Box-and-whisker plots — quartiles, IQR: Foundations

This page assumes you know nothing. Before you can read the parent note on Box-and-whisker plots — quartiles, IQR, you need a small toolkit of ideas. We build each one from zero, anchor it to a picture, and say why the topic needs it. Every later idea leans only on earlier ones.

0. "Data" and "a value" — what are we even holding?

Before symbols, the raw material.

Picture: imagine each value as a bead. Your whole data set is a handful of loose beads.

$n$ is our very first symbol.

1. Sorting — turning a pile into a queue

The single most important move in this whole topic.

Figure — Box-and-whisker plots — quartiles, IQR

Look at the figure. On top, the beads are a jumbled pile (unsorted). On the bottom, the same beads stand in a queue, shortest on the left, tallest on the right. That rearrangement is called sorting ascending.

2. Position vs value — the two things every bead has

This is subtle and worth its own picture.

Each bead in the sorted queue has two different numbers:

its position — where it stands in line: 1st, 2nd, 3rd, … (shown below the line in yellow).
its value — what number it is (shown inside the bead in blue).

3. Fractions of the way along — "half", "a quarter", "three-quarters"

A box plot marks the queue at $\frac{1}{2}$ , $\frac{1}{4}$ , $\frac{3}{4}$ of the way. So we must be crystal clear what a fraction of a queue means.

Picture: stand in the sorted queue. Walk until exactly half the people are behind you — that spot is the halfway mark. Walk only a quarter of the crowd — that's the quarter mark.

4. The median $Q_{2}$ — the halfway value

Now we can name the halfway mark.

How to find it depends on whether $n$ is odd or even — because "the exact middle" only exists when the count is odd.

$n$ odd (top row, 7 beads): there is a lone middle bead. Its position is $\frac{n + 1}{2} = \frac{7 + 1}{2} = 4$ , so the median is the 4th value.
$n$ even (bottom row, 6 beads): no single bead is in the middle — the centre falls between two beads (3rd and 4th). The median is their average.

5. Quartiles $Q_{1}$ and $Q_{3}$ — the quarter and three-quarter values

"Quart" means four. Quartiles are the marks that cut the queue into four equal-count groups.

These three symbols are exactly the 25th, 50th, and 75th Percentiles and Quantiles — quartiles are just percentiles at nice quarter-marks.

6. Subtraction as "distance" — building the IQR

The parent note's spread measure is $Q_{3} - Q_{1}$ . Why subtract?

7. The multiplier — " $1.5 \times IQR$ " and fences

Last symbol to unpack: the number $1.5$ and the word "fence".

How these feed the topic

Equipment checklist

Recall Can you answer each without peeking?

What does the symbol $n$ stand for? ::: The count of how many values are in the data set. What must you always do first (Step 0) with any data before finding quartiles? ::: Sort it ascending (smallest to largest). What is the difference between a value's position and its value? ::: Position is where it stands in the sorted queue (1st, 2nd, …); value is the actual number it is. What does "a quarter of the way along" the queue mean? ::: One bead in four is to your left, three in four to your right. What is the median $Q_{2}$ and how does finding it differ for odd vs even $n$ ? ::: The halfway value; for odd $n$ it's the single middle value at position $(n + 1) /2$ , for even $n$ it's the average of the two middle values. What are $Q_{1}$ and $Q_{3}$ in one phrase each? ::: $Q_{1}$ = median of the lower half; $Q_{3}$ = median of the upper half. Why does $b - a$ measure spread? ::: It is the distance/gap on the number line — bigger gap means more spread out. Write the IQR formula. ::: $IQR = Q_{3} - Q_{1}$ . Why multiply the IQR (rather than use a fixed number) to set outlier fences? ::: So the "far" yardstick scales with the data's own spread; $1.5 IQR$ beyond each edge.