Box Plot: Median, Quartiles, and Outliers

What is a box plot?

A box plot (box-and-whisker plot) displays the five-number summary of a dataset at a glance: minimum, Q1, median, Q3, maximum. It shows the center, spread, and skewness of data.

How to construct a box plot — step by step

Step 1 — Sort the data. Example: $\{2, 3, 4, 5, 5, 6, 7, 8, 100\}$.

Step 2 — Find the median (Q2). Middle value: $5$.

Step 3 — Find Q1 (median of lower half). Lower half: $\{2, 3, 4, 5\}$. Q1 $= (3+4)/2 = 3.5$.

Step 4 — Find Q3 (median of upper half). Upper half: $\{6, 7, 8, 100\}$. Q3 $= (7+8)/2 = 7.5$.

Step 5 — Compute the IQR. $\text{IQR} = Q3 - Q1 = 7.5 - 3.5 = 4$.

Step 6 — Find the fences (outlier boundaries): Lower fence $= Q1 - 1.5 \times \text{IQR} = 3.5 - 6 = -2.5$. Upper fence $= Q3 + 1.5 \times \text{IQR} = 7.5 + 6 = 13.5$.

Step 7 — Identify outliers. Any value below $-2.5$ or above $13.5$ is an outlier. Here: $100$ is an outlier!

Step 8 — Draw the box from Q1 to Q3, with a line at the median. Whiskers extend to the most extreme non-outlier values (2 and 8). Outliers are plotted as individual dots (100).

What box plots reveal

• Skewness: If the median line is closer to Q1 → right-skewed. Closer to Q3 → left-skewed.
• Spread: A wide box (large IQR) means more variability.
• Outliers: Dots beyond the whiskers flag unusual values.

Try it in the visualization

Toggle the outlier on/off. Watch how the box, whiskers, and median change. The data points animate into sorted order, then the box forms around the quartiles.