Quartiles, Box Plots and Outliers

Quartiles are simply the three values that split sorted data into four equal-sized groups; the name just means "quarters." $Q_1$ (a quarter of the data below it), the median (half below) and $Q_3$ (three quarters below). The IQR, $Q_3 - Q_1$, is the width of the middle half of the data, and the fences are that width times 1.5, measured out from each side of the box. It's all just sorting and finding middles, the same skill as finding a median, a few times over.

Finding quartiles: sort the data and find the median. $Q_1$ is the median of the lower half and $Q_3$ the median of the upper half (leave out the overall median when $n$ is odd).

Percentiles: the $p$th percentile has $p\%$ of the data at or below it. $Q_1$, the median and $Q_3$ are the 25th, 50th and 75th percentiles.

Box plots: a box from $Q_1$ to $Q_3$ with a line at the median, and whiskers out to the smallest and largest values that aren't outliers. Outliers are plotted as separate points.

Why build the fences from the IQR? Remember that the median shrugs off extreme values while the mean doesn't. Quartiles are medians of halves, so they inherit the same toughness. That makes the IQR a fair yardstick for judging outliers: if we used the standard deviation instead, a wild value would inflate the very yardstick that's supposed to catch it.

Outliers are values beyond the fences. They deserve a second look: data-entry errors, or the most interesting observations in the set.

A common mistake. It's tempting to draw the whiskers all the way to the minimum and maximum every time. That's only right when nothing lies past the fences. When something does, the whisker stops at the most extreme value inside the fence, and the outlier gets its own dot.

Box plots compare groups side by side: salaries by department, test scores by school, response times before and after a code change.