statistical test called Hartigan's dip test.

If the data smells like one group: It draws a standard-looking picture that combines the best parts of old tools: a box (for the summary), a smooth curve (like a violin shape to show density), and a "rug" (little lines at the bottom showing every single data point).
If the data smells like a mix: It knows something is up. It realizes the crowd isn't uniform.

2. The "Smart Sorter" (Clustering)

If the data is a mix, the bixplot doesn't just guess; it uses a special clustering algorithm to sort the data into separate, neat piles.

The "No-Inside" Rule: Imagine you are sorting marbles into jars. A normal sorter might put a red marble inside a jar of blue marbles just because it's close. The bixplot's sorter is stricter: it ensures that no jar has a marble "inside" another jar's territory. The groups must be distinct, side-by-side blocks.
The "Minimum Size" Rule: It also refuses to make tiny, lonely jars. If a group is too small (like just one or two data points), it gets merged back in. This prevents the tool from seeing "ghost groups" that are just random noise.

3. The Final Picture

Once the data is sorted, the bixplot draws a new kind of map:

Separate Bodies: Instead of one big shape, you see separate "bodies" for each group, often in different colors.
Density Curves: The width of these bodies shows how crowded that group is.
The Rug: At the bottom, you see little lines for every single data point. This is crucial because it lets you see exactly where the points fall, including any isolated ones that might be hiding between the groups.
Color Coding: You can even color these little lines based on other information (like the sex of a penguin or the species of a flower), helping you see how different variables relate to the groups.

Real-World Examples from the Paper

The authors tested this on real data to show how it finds things other tools miss:

Fish: They looked at how long fish waited before exploring. The old tools hinted at two groups, but the bixplot clearly showed two distinct "waiting styles."
Penguins: They measured the beak lengths of penguins from different islands. On one specific island, the bixplot revealed that the penguins weren't just one group; there were actually two distinct size groups (likely males and females), which the old boxplots blended together.
Flowers (Iris): When looking at petal widths, the bixplot suggested there might be three distinct groups, helping to separate the flower species more clearly than before.

Why It Matters

The authors emphasize that the bixplot is an exploratory tool, not a final verdict. It's like a flashlight in a dark room. It helps you see that there might be three distinct groups of furniture in the room, but it doesn't prove why they are there or if they are definitely three separate categories in the real world. It just says, "Hey, look here, the data looks like it has two or three peaks, let's investigate further."

The Bottom Line

The bixplot is a smarter, more flexible version of the classic boxplot. It combines the summary power of the box, the shape of the violin plot, and the detail of the rug plot. Its superpower is that it automatically detects when data is a "mixture" of different groups and separates them visually, so you don't miss the hidden stories in your data.

The authors have made this tool available for free in both Python and R, so anyone can start using it to "unmix" their data.

Technical Summary: The Bixplot

Problem Statement
The authors identify a significant limitation in the standard boxplot, a cornerstone of Exploratory Data Analysis (EDA) introduced by Tukey (1977). While effective for unimodal distributions, the boxplot implicitly assumes unimodality and cannot visually represent bimodal or multimodal data. Existing extensions, such as the vaseplot (Benjamini, 1988), violin plot (Hintze and Nelson, 1998), and beanplot (Kampstra, 2008), improve upon the boxplot by displaying density curves or individual data points. However, these methods do not explicitly detect or separate distinct modes. Consequently, summary statistics within these plots (e.g., median and quartiles) may be misleading if the data arises from a mixture of subpopulations, and the visual representation may fail to clearly distinguish between overlapping clusters.

