Measuring spread and reading the data

Displaying a series without betraying it

A statistical summary can be accurate and still mislead. Reading a series also means spotting what a graph or an indicator conceals.

Choosing the right display

Type of data                   Suitable display
----------------------------   -------------------------------
qualitative (categories)       bar chart, pie chart
quantitative discrete          bar/stick chart
quantitative continuous        HISTOGRAM (areas proportional)
comparing several series       side-by-side box plots
relation between two variables scatter plot
change over time               line chart

A trap specific to the histogram: when classes have unequal widths, it is the area of each rectangle that must be proportional to the count, not its height. Otherwise a class twice as wide looks twice as populated as it is.

Graphs that lie

The truncated axis. The commonest device: not starting at zero.

   full axis                      truncated axis
   100 |                          52 |        █
       |     █  █                    |     █  █
    50 |  █  █  █                 51 |  █  █  █
       |  █  █  █                    |  █  █  █
     0 +----------                50 +----------
    a 2 % increase                the same, made dramatic

Both charts show exactly the same data. The second is not false, it is framed to suggest a major change. The reflex is to look at the scale before the shape of the bars.

A pie chart with too many slices makes comparison impossible: the eye compares angles badly. Beyond five categories, a bar chart is preferable.

3D effects distort proportions: slices at the front look larger.

What a single indicator does not say

The mean hides the spread. "One foot in ice, the other in embers: on average, all is well." Two classes with mean 11 can have opposite profiles — one homogeneous, the other polarised between 5 and 17.

The mean hides the shape. Two series with the same mean and standard deviation can have very different distributions (bimodal versus unimodal). Only a plot reveals it: always display the data before summarising them.

Simpson's paradox. A trend can reverse depending on whether groups are aggregated.

Treatment A better than B among men     ✔
Treatment A better than B among women   ✔
Treatment A WORSE than B overall        ✘ (and yet true)

This happens when the groups are of very unequal sizes. It is the sharpest reminder that an aggregate figure is not neutral: the chosen breakdown is part of the result.

The sound approach

1. DISPLAY the data (histogram, box plot)
2. identify the SHAPE: symmetric? skewed? several peaks?
3. choose indicators CONSISTENT with that shape
      symmetric  -> mean + standard deviation
      skewed     -> median + interquartile range
4. examine ATYPICAL values without deleting them by default
5. report location AND spread, never one without the other

Summary