Editing Effect size (section)

==== Effect size for ordinal data ====
'''Cliff's delta''' or <math>d</math>, originally developed by [[Norman Cliff]] for use with ordinal data,<ref name="Cliff1993">{{cite journal | last=Cliff | first=Norman | title=Dominance statistics: Ordinal analyses to answer ordinal questions | year=1993 | journal=Psychological Bulletin | volume=114 | pages=494–509 | issue=3 | doi=10.1037/0033-2909.114.3.494}}</ref>{{Dubious|date=May 2024|reason=I'm at least 80% sure this is just a weird name for Kendall's tau.}} is a measure of how often the values in one distribution are larger than the values in a second distribution.  Crucially, it does not require any assumptions about the shape or spread of the two distributions.

The sample estimate <math>d</math> is given by:
<math display="block">d = \frac{\sum_{i,j} [x_i > x_j] - [x_i < x_j]}{mn}</math>
where the two distributions are of size <math>n</math> and <math>m</math> with items <math>x_i</math> and <math>x_j</math>, respectively, and <math>[\cdot]</math> is the [[Iverson bracket]], which is 1 when the contents are true and 0 when false.

<math>d</math> is linearly related to the [[Mann–Whitney U test|Mann–Whitney U statistic]]; however, it captures the direction of the difference in its sign.  Given the Mann–Whitney <math>U</math>, <math>d</math> is:
<math display="block">d = \frac{2U}{mn} - 1</math>