Spread without mean

First proposal: Focus on prediction, not “typical.” 85% summary interval.

How to define “spread” without a mean to refer it to?

• Could use length of the interval.

Second proposal: Define spread as the average difference between pairs of individuals.

Formulas:

\[D_2 = \sqrt{\frac{1}{n} \frac{1}{n-1} \sum_{i=1}^n \sum_{j=1}^n (x_i - x_j)^2}\]

… or …

\[D_1 = \frac{1}{n} \frac{1}{n-1} \sum_{i=1}^n \sum_{j=1}^n |x_i - x_j|\]

For fun, let’s see what these are for a standard normal distribution:

## [1] 1.123165

## [1] 1.412621

For a normal distribution,

• \(D_2\) is \(\sqrt{2} \times\) standard deviation

• \(D_1\) is 20% smaller.

• Either can be used in what follows.

More fun: The \(n-1\) term, obscure in the standard deviation, arises naturally here. In summing the \(n^2\) terms, there will be \(n\) of them that are \(x_i - x_i\) and so must always be exactly zero.

The 85% summary interval

The length of the 85% summary interval is \(2 D_2\).