How Violin Plots Expose the Hidden Bimodality That Summary Statistics Erase

The binding constraint for summary statistics—metrics like the mean, variance, and correlation coefficient—is that the underlying data must be unimodal and roughly symmetric. When that condition holds, a handful of numbers can perfectly describe a population's behavior. But when it fails, those exact same numbers become a mathematical illusion, hiding multimodal peaks, extreme skew, or clustered subgroups. In modern data analysis, this constraint is routinely violated, making visualization the only reliable diagnostic tool to determine if the statistics are lying. Without a visual check, analysts risk building models on data that looks normal in a spreadsheet but behaves erratically in reality.[7]

The most famous proof of this failure originated in 1973 with statistician Francis Anscombe, who published Anscombe's Quartet to demonstrate how four wildly different datasets could share identical statistical properties. This lesson was updated for modern data science in 2016 when designer Alberto Cairo created a dataset of 142 observations. Cairo's dataset possessed standard, unremarkable summary statistics, but when plotted on a scatterplot, the points formed the distinct shape of a Tyrannosaurus rex. Inspired by this visual trick, Autodesk researchers Justin Matejka and George Fitzmaurice published the 'Datasaurus Dozen' in 2017, taking the concept to its mathematical extreme.[1][2]

The Autodesk team used an optimization algorithm called simulated annealing to create 12 additional datasets that matched the dinosaur's metrics. As they wrote in their 2017 paper, 'new datasets are iteratively generated from a seed dataset through random perturbations of individual data points.' The resulting 13 datasets share identical summary statistics to two decimal places: the mean of the x-variable is exactly 54.26, the mean of the y-variable is 47.83, and the correlation between them is −0.06. The linear regression coefficient of determination ($R^2$) is a flat 0.004 across the board, and the sample variance for x is locked at 16.76.[1][2]

Relying on these numbers alone, a machine learning model or a statistical summary would treat all 13 datasets as perfectly identical. Yet when plotted visually, one dataset forms a five-pointed star, another forms a bullseye, and a third forms parallel horizontal lines. The summary statistics erase the geometry of the data entirely. To solve this masking problem in one-dimensional continuous data, analysts increasingly rely on Kernel Density Estimation (KDE) and violin plots to visualize the true shape of a distribution before applying any predictive modeling.[1][2][4]

A traditional box plot only displays the five-number summary—the minimum, first quartile, median, third quartile, and maximum. Because it relies entirely on these summary metrics, a box plot cannot reveal if data is clustered into multiple peaks; a bimodal distribution looks exactly like a normal distribution with a wide variance. Enter the violin plot. By mirroring a density curve on both sides of a central axis, it shows exactly where the values cluster. As researchers at Orange Data Mining explain, 'where the violin is fatter, there are more data points in the neighborhood. And where it is thinner, there are less.'[3][6]

The violin plot achieves this clarity by applying Kernel Density Estimation. KDE works by placing a smooth curve, known as a kernel function (typically a Gaussian bell curve), over every single data point and summing them up to create a continuous probability density function. Instead of slicing data into rigid, arbitrary bins like a traditional histogram, KDE provides a fluid estimate of where values are most concentrated. This mathematical smoothing easily exposes bimodal or multimodal distributions that would otherwise remain hidden inside a standard standard deviation metric.[3][4]

However, KDE introduces its own binding constraint: the bandwidth parameter, often denoted mathematically as $h$. The bandwidth controls the width of the individual kernel functions placed over each of the 142 data points. If the bandwidth is set too small, the resulting density plot is severely undersmoothed. It appears as a bumpy, noisy series of individual spikes that overfit the specific sample points, failing to generalize the broader trend and creating false peaks that do not actually exist in the wider population.[4][5]

Conversely, if the bandwidth is too large, the plot becomes oversmoothed, which is the most dangerous failure mode in exploratory data analysis. An oversmoothed KDE will merge distinct peaks together, forcing a bimodal distribution to look like a wide, smooth unimodal bell curve. In doing so, a poorly tuned KDE recreates the exact problem it was meant to solve: it hides the true shape of the data behind an algorithmic assumption, masking the very multimodality the analyst was trying to find.[5]

This creates a paradox in evidence-based data analysis. Summary statistics are perfectly objective but easily fooled by geometry, while KDE reveals geometry but relies on a subjective bandwidth parameter. While algorithms like Scott’s rule or Silverman’s rule attempt to automate bandwidth selection, the evidence demonstrates that neither summary statistics nor automated density plots can be trusted in isolation. The only robust approach is to plot the raw data points alongside the density estimation, testing multiple bandwidths to ensure the visual shape is a property of the data, not an artifact of the smoothing parameter.[4][7]