Boxplots and Quartiles
Summarising data with five numbers
A long list of data is hard to take in at a glance, so statisticians summarise it. One of the most useful summaries is the five-number summary: the minimum, the first quartile (called Q1), the median, the third quartile (Q3), and the maximum. The median is the middle value, splitting the data in half; the quartiles are the middle values of each half, splitting the data into quarters. These five numbers between them describe where the data starts and ends, where its centre lies, and how its middle bulk is spread, all without listing every value. They are the foundation of the boxplot, a diagram designed to show exactly this summary in a form the eye can read instantly.
Drawing a boxplot
A boxplot turns the five-number summary into a picture drawn against a number line. A rectangular box is drawn from the first quartile to the third quartile, so the box contains the middle half of the data. A line is drawn inside the box at the median, showing the centre. Then two lines called whiskers extend from the box, one out to the minimum and one out to the maximum, marking the full extent of the data. The result is a clear visual: the box shows where the bulk of the data lies, the median line shows the centre, and the whiskers show the reach. Because every boxplot follows the same construction, once you can read one you can read any.
What the quartiles mean
The key to reading a boxplot is understanding the quartiles. They divide the sorted data into four parts, each holding about a quarter, that is 25 percent, of the values. So about a quarter of the data lies below Q1, a quarter between Q1 and the median, a quarter between the median and Q3, and a quarter above Q3. A crucial subtlety is that the quartiles are about counts of values, not equal distances: if the data is bunched in one region and sparse in another, the four parts will have very different widths on the number line even though each contains the same number of values. A narrow box section means the data is densely packed there; a wide one means it is spread out.
Spread and outliers
The width of the box has its own name and use: the interquartile range, or IQR, is Q3 minus Q1, and it measures the spread of the middle half of the data. Because it ignores the extreme quarters, the IQR is a measure of spread that is not thrown off by a single unusual value, in the way the full range can be. The IQR also gives an objective test for outliers, values that sit surprisingly far from the rest. A common rule flags any value more than 1.5 times the IQR beyond a quartile as an outlier. This turns a vague sense that a value looks odd into a definite calculation, and boxplots are often drawn with such outliers marked as separate points beyond the whiskers.
Comparing distributions
Where boxplots truly shine is in comparing two or more groups. Drawn on the same scale, one above the other, two boxplots let you compare distributions at a glance in a way that raw lists never could. If one box sits further along the scale than the other, that group generally has larger values, seen most clearly in the position of its median. If one box is wider, that group has a greater spread in its middle half. You can compare centres through the median lines, spreads through the box widths, and reach through the whiskers, all at once. This makes parallel boxplots a favourite tool for questions like whether one class scored higher than another, or whether one method gives more consistent results.
Shape: symmetric or skewed
Shape is the fourth thing to discuss about a distribution, alongside its centre, spread and outliers, and a boxplot shows it through balance. When the median sits near the middle of the box and the two whiskers are about the same length, the distribution is roughly symmetric: the values spread out about equally on both sides of the centre. When one side is stretched, the distribution is skewed toward that side. The data used throughout this unit is an example. Its left whisker runs only from 2 to 5, while its right whisker stretches from 12 all the way to 21, and the median of 8 sits a little nearer Q1 than Q3. The values bunch at the low end and trail off in a long tail toward high values, so the distribution is skewed to the right, also called positively skewed. Group B, by contrast, has whiskers of 3 and 4 and its median near the middle of its box, so it is roughly symmetric. Shape matters in context: for skewed data such as incomes or waiting times, the long tail pulls the mean toward it (here the mean is 98 ÷ 11, about 8.9, above the median of 8), so the median is usually the better centre to report.