How to read a box plot

A box plot packs five numbers and any outliers into one small picture. Once you know what each part means, you can read the center, spread and shape of a data set at a glance.

By Ahmad Saleem, M.Sc. Mathematics & Statistics · Updated

Skip the manual work Use the free box plot maker Make box-and-whisker plots for one or more groups, with the five-number summary and outliers marked.

The parts of a box plot

Here is a box plot of how long nine customers waited, in minutes: 3, 5, 7, 8, 9, 11, 13, 15 and 30.

Labeled box plot A horizontal box plot with the minimum at 3, Q1 at 6, the median at 9, Q3 at 14, the upper whisker at 15 and an outlier at 30. The box from 6 to 14 is labeled IQR, the middle 50% of the data. Minimum 3 Q1 6 Median 9 Q3 14 Upper whisker Outlier 30 IQR: the middle 50% 0 5 10 15 20 25 30
Waiting times in minutes. The 30-minute wait is plotted as a separate dot because it is an outlier.
  • The box runs from the first quartile (Q1 = 6) to the third quartile (Q3 = 14). The middle half of the customers waited between 6 and 14 minutes.
  • The line inside the box is the median (9 minutes): half waited less, half waited more.
  • The whiskers reach out to the smallest and largest values that aren't outliers: 3 and 15 minutes.
  • The dot is an outlier: a value unusually far from the rest. Here, the 30-minute wait.

Read it in four steps

  1. Center: find the median line. That's the typical value.
  2. Spread: measure the box. Its length is the interquartile range (IQR = Q3 − Q1 = 8 minutes). A long box means the middle values vary a lot.
  3. Shape: compare the two halves of the box and the two whiskers. Equal halves mean a roughly symmetric distribution; a longer right side means right skew.
  4. Outliers: look for dots beyond the whiskers and ask whether each is an error or a genuine unusual case.

How outliers are decided

Most box plots use the 1.5 × IQR rule. Anything more than 1.5 IQRs beyond the box counts as an outlier:

  1. Interquartile range
    IQR = 14 − 6 = 8
  2. Fences: 1.5 × 8 = 12 beyond each end of the box
    6 − 12 = −6   and   14 + 12 = 26
  3. 30 is above 26, so it's an outlier; the whisker stops at the next value, 15
    Outliers: 30

The fences themselves aren't drawn; they only decide where the whiskers end.

Spotting skew

  • Symmetric: the median sits near the middle of the box and the whiskers are about the same length.
  • Skewed right: the median is closer to the left of the box and the right whisker is longer. Large values stretch out the top. Common for incomes, house prices and waiting times.
  • Skewed left: the mirror image: a longer left whisker, with the median toward the right of the box. Common for scores on an easy test.

In the example, the left half of the box (6 to 9) is shorter than the right half (9 to 14), and there's a high outlier, so waiting times are skewed to the right. A histogram of the same data would show a long tail on the right.

Comparing groups

Box plots are at their best side by side. When you compare two or more:

  1. Compare the medians to see which group is typically higher.
  2. Compare the box lengths to see which group is more consistent.
  3. Check the overlap. If one group's box sits entirely above the other's, the difference is clear. If the boxes overlap a lot, the difference may be due to chance; a t-test can check that formally.

To make your own, paste your groups into the box plot maker. It draws the plot, lists the five-number summary for every group and shows the working for the quartiles.

Why quartiles sometimes differ between tools

There are several accepted ways to calculate quartiles, so Q1 and Q3 can differ slightly between a textbook, a TI-84, Excel and other software, especially with small data sets. The most common classroom method, also used by TI-83/84 calculators, takes the median of the lower and upper halves of the data. Excel's QUARTILE.INC interpolates between values and QUARTILE.EXC uses yet another formula. None is wrong; just use the method your course or software expects. The box plot maker lets you choose.

Frequently asked questions

What does the line in the middle of a box plot mean?

It is the median: half of the values are below it and half are above. It is not the mean. Some box plots also mark the mean with an × or a dot; when the mean sits away from the median, the data are skewed toward the mean’s side.

What percentage of the data is in each part of a box plot?

Roughly 25% of the values fall in each of the four sections: lower whisker, lower half of the box, upper half of the box and upper whisker (counting outliers with their whisker). So the box always holds the middle 50%, however long or short it looks.

Why is one whisker longer than the other?

A longer whisker means the values on that side are more spread out, which usually signals skew in that direction. A long upper whisker with the median near the bottom of the box is the classic right-skewed shape, common for incomes, prices and waiting times.

Do box plots show the sample size?

No, and that is one of their limits: a box plot of 8 values looks just as solid as one of 8,000. Always report n alongside the plot. With very small samples, show the individual points too, for example with a dot plot.