Quartiles and box plots

The median cuts the series in two. Cutting it in four gives a much finer description of the spread.

Quartiles

   Q1  : 25 % of the values are at or below it
   Q2  : the MEDIAN, 50 %
   Q3  : 75 %

   min        Q1       Q2       Q3        max
    |---------|--------|--------|---------|
      25 %      25 %     25 %     25 %

Each quarter contains the same number of values — but not necessarily the same width: a narrow quarter signals concentration, a wide one dispersion.

Computing them

Sort the series, then locate the ranks. The usual French school convention:

Q1: value at rank n/4,  rounded up
Q3: value at rank 3n/4, rounded up

Series: 2  3  5  7  8  10  12          n = 7

rank of Q1: 7/4 = 1.75  ->  rank 2  ->  Q1 = 3
median    : rank 4      ->  Q2 = 7
rank of Q3: 21/4 = 5.25 ->  rank 6  ->  Q3 = 10

Other conventions exist (interpolation, spreadsheet definitions) and sometimes give slightly different values. What matters is stating which one is used.

The box plot

This is the display condensing the five key numbers — minimum, Q1, median, Q3, maximum:

        Q1      Q2        Q3
  min    |       |         |      max
   |-----[=======|=========]-------|
   2     3       7        10       12
    \___/ \_____________/ \______/
   whisker      BOX        whisker
              (50 % of the values)

The box holds the central half of the data; the whiskers reach the extreme values. Its strength is making skewness immediately visible:

Symmetric series              Series stretched to the right
   |---[===|===]---|             |-[==|===]-----------|
                                        (salaries: long right whisker)

A box shifted left with a long right whisker is the typical signature of an income or waiting-time distribution.

Spotting outliers

A common convention treats as atypical any value more than 1.5 × IQR beyond the edges of the box, where IQR = Q3 - Q1:

thresholds:  Q1 - 1.5 × IQR     and     Q3 + 1.5 × IQR

   |-----[=====|=====]-----|        •          <- isolated point
                                  beyond the threshold: examine it

A value flagged this way is not necessarily wrong: it may be a data-entry error or the most interesting observation of the study. The rule is there to draw attention, never to delete a data point automatically.

Deciles

For large samples, we cut into ten: D1 (10 %), …, D9 (90 %). The interdecile ratio D9/D1 is the most used inequality indicator in economics — it compares the threshold of the richest 10 % with that of the poorest 10 %.

Summary

• Quartiles Q1, Q2 (median), Q3 split the series into four parts of equal size.
• The box plot summarises five numbers: min, Q1, median, Q3, max.
• The box holds the central 50 % of the values.
• IQR = Q3 - Q1; beyond 1.5 × IQR from the edges, a value is called atypical.
• An atypical value is examined, not automatically removed.
• Deciles and the D9/D1 ratio measure inequality.