In
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, the Kendall rank correlation coefficient, commonly referred to as Kendall's τ coefficient (after the Greek letter
τ, tau), is a
statistic
A statistic (singular) or sample statistic is any quantity computed from values in a sample which is considered for a statistical purpose. Statistical purposes include estimating a population parameter, describing a sample, or evaluating a hypo ...
used to measure the
ordinal association between two measured quantities. A τ test is a
non-parametric
Nonparametric statistics is the branch of statistics that is not based solely on parametrized families of probability distributions (common examples of parameters are the mean and variance). Nonparametric statistics is based on either being distri ...
hypothesis test
A statistical hypothesis test is a method of statistical inference used to decide whether the data at hand sufficiently support a particular hypothesis.
Hypothesis testing allows us to make probabilistic statements about population parameters.
...
for statistical dependence based on the τ coefficient.
It is a measure of
rank correlation
In statistics, a rank correlation is any of several statistics that measure an ordinal association—the relationship between rankings of different ordinal variables or different rankings of the same variable, where a "ranking" is the assignment o ...
: the similarity of the orderings of the data when
ranked
A ranking is a relationship between a set of items such that, for any two items, the first is either "ranked higher than", "ranked lower than" or "ranked equal to" the second.
In mathematics, this is known as a weak order or total preorder of ...
by each of the quantities. It is named after
Maurice Kendall, who developed it in 1938, though
Gustav Fechner
Gustav Theodor Fechner (; ; 19 April 1801 – 18 November 1887) was a German physicist, philosopher, and experimental psychologist. A pioneer in experimental psychology and founder of psychophysics (techniques for measuring the mind), he inspired ...
had proposed a similar measure in the context of
time series
In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Exa ...
in 1897.
Intuitively, the Kendall correlation between two variables will be high when observations have a similar (or identical for a correlation of 1)
rank
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as:
Level or position in a hierarchical organization
* Academic rank
* Diplomatic rank
* Hierarchy
* ...
(i.e. relative position label of the observations within the variable: 1st, 2nd, 3rd, etc.) between the two variables, and low when observations have a dissimilar (or fully different for a correlation of −1) rank between the two variables.
Both Kendall's
and
Spearman's can be formulated as special cases of a more
general correlation coefficient
In statistics, a rank correlation is any of several statistics that measure an ordinal association—the relationship between rankings of different ordinal variables or different rankings of the same variable, where a "ranking" is the assignment ...
.
Definition
Let
be a set of observations of the joint random variables ''X'' and ''Y'', such that all the values of (
) and (
) are unique (ties are neglected for simplicity). Any pair of observations
and
, where
, are said to be ''
concordant
Concordance may refer to:
* Agreement (linguistics), a form of cross-reference between different parts of a sentence or phrase
* Bible concordance, an alphabetical listing of terms in the Bible
* Concordant coastline, in geology, where beds, or ...
'' if the sort order of
and ''
'' agrees: that is, if either both
and
holds or both