Measuring spread

Return, risk and pooling

Expected value and standard deviation are not school exercises: they are the two numbers underpinning insurance, finance and quality control.

The return / risk pair

Indicator	What it measures
$E(X)$	what is gained on average — the return
$\sigma(X)$	how much it varies from one instance to the next — the risk

A financial investment almost always boils down to this pair. Two products with the same expected value are not equivalent: at equal return, the one with the smaller standard deviation is preferable — the basic principle of portfolio management.

        return E(X)
             ^
             |        • equity fund (high E, high σ)
             |
             |   • bonds (medium E, low σ)
             | • savings account (low E, σ ≈ 0)
             +---------------------------> risk σ(X)

Insurance, explained by expectation

A company insures 100,000 drivers. For each, the probability of a claim costing €10,000 is 1 %.

$E(\text{cost per policy, in euros}) = 0.01 \times 10\,000 = 100$

The premium must therefore exceed €100 to cover claims, costs and margin. But why does insurance work at all, when the expectation is unfavourable to the insured?

Because the two parties are not looking at the same number:

the insured faces a huge risk at their own scale: their individual $\sigma$ is very large, and one claim can ruin them. They pay to remove that variance;
the insurer pools 100,000 independent risks. Their average cost concentrates around €100: the relative standard deviation shrinks as $\tfrac{1}{\sqrt{n}}$ .

This is the law of large numbers turned into a business model: individual risk vanishes through aggregation.

Chebyshev's inequality

It gives quantitative meaning to the standard deviation, assuming nothing about the distribution:

$P\left(\,|X - E(X)| \geq k\sigma\,\right) \leq \frac{1}{k^2}$

$k$	Proportion of values more than $k\sigma$ from the mean
$2$	at most 25 %
$3$	at most 11 %
$10$	at most 1 %

This bound is universal but crude. For a normal distribution, the actual proportion beyond $2\sigma$ is 5 %, not 25 %. So it mainly serves to guarantee an order of magnitude when the distribution is unknown.

Quality control

A production line targets a length of 100 mm with $\sigma = 0.2$ mm. Parts outside $[\mu - 3\sigma \,;\, \mu + 3\sigma]$ , that is $[99.4 \,;\, 100.6]$ , are rejected.

      99.4        100        100.6
        |----------|----------|
            accepted zone
   reject <--                --> reject

The famous « six sigma » goal consists in making tolerances so wide relative to $\sigma$ that the defect rate falls to a few parts per million. All the industrial work then consists in reducing $\sigma$ , not in moving the mean — the latter is easy to adjust, dispersion is not.

Summary