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 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 In mathematics, two sequences of numbers, often experimental data, are proportional or directly proportional if their corresponding elements have a constant ratio, which is called the coefficient of proportionality or proportionality constan ...

, and is given by

y = k/x

where ''k'' > 0 is a constant. In a Cartesian plane this relationship is displayed as a hyperbola with ''y'' decreasing as ''x'' increases.The derivative

\ y \prime = \frac \

is negative for positive real numbers ''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-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 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)

See also

References

External links