Summarising a series by its location

Mean, median and mode

Faced with a series of measurements — salaries, marks, durations — the first task is to summarise it with a few numbers. Measures of location answer the question: "around which value do the data sit?"

The mean

              sum of the values
   mean  =  --------------------
             number of values

For a series where values repeat, we use the counts:

Marks:      8   ×3      12  ×5      16  ×2

            8×3 + 12×5 + 16×2       24 + 60 + 32       116
mean    = --------------------- = --------------- = ------- = 11.6
                 3 + 5 + 2               10            10

This is a weighted mean: each value counts in proportion to its frequency. Beware the classic error — averaging the values (8 + 12 + 16)/3 = 12 — which ignores that the mark 12 was obtained five times.

The mean has a mechanical property: it is the series' balance point.

        8        12          16
        |         |           |
    ----▲---------▲-----------▲----
             ^
           11.6      <- the pivot balancing the scales

The median

The median splits the ordered series into two halves of equal size: half the values are below it, half above.

Computing it requires sorting first — the step most often forgotten.

Odd-sized series (7 values):  2  3  5  7  8  10  12
                                       ^
                              the 4th value -> median = 7

Even-sized series (8 values):  12  15  15  18 | 20  22  25  30
                                              ^
                              mean of the two middle ones:
                              (18 + 20)/2 = 19

The mode

The mode is the most frequent value. It is the only usable indicator for qualitative data (favourite colour, political party), where mean and median make no sense. A series may have several modes.

Mean or median? The decisive point

Both summarise location, but they react entirely differently to extreme values.

Salaries in a small firm (in euros):

   1500   1600   1700   1800   20000
                                 ^^^^^ the director

   mean   = 26,600 / 5 = €5320
   median = €1700

Four employees out of five earn less than the mean. Announcing "the average salary is €5320" is accurate and yet deeply misleading.

The mean is SENSITIVE to extreme values: it absorbs them in full.
The median is ROBUST: it depends only on rank, not on magnitude.

Multiplying the director's salary tenfold would push the mean above €40,000; the median would not move by a cent.

The practical rule: on a symmetric series, mean and median coincide and the choice does not matter. On a skewed series — salaries, wealth, waiting times — the median better describes a "typical" person's situation. This is why statistical offices publish both and highlight the median salary.

Summary