Methodology
The paper proposes the bixplot, a visualization tool designed to detect and display unimodal, bimodal, and multimodal structures in univariate data. The construction of a bixplot involves a specific algorithmic pipeline:

Unimodality Testing:
- The algorithm first assesses the number of non-missing values ( $n$ ). If $n$ is small (specifically $n < 2 \times \text{minN}$ , where $\text{minN}=15$ by default) or if the user forces a single cluster ( $k_{max}=1$ ), the data is treated as unimodal.
- For larger datasets, Hartigan's dip test (Hartigan and Hartigan, 1985) is applied to test the null hypothesis of unimodality. The test computes the maximum vertical distance between the empirical distribution function and the closest unimodal distribution. If the $p$ -value is below a significance level (default 0.01), unimodality is rejected.
Constrained Univariate Clustering:
- If the data is deemed non-unimodal, a specialized clustering algorithm is employed to separate the data into contiguous clusters (intervals where no cluster members are nested inside another).
- The method utilizes $k$ -medoids clustering (Partitioning Around Medoids, PAM) to select actual data points as cluster centers.
- Crucially, the authors implement a constrained version of $k$ -medoids. To ensure clusters are meaningful and support density estimation, the algorithm enforces a constraint that each cluster must contain at least a minimum number of unique values ( $\text{clusMinN}$ , default 3). This prevents isolated points or tied values from forming spurious clusters. The algorithm uses a transport method to reassign points iteratively, minimizing the total deviation while satisfying the constraints.
Cluster Selection:
- The algorithm evaluates cluster counts ( $k$ ) from 2 up to a user-defined maximum ( $k_{max}$ , default 7).
- The optimal number of clusters is selected based on the highest silhouette score (Rousseeuw, 1987), which measures how well points fit within their assigned clusters compared to other clusters.
Visualization Construction:
- Unimodal Data: Displays a standard boxplot superimposed on a kernel density estimate (similar to a violin plot) and a rug plot (beanplot style) showing individual data points.
- Multimodal Data: The data is split into the identified clusters. Each cluster is displayed as a separate "body" with its own density curve, boxplot, and rug plot.
- Sizing Options: The area of the cluster bodies can be proportional to the cluster size, equalized across clusters, or set to equal width.
- Coloring: Rug lines can be colored based on continuous or categorical external variables to reveal relationships.

Key Contributions

The Bixplot Visualization: A new graphical display that combines the boxplot, density trace (violin), and rug plot (bean), specifically adapted to handle multimodal data by separating it into distinct, contiguous clusters.
Constrained $k$ -medoids Algorithm: A novel adaptation of the $k$ -medoids method that enforces a minimum number of unique values per cluster and ensures cluster contiguity, specifically designed for the purpose of identifying modes in univariate data.
Software Implementation: The authors provide robust implementations in both Python (package bixplot) and R (function bixplot() in the classmap package). These implementations are designed with practical defaults to allow for short function calls while offering extensive customization options for colors, line widths, and sizing.

Results and Applications
The paper illustrates the utility of the bixplot through several real-world datasets:

Fish Behavior: Detected bimodality in latency times across experimental rounds, showing shifts in mode centers.
Palmer Penguins: Revealed bimodality in bill lengths for penguins on Dream Island (distinguishing subgroups not visible in standard violin plots) and allowed for direct comparison of sexes within islands using "half" bixplots.
Iris Data: Identified that petal length and width contain more structure (bimodal/multimodal) than sepal dimensions, aiding in the separation of species without prior knowledge of class labels.
Kingfisher Data: Demonstrated the ability to handle complex multimodal data (7 clusters in wing length), though noting that visual distinction becomes difficult with high numbers of clusters.
Color Mapping: Successfully visualized relationships between rug plots and external variables (e.g., body mass in penguins, species in iris data).

Significance and Claims
The authors position the bixplot strictly as an exploratory tool intended to assist user intuition and reveal potential subgroups within data. They explicitly state that the bixplot is not a confirmatory method; the identification of clusters (e.g., three clusters in a variable) is an indication of structure, not a statistical confirmation of the population's exact composition. The method avoids probabilistic assumptions (such as Gaussianity) required by inferential mixture models, making it suitable for a broader range of data distributions. The primary significance lies in its ability to let the data "speak" regarding modality, addressing a specific gap in existing EDA tools where standard boxplots fail and density plots lack explicit cluster separation.