Range, interquartile range and standard deviation

Two series can share a mean and have nothing in common. Measures of spread say how far the values stray from the centre.

The problem

Series A:  9  10  10  10  11        mean 10
Series B:  0   5  10  15  20        mean 10

Same mean, same median — yet two opposite realities: a homogeneous group, a very heterogeneous one.

The range

range = maximum - minimum

Series A: 11 - 9  = 2
Series B: 20 - 0  = 20

Simple, but fragile: it depends only on the two extreme values, so a single outlier blows it up.

The interquartile range

IQR = Q3 - Q1

This is the spread of the central half of the data. By construction it ignores the lowest 25 % and the highest 25 %, which makes it robust to extremes. It is the natural companion of the median.

The standard deviation

The most used one, because it accounts for all the values.

1. compute deviations from the mean       xi - mean
2. square them                            (xi - mean)²
3. average them  ->  the VARIANCE
4. take the root ->  the STANDARD DEVIATION

Series: 2  3  5  7  8  10  12       mean = 47/7 ≈ 6.71

variance          ≈ 11.35
standard deviation ≈ 3.37

Why square them? Because the sum of the plain deviations is always zero — negative ones exactly offset positive ones. Squaring removes that problem, and penalises large deviations more. The final root brings the result back into the unit of the data: euros, not euros².

       mean
         |
   ------●------            SMALL standard deviation: tight values
      ●  ●  ●

   ●-----●-----●            LARGE standard deviation: scattered values

How to read it

The standard deviation reads as a typical deviation from the mean. For a bell-shaped distribution, the three-sigma rule gives instant landmarks: about 68 % of values within mean ± 1 standard deviation, 95 % within ± 2.

To compare series in different units, use the coefficient of variation:

CV = standard deviation / mean          (unitless, often as a %)

A CV of 5 % indicates a very homogeneous series, in any field.

The choice table

                     goes with        robust to extremes?
------------------   --------------   -----------------------
range                min/max          no, very sensitive
interquartile range  MEDIAN           YES
standard deviation   MEAN             no

The consistency rule: report median and interquartile range, or mean and standard deviation. Mixing a robust indicator with a sensitive one produces an incoherent summary.

Summary

• The range max - min is simple but very sensitive to extremes.
• The interquartile range Q3 - Q1 describes the central half: robust.
• The standard deviation uses every value; we square because plain deviations sum to zero.
• It is expressed in the unit of the data, unlike the variance.
• The coefficient of variation σ/mean compares series in different units.
• Pair median + IQR, or mean + standard deviation.