Box and Whisker Plot

A box and whisker plot compresses a whole data set into five landmarks. Built from the five-number summary, it lets you see center, spread, and skew in a single glance without scanning a long list of values.

The Five-Number Summary And IQR

A box and whisker plot is built from five values: the minimum, the first quartile $Q_1$, the median, the third quartile $Q_3$, and the maximum. The box runs from $Q_1$ to $Q_3$, so it contains the middle $50\%$ of the data, and the line inside the box is the median. The whiskers show how far the data extends beyond that middle half.

The width of the box is the interquartile range:

A larger $IQR$ means the middle half of the data is more spread out. Many box plots also use the $1.5 \times IQR$ rule to mark possible outliers; in that version, the whiskers stop at the most extreme non-outlier values instead of the absolute minimum and maximum.

Why The Box Sits Where It Does

The quartiles are not arbitrary. They are the cut points that split ordered data into four equal parts, so the box from $Q_1$ to $Q_3$ literally brackets the central half of the values. The median splits that central half again, which is why its position inside the box hints at skew: if it sits off-center, the data is not balanced around the middle.

This is also why two correct box plots of the same data can differ. The whisker rule, not the data, decides how far the tails are drawn.

Worked Example: From Data To Box Plot

Use the ordered data set

There are $8$ values, so the median is the average of the two middle values:

Because there is an even number of data points, split the list into two equal halves. The lower half is $3, 5, 6, 7$, so

The upper half is $8, 9, 12, 15$, so

That gives the five-number summary:

Now compute the interquartile range:

If you use the common $1.5 \times IQR$ outlier rule, the fences are

and

All the data values fall between $-2$ and $18$, so there are no possible outliers under that rule. For this data set, the box would run from $5.5$ to $10.5$, the median line would be at $7.5$, and the whiskers would reach $3$ and $15$.

Practice: Read And Build One Yourself

Take the sorted set $4, 6, 6, 9, 11, 14, 20$ and write its five-number summary before drawing anything. With seven values the median is the fourth value, $9$; the lower half $4, 6, 6$ gives $Q_1 = 6$, and the upper half $11, 14, 20$ gives $Q_3 = 14$, so $IQR = 8$. Check your work: the box should run $6$ to $14$ with the median line at $9$.

To read any finished plot, start with the median line to locate the center, compare the box width and whisker lengths for spread, then look for asymmetry that signals skew.

Calculation Pitfalls To Watch

The most common slip is reading the edges of the box as the minimum and maximum. They usually represent $Q_1$ and $Q_3$, not the endpoints of the full data set.

A second pitfall is assuming every box plot uses the same whisker rule. Some whiskers extend to the true minimum and maximum; others stop at the most extreme non-outlier values under the $1.5 \times IQR$ rule.

A third is forgetting that quartiles depend on ordered data. If the values are not sorted first, the quartiles and median will be wrong. Box and whisker plots are most useful exactly when outliers or skew matter, because the median and quartiles stay more stable than the mean alone.