In statistics, 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 statisti ...

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 used ...

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 vectors are viewed as points on a sphere, and the

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 In mathematics, antipodal points of a sphere are those diametrically opposite to each other (the specific qualities of such a definition are that a line drawn from the one to the other passes through the center of the sphere so forms a true d ...

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 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

y = k/x

where ''k'' > 0 is a constant. In a Cartesian plane 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, c ...

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. ...

\ y \prime = \frac \

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 f ...

''x'' and as well for negative real numbers. Thus the slope is everywhere negative except at the singularity ''x'' = 0. In finance, an inverse correlation between the returns 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.

References

{{Reflist

External links

* Michael Palme
Testing for correlation
from Oklahoma State University–Stillwater Independence (probability theory)

See also

References

External links