In
statistics
Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...
, 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
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 ...
between two measured quantities. A τ test is a
non-parametric hypothesis test 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 by each of the quantities. It is named after
Maurice Kendall
Sir Maurice George Kendall, FBA (6 September 1907 – 29 March 1983) was a prominent British statistician. The Kendall tau rank correlation is named after him.
Education and early life
Maurice Kendall was born in Kettering, Northampton ...
, 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 ins ...
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. Ex ...
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 (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 o ...
.
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'' if the sort order of
and ''
'' agrees: that is, if either both
and
holds or both