![IdempotentCosineAngle](https://upload.wikimedia.org/wikipedia/commons/3/37/IdempotentCosineAngle.jpg)
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 ...
, there is a negative relationship or inverse relationship between two variables if higher values of one variable tend to be associated with lower values of the other. A negative relationship between two variables usually implies that the
correlation
In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistics ...
between them is negative, or — what is in some contexts equivalent — that the
slope
In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is use ...
in a corresponding graph is negative. A negative correlation between variables is also called anticorrelation or inverse correlation.
Negative correlation can be seen geometrically when two normalized
random vector
In probability, and statistics, a multivariate random variable or random vector is a list of mathematical variables each of whose value is unknown, either because the value has not yet occurred or because there is imperfect knowledge of its value ...
s are viewed as points on a sphere, and the
correlation
In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistics ...
between them is the
cosine of the arc of separation of the points on the sphere. When this arc is more than a quarter-circle (θ > π/2), then the cosine is negative.
Diametrically opposed points represent a correlation of –1 = cos(π). Any two points not in the same hemisphere have negative correlation.
An example would be a negative
cross-sectional
Cross-sectional data, or a cross section of a study population, in statistics and econometrics, is a type of data collected by observing many subjects (such as individuals, firms, countries, or regions) at the one point or period of time. The anal ...
relationship between illness and vaccination, if it is observed that where the incidence of one is higher than average, the incidence of the other tends to be lower than average. Similarly, there would be a negative
temporal relationship between illness and vaccination if it is observed in one location that times with a higher-than-average incidence of one tend to coincide with a lower-than-average incidence of the other.
A particular inverse relationship is called
inverse proportionality, and is given by
where ''k'' > 0 is a
constant. In a
Cartesian plane
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
this relationship is displayed as a
hyperbola
In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, cal ...
with ''y'' decreasing as ''x'' increases.
[The ]derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
is negative for positive real numbers
In mathematics, the set of positive real numbers, \R_ = \left\, is the subset of those real numbers that are greater than zero. The non-negative real numbers, \R_ = \left\, also include zero. Although the symbols \R_ and \R^ are ambiguously used fo ...
''x'' and as well for negative real numbers. Thus the slope is everywhere negative except at the singularity ''x'' = 0.
In
finance
Finance is the study and discipline of money, currency and capital assets. It is related to, but not synonymous with economics, the study of production, distribution, and consumption of money, assets, goods and services (the discipline of fina ...
, an inverse correlation between the
returns
Return may refer to:
In business, economics, and finance
* Return on investment (ROI), the financial gain after an expense.
* Rate of return, the financial term for the profit or loss derived from an investment
* Tax return, a blank document or t ...
on two different assets enhances the
risk
In simple terms, risk is the possibility of something bad happening. Risk involves uncertainty about the effects/implications of an activity with respect to something that humans value (such as health, well-being, wealth, property or the environme ...
-reduction effect of
diversifying by holding them both in the same portfolio.
See also
*
Diminishing returns
In economics, diminishing returns are the decrease in marginal (incremental) output of a production process as the amount of a single factor of production is incrementally increased, holding all other factors of production equal ( ceteris paribu ...
References
{{Reflist
External links
* Michael Palme
Testing for correlationfrom
Oklahoma State University–Stillwater
Oklahoma State University–Stillwater (officially Oklahoma State University; informally Oklahoma State, OK State, OSU) is a public land-grant research university in Stillwater, Oklahoma. OSU was founded in 1890 under the Morrill Act. Originall ...
Independence (probability theory)