Box plots and violin plots are powerful tools for visualizing univariate data. They help us understand the spread, central tendency, and shape of distributions. These plots are essential for comparing groups and spotting outliers in datasets.

Both plot types offer unique insights into data. Box plots are great for quick summaries and outlier detection. Violin plots show the full distribution shape, revealing hidden patterns. Choosing between them depends on your data and what you want to highlight.

Box plots for univariate data

Definition and components

• Box plots, also known as box-and-whisker plots, display the distribution of univariate data based on five key statistics: minimum, first quartile, median, third quartile, and maximum
• The box represents the interquartile range (IQR), which contains the middle 50% of the data (Q1 to Q3)
• The line inside the box represents the median, the middle value of the dataset when arranged in ascending or descending order
• Whiskers extend from the box to the minimum and maximum values within 1.5 times the IQR
• Points beyond the whiskers are considered outliers and plotted individually

Advantages and use cases

• Box plots provide a quick visual summary of the center (median), spread (IQR), symmetry of a distribution, and presence of outliers
• They are particularly useful for comparing multiple distributions side-by-side (different treatment groups in a medical study)
• Box plots are more concise than histograms or density plots, making them suitable for small datasets or when simplicity is desired
• They are commonly used in exploratory data analysis and for identifying potential issues in the data (extreme outliers)

Interpreting box plot statistics

Five-number summary

• To create a box plot, calculate the five-number summary: minimum, first quartile (Q1), median, third quartile (Q3), and maximum
• Determine the IQR by subtracting Q1 from Q3 ($IQR = Q3 - Q1$)
• Identify outliers as data points below $Q1 - 1.5 × IQR$ or above $Q3 + 1.5 × IQR$

Drawing and interpreting box plots

• Draw a box from Q1 to Q3, with a line inside representing the median
• Extend whiskers from the box to the minimum and maximum values, excluding outliers
• Plot outliers individually as points beyond the whiskers
• Interpret the box plot by examining the median position (central tendency), box size (IQR, variability), whisker lengths (range), and outliers (extreme values)
• Compare multiple box plots to assess differences in distribution shape, center, and spread between groups (test scores of different classes)

Violin plots for data density

Definition and components

• Violin plots combine a box plot and a kernel density plot, displaying both summary statistics and the probability density of the data
• The width of the violin plot at any point represents the density of data at that value, providing a more detailed view of the distribution shape compared to box plots
• The symmetry or asymmetry of a violin plot indicates the skewness of the distribution (asymmetric plots suggest a skewed distribution)
• Overlaying a box plot within the violin plot combines the advantages of both techniques, showing summary statistics and the full data distribution

Advantages and use cases

• Violin plots are particularly useful for comparing the distribution of data across multiple categories or groups (income distribution by country), as they reveal differences in peaks, valleys, and bumps
• They provide more information about the distribution shape than box plots, especially for large datasets or complex distributions
• Violin plots are effective for visualizing multimodal distributions (exam scores with distinct peaks for different grade clusters)
• They can be used to identify potential subgroups or patterns within the data (bimodal distribution of heights suggesting gender differences)

Customizing violin plots

Kernel density estimation (KDE)

• To create a violin plot, compute the kernel density estimate (KDE) for the data, a non-parametric way to estimate the probability density function
• Mirror the KDE along the vertical axis to create the symmetric violin shape
• Adjust the bandwidth of the KDE to control the smoothness of the distribution curve (smaller bandwidth results in a more detailed but potentially noisier plot)

Enhancing violin plots

• Add a box plot inside the violin plot to display summary statistics, such as median, quartiles, and outliers
• When comparing multiple distributions, align the violin plots vertically or horizontally to facilitate visual comparison of distribution shapes and summary statistics
• Use different colors, transparency, or split violin plots to distinguish between categories or groups and highlight differences in distribution characteristics (gender differences in age distribution)
• Customize the appearance of the violin plots (line width, fill color) to improve readability and aesthetics

Box plots vs Violin plots

Choosing the appropriate plot

• Box plots are best suited for displaying summary statistics and identifying outliers in a concise manner, especially when comparing multiple distributions
• Violin plots are more informative when the goal is to visualize the entire distribution shape, including peaks, valleys, and bumps, and to compare the density of data across different values
• For large datasets or complex distributions, violin plots may be preferred over box plots to provide a more detailed representation of the data
• When the focus is on summary statistics and outliers, or when simplicity is desired, box plots may be more appropriate than violin plots

Audience and purpose considerations

• Consider the audience and purpose of the visualization when choosing between box plots and violin plots
• Box plots are more commonly used and easily interpretable, while violin plots may require more explanation but offer a richer understanding of the data distribution
• In some cases, combining box plots and violin plots can provide a comprehensive summary of both distribution shape and key statistics, catering to a wider range of communication goals (presenting both summary statistics and distribution density)
• Tailor the choice of plot to the specific data properties, research questions, and intended message of the visualization (comparing means vs. exploring distribution differences)