A box plot (box-and-whisker plot) displays the distribution of data using a five-number summary: minimum, Q1, median, Q3, and maximum. It helps identify outliers and compare groups visually.

What is the difference between a box plot and a violin plot?

A box plot shows summary statistics only. A violin plot shows the full density distribution shape using kernel density estimation, revealing multi-modal patterns that box plots hide.

How do you detect outliers?

Outliers are detected using the Tukey fences method: values below Q1 − 1.5×IQR or above Q3 + 1.5×IQR are flagged. The Z-score method also identifies values more than 3 standard deviations from the mean.

Learn Statistics & Data Visualization – Box Plots, Violin Plots & More

What is a Box Plot?

A box plot (also called a box-and-whisker plot) is a standardized way of displaying the distribution of data based on a five-number summary: minimum, first quartile (Q1), median, third quartile (Q3), and maximum.

Invented by John Tukey in the 1970s, the box plot is built on the five-number summary:

Statistic	Description
Minimum (Min)	The smallest data point, excluding outliers
First Quartile (Q1)	25th percentile — 25% of data falls below Q1
Median (Q2)	50th percentile — the middle value of the sorted data
Third Quartile (Q3)	75th percentile — 75% of data falls below Q3
Maximum (Max)	The largest data point, excluding outliers
IQR	Q3 − Q1 — the spread of the middle 50% of data
Whiskers	Lines from the box extending to min and max (non-outlier range)
Outliers	Points beyond 1.5 × IQR from Q1 or Q3 (Tukey fences)

How to Read a Box Plot

Step by Step

Locate the median line inside the box — this is the middle value of your dataset.
Look at the box width (IQR) — a wider box means more variability in the middle 50% of data.
Check whisker lengths — asymmetric whiskers indicate skewness.
Identify outliers as dots beyond the whiskers.
Compare boxes when multiple datasets are plotted — look at median shifts and IQR changes.

What the Shape Tells You

Symmetric: Median centered in box, whiskers roughly equal length
Right-skewed: Median closer to Q1, upper whisker longer
Left-skewed: Median closer to Q3, lower whisker longer
Wide IQR: High variability in the middle 50%
Narrow IQR: Data tightly clustered around the median

Box Plot vs Violin Plot

Key Differences

Feature	Box Plot	Violin Plot
Display	Summary statistics	Full distribution shape
Density	Not shown	KDE curve on each side
Multi-modality	Cannot show (only hints)	Reveals multiple peaks clearly
Outliers	Marked explicitly	Visible in tails
Best for	Quick comparisons	Distribution analysis
Space	Compact	Wider

When to Use Each

Use a box plot when you need a clean, compact summary and want to compare many groups at once.
Use a violin plot when you need to see the full distribution shape, especially to spot bimodal or multi-modal patterns.

Outlier Detection Methods

IQR Method (Tukey Fences)

The most common method. Any data point outside [Q1 − 1.5×IQR, Q3 + 1.5×IQR] is flagged as an outlier.

Z-Score Method

Values with a Z-score greater than 3 (or less than −3) are considered outliers. This assumes the data is approximately normally distributed.

Which Method to Use?

IQR: Robust, non-parametric. Works well for skewed data.
Z-score: Works best for normally distributed data.
Use both for a comprehensive analysis.

Statistics Glossary

Mean: The arithmetic average: sum of all values divided by count.
Median: The middle value when data is sorted. 50th percentile.
Mode: The most frequently occurring value in a dataset.
Quartile: Values that divide sorted data into four equal parts (Q1, Q2, Q3).
IQR (Interquartile Range): Q3 minus Q1. Measures the spread of the middle 50%.
Standard Deviation: Square root of the average squared deviation from the mean. Most common measure of spread.
Variance: The average of squared deviations from the mean (σ²). Standard deviation squared.
Skewness: Measures asymmetry. Positive = tail to right, negative = tail to left.
Kurtosis: Measures 'tailedness'. High kurtosis = heavy tails, more outliers.
Percentile: The value below which a given percentage of observations fall.
KDE (Kernel Density Estimation): A non-parametric way to estimate the probability density function of a variable.
Correlation: A measure of the linear relationship between two variables (ranges from -1 to +1).
Tukey Fences: Boundary for outlier detection: Q1−1.5×IQR and Q3+1.5×IQR.
Normal Distribution: A symmetric bell-shaped distribution defined by mean and standard deviation.