1.2 When Parametric Inference Will Not Do
- (i).
- The assumption may be unreasonable. Examination marks, for example, are bounded and frequently skewed, and no standard family fits them naturally.
- (ii).
- Even with precise measurements, assuming normality asserts properties — symmetry, unbounded support, specific tail behaviour — that the data may visibly contradict.
- (iii).
- The data may not be measurements at all. For ranked or ordered categorical data there is no numerical scale on which a mean is meaningful.
Definition 1.2. A statistical method is distribution-free if the null distribution of its test statistic does not depend on the distribution from which the sample was drawn.
Note 1.3. “Distribution-free” and “non-parametric” are used interchangeably, and the term is slightly misleading in the same way the first note warned about. It does not say the test statistic has no distribution — it has a perfectly definite one. It says the method may be applied whatever the population distribution happens to be.
Non-parametric methods make weaker assumptions, but they are not preferred in all situations. Breadth of applicability is bought at a price: when a parametric assumption is justified, the parametric test extracts more information from the same data and is more powerful. The sensible rule is to use the parametric method when its assumptions hold and the non-parametric one when they do not.
Most of the tests here concern measures of location — the mean, and more often the median — though measures of dispersion such as the variance and standard deviation also arise.
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