Box Plots
A box plot squeezes an entire data set into five numbers drawn over a number line: the minimum, , the median, , and the maximum. That's called the five-number summary, and once you can point to all five, every box plot question is a read-and-subtract problem.
The picture itself is simple: a box from to with a line inside it at the median, and two whiskers reaching out to the minimum and maximum. Each of the four pieces — whisker, half-box, half-box, whisker — holds about of the data.
The five-number summary
Reading left to right: the left whisker tip is the minimum, the left edge of the box is , the line inside the box is the median, the right edge of the box is , and the right whisker tip is the maximum.
The most common mistake is reading the median off an edge of the box. The median is the line inside the box — the edges are the quartiles and .
IQR and range from the picture
Two spread measures come straight off the plot. The range is the full stretch: maximum minimum, whisker tip to whisker tip. The IQR is the width of the box: .
The box holds the middle of the data. A short box means the middle half of the data is packed tightly together; a long box means it's spread out.
What a box plot can and can't tell you
A box plot shows center (the median), spread (range and IQR), and how the data leans — if the median line sits close to one edge of the box, the data bunches on that side.
It can't tell you individual values or even how many values are in the data set. Each section holds about a quarter of the data no matter how long or short it looks — a long whisker means that quarter is spread out, not that it holds more values.
Worked examples
Example 1: read the five-number summary
A box plot has whiskers ending at and , a box from to , and a line inside the box at . Write the five-number summary.
Answer: Minimum , , median , , maximum
Example 2: find the IQR
Using the same box plot — box from to — find the IQR.
Answer: The IQR is
Example 3: find the range
Using the same box plot — whiskers ending at and — find the range.
Answer: The range is
Try one yourself
Common questions
Is the line inside the box the mean or the median?
The median. A box plot is built entirely from position-based numbers — minimum, quartiles, median, maximum — and the mean isn't one of them. You can't find the mean from a box plot.
What percent of the data is inside the box?
About — the middle half, from to . Each whisker holds about more, which accounts for all of the data.
Does a longer whisker mean more data on that side?
No — this is the classic trap. Each whisker holds about a quarter of the data regardless of length. A long whisker means that quarter is stretched over a wide range of values, not that it contains more of them.
How do I compare two box plots drawn on the same scale?
Compare centers first: whichever median sits further right is the higher-scoring set overall. Then compare spreads: the narrower box (smaller IQR) is the more consistent set. Those two sentences answer almost every comparison question.
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