In
statistics
Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...
, 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 hypot ...
used to measure the
ordinal association between two measured quantities. A τ test is a
non-parametric
Nonparametric statistics is a type of statistical analysis that makes minimal assumptions about the underlying distribution of the data being studied. Often these models are infinite-dimensional, rather than finite dimensional, as in parametric sta ...
hypothesis test
A statistical hypothesis test is a method of statistical inference used to decide whether the data provide sufficient evidence to reject a particular hypothesis. A statistical hypothesis test typically involves a calculation of a test statistic. ...
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 data, ordinal variables or different rankings of the same variable, where a "ranking" is t ...
: the similarity of the orderings of the data when
ranked 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 inspi ...
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. ...
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
A rank is a position in a hierarchy. It can be formally recognized—for example, cardinal, chief executive officer, general, professor—or unofficial.
People Formal ranks
* Academic rank
* Corporate title
* 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 ...
. Its notions of
concordance and discordance also appear in other areas of statistics, like the
Rand index
The Rand index or Rand measure (named after William M. Rand) in statistics, and in particular in data clustering, is a measure of the similarity between two data clusterings. A form of the Rand index may be defined that is adjusted for the chance ...
in
cluster analysis
Cluster analysis or clustering is the data analyzing technique in which task of grouping a set of objects in such a way that objects in the same group (called a cluster) are more Similarity measure, similar (in some specific sense defined by the ...
.
Definition

Let
be a set of observations of the joint random variables ''X'' and ''Y'', such that all the values of (
) and (
) are unique. (See the section
#Accounting for ties for ways of handling non-unique values.) 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 la ...
'' if the sort order of
and ''
'' agrees: that is, if either both
and
holds or both