Alec's ordered data test

Notes

Consider some data $\big(x_i\big)_{i\eq 1}^n$ which are of the ordered type in Alec's taxonomy of measures, meaning we have the $<$ and $>$ operators forming a total ordering, and surely $\eq$ and $\neq$ to (from the present type)

We may speak of the median with this type of measuring unit

Consider the following null hypothesis:

• $\text{H}_0:$ $\text{median}\eq m$

and the following three alternate hypotheses (we shall design the test, but I'm not sure which will work due to "can't read my own handwriting" issues)

1. $\text{H}_1:$ $\text{median}< m$
2. $\text{H}_2:$ $\text{median}> m$
3. $\text{H}_3:$ $\text{median}\neq m$

Principle of test

Our goal is to produce a useful method for hypothesis testing by looking at how many are above and below the median. We set this out generally:

Notes for notes

For an ordered data type which is not additive we cannot take any meaning from the difference between values, only $>$ or $<$.

The concept is to remove the median value (of the sample, which works if n is odd) and then for any remaining item we'd expect a 50/50 chance it's above or below our median. $\sim\text{Bin}(n-1,\frac{1}{2})$