Five-Number Summary and Box Plots

You have a year of daily P&L for a strategy and want to describe it in a sentence before diving into anything fancier. "The mean daily P&L was $1,200" tells you almost nothing on its own — a single $500,000 blow-up day and 250 quiet days averaging $800 can produce the exact same mean as 250 consistently solid $1,200 days, yet those are completely different risk profiles. The mean is easily dragged around by a few extreme values, and standard deviation shares the same weakness. What you often want first is a description built entirely from positions in the sorted list — which value is in the middle, which values mark off the middle chunk, which values are the extremes — because those don't budge much just because one day was wild.

An analogy: a classroom's height lineup

Line up thirty students from shortest to tallest. Five landmarks describe the whole line without you needing to write down all thirty heights: the shortest student, the tallest student, the student standing exactly in the middle, and two more students — one a quarter of the way along the line, one three-quarters of the way along. Those five positions tell you the full spread of the line, where the crowded middle sits, and how far the extremes stick out, all without being thrown off if one kid happens to be freakishly tall. That's the whole idea of the five-number summary: describe the shape of a batch of numbers using landmarks in the sorted order, not an average that any one extreme value can drag around.

The five numbers, one symbol at a time

Sort your $n$ observations from smallest to largest. The five landmarks are:

• Minimum, $\min$ — the smallest value.
• First quartile, $Q_1$ — the value one-quarter of the way through the sorted list (25% of observations lie below it).
• Median, $Q_2$ — the value exactly in the middle (50% of observations lie below it).
• Third quartile, $Q_3$ — the value three-quarters of the way through (75% of observations lie below it).
• Maximum, $\max$ — the largest value.

The middle 50% of the data, from $Q_1$ to $Q_3$, is called the interquartile range:

$$IQR = Q_3 - Q_1 .$$

In plain English: the IQR is the width of the box that contains the "typical" middle half of your observations, ignoring the bottom and top quarters entirely — a spread measure that a single wild outlier can barely move, unlike the standard deviation.

A box plot draws this directly: a box from $Q_1$ to $Q_3$ with a line at the median, and whiskers extending out to the most extreme point still within

$$[\,Q_1 - 1.5 \times IQR,\;\; Q_3 + 1.5 \times IQR\,] ,$$

with any points beyond that range plotted individually as dots — flagged as candidate outliers. In plain English: the whiskers stretch to cover "normal-looking" spread beyond the box, using 1.5 box-widths as a conventional cutoff, and anything further out gets drawn separately so it doesn't silently stretch the whisker to hide how far away it really is.

Worked example 1: eleven days of P&L by hand

Sorted daily P&L ($000s): $-8, -3, -1, 0, 2, 4, 5, 6, 9, 12, 40$. With $n=11$, the median is the 6th value: $Q_2 = 4$. $Q_1$ is the median of the lower half (values $-8,-3,-1,0,2$): the middle of those five is $Q_1 = -1$. $Q_3$ is the median of the upper half ($5,6,9,12,40$): $Q_3 = 9$. So:

$$IQR = 9 - (-1) = 10 .$$

The whisker fence is $Q_1 - 1.5 \times 10 = -16$ on the low side and $Q_3 + 1.5 \times 10 = 24$ on the high side. Every value except 40 falls inside $[-16, 24]$, so the upper whisker stops at 12 (the largest point still inside the fence) and 40 is flagged as an outlier, plotted as an isolated dot. The five-number summary — $(-8, -1, 4, 9, 40)$ — immediately tells you the typical day nets a few thousand dollars, half the days fall between −$1,000 and $9,000, and there's one exceptional $40,000 day sitting well outside the normal pattern, all without a mean or standard deviation calculation that $40,000 day would have dominated.

Worked example 2: comparing two strategies side by side

Two strategies, summarized:

	Min	$Q_1$	Median	$Q_3$	Max	IQR
Strategy A	$-8$	$-1$	$4$	$9$	$40$	$10$
Strategy B	$-2$	$1$	$3$	$5$	$9$	$4$

Strategy A has a higher median ($4$ vs $3$) but a much wider IQR ($10$ vs $4$) — its typical middle-half spread is 2.5× wider. Strategy B's box sits in a tight band from 1 to 5, meaning most days cluster close together, while A's box from $-1$ to $9$ shows a much more dispersed "normal" outcome even before considering the $40,000 outlier day. If you only compared the two means, you might miss that A's slightly higher average return comes with meaningfully more day-to-day variability in the typical (non-outlier) range — exactly the comparison a five-number summary is built to surface at a glance.

What this means in practice

Box plots are the fastest way to eyeball dozens of strategies, factors, or backtests side by side — lining up their boxes reveals which ones are tightly consistent versus wildly dispersed, and which have outlier days worth investigating individually, before you compute a single formal statistic. Because the five-number summary is built from sorted positions rather than sums, it's naturally robust to the exact kind of fat-tailed, outlier-prone data that finance produces constantly, which is precisely where mean-and-standard-deviation summaries are least trustworthy. It's a description, not an inference — a box plot tells you what the data looks like, not whether a difference between two groups is statistically significant, for which you'd reach for a formal test.