# Negative relationship

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Negative_relationship
> Markdown URL: https://mediated.wiki/source/Negative_relationship.md
> Source: https://en.wikipedia.org/wiki/Negative_relationship
> Source revision: 1345464315
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

{{Short description|Higher values of one variable leading to lower values of the other}}
right|thumb|300px|When t > π /2 or t < – π /2, then cos(t) < 0.

In [statistics](/source/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](/source/correlation) between them is negative, or — what is in some contexts equivalent — that the [slope](/source/slope) in a corresponding graph is negative. A '''negative correlation''' between variables is also called '''inverse correlation'''.

Negative correlation can be seen geometrically when two normalized [random vector](/source/random_vector)s are viewed as points on a sphere, and the [correlation](/source/Pearson_correlation_coefficient) between them is the [cosine](/source/cosine) of the [circular arc](/source/circular_arc) of separation of the points on a [great circle](/source/great_circle) of the sphere.<ref>R. J. Rummel [http://www.hawaii.edu/powerkills/UC.HTM Understanding Correlation] from [University of Hawaii](/source/University_of_Hawaii)</ref> When this arc is more than a quarter-circle (θ > π/2), then the cosine is negative. [Diametrically opposed](/source/Diametrically_opposed) points represent a correlation of –1 = cos(π), called '''anti-correlation'''. Any two points ''not'' in the same hemisphere have negative correlation.

An example would be a negative [cross-sectional](/source/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](/source/time_series) 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](/source/Proportionality_(mathematics)), and is given by <math>y = k/x </math> where ''k'' > 0 is a [constant](/source/constant_(mathematics)). In a [Cartesian plane](/source/Cartesian_plane) this relationship is displayed as a [hyperbola](/source/hyperbola) with ''y'' decreasing as ''x'' increases.<ref>The [derivative](/source/derivative) <math>\ y \prime = \frac{-k}{x^2} \ </math> is negative for [positive real numbers](/source/positive_real_numbers) ''x'' and as well for negative real numbers. Thus the slope is everywhere negative except at the [singularity](/source/singular_point_of_a_curve) ''x'' = 0.</ref>

In [finance](/source/finance), an inverse correlation between the [returns](/source/rate_of_return) on two different assets enhances the [risk](/source/financial_risk)-reduction effect of  [diversifying](/source/diversification_(finance)) by holding them both in the same portfolio.

==See also==
* [Diminishing returns](/source/Diminishing_returns)

==References==
{{Reflist}}

==External links==
* Michael Palmer [http://ordination.okstate.edu/STATS.htm Testing for correlation] from [Oklahoma State University–Stillwater](/source/Oklahoma_State_University%E2%80%93Stillwater)

Category:Independence (probability theory)

---
Adapted from the Wikipedia article [Negative relationship](https://en.wikipedia.org/wiki/Negative_relationship) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Negative_relationship?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.