What Is an Inverse Correlation?

An inverse correlation, also known as negative correlation, is a contrary relationship between two variables such that they move in opposite directions. For example, with variables A and B, as A increases, B decreases, and as A decreases, B increases. In statistical terminology, an inverse correlation is denoted by the correlation coefficient "r" having a value between -1 and 0, with r = -1 indicating perfect inverse correlation.

Graphing Inverse Correlation

Two sets of data points can be plotted on a graph on an x and y-axis to check for correlation. This is called a scatter diagram, and it represents a visual way to check for a positive or negative correlation. The graph below illustrates a strong negative correlation between two sets of data points plotted on the graph.

Scatter Plot diagram
Scatter Plot diagram. Investopedia

Example of Calculating Inverse Correlation

Correlation can be calculated between two sets of data to arrive at a numerical result. The resulting statistic is used in a predictive manner to estimate metrics like the risk reduction benefits of portfolio diversification and other important data. The example presented below shows how to calculate the statistic.

Assume an analyst needs to calculate the degree of correlation between the following two data sets:

  • X: 55, 37, 100, 40, 23, 66, 88
  • Y: 91, 60, 70, 83, 75, 76, 30

There are three steps involved in finding the correlation. First, add up all the X values to find SUM(X), add up all the Y values to find SUM(Y) and multiply each X value with its corresponding Y value and sum them to find SUM(X,Y):

  • SUM(X)=55+37+100+40+23+66+88=409\text{SUM}(X) = 55 + 37 + 100 + 40 + 23 + 66 + 88 = 409SUM(X)=55+37+100+40+23+66+88=409
  • SUM(Y)=91+60+70+83+75+76+30=485\text{SUM}(Y) = 91 + 60 + 70 + 83 + 75 + 76 + 30 = 485SUM(Y)=91+60+70+83+75+76+30=485
  • SUM(X,Y)=(55×91)+(37×60)+(100×70)++(88x×30)=26,926\text{SUM}(X,Y) = (55 \times 91) + (37 \times 60) + (100 \times 70) + \dotso + (88 x\times 30) = 26,926SUM(X,Y)=(55×91)+(37×60)+(100×70)++(88x×30)=26,926

The next step is to take each X value, square it and sum up all these values to find SUM(x2). The same must be done for the Y values:

SUM(X2)=(552)+(372)+(1002)++(882)=28,623\text{SUM}(X^2) = (55^2) + (37^2) + (100^2) + \dotso + (88^2) = 28,623SUM(X2)=(552)+(372)+(1002)++(882)=28,623

SUM(Y2)=(912)+(602)+(702)++(302)=35,971\text{SUM}(Y^2) = (91^2) + (60^2) + (70^2) + \dotso + (30^2) = 35,971SUM(Y2)=(912)+(602)+(702)++(302)=35,971

Noting there are seven observations, n, the following formula can be used to find the correlation coefficient, r:

r=[n×(SUM(X,Y)(SUM(X)×(SUM(Y))][(n×SUM(X2)SUM(X)2]×[nxSUM(Y2)SUM(Y)2)]r = \frac{[n \times (\text{SUM}(X,Y) - (\text{SUM}(X) \times ( \text{SUM}(Y) ) ]} {\sqrt{[(n \times \text{SUM}(X^2) - \text{SUM}(X)^2 ] \times [n x \text{SUM}(Y^2) - \text{SUM}(Y)^2)]}}r=[(n×SUM(X2)SUM(X)2]×[nxSUM(Y2)SUM(Y)2)][n×(SUM(X,Y)(SUM(X)×(SUM(Y))]

In this example, the correlation is:

  • r=(7×26,926(409×485))((7×28,6234092)×(7×35,9714852))r = \frac{(7 \times 26,926 - (409 \times 485))} {\sqrt{((7 \times 28,623 - 409^2) \times (7 \times 35,971 - 485^2))}}r=((7×28,6234092)×(7×35,9714852))(7×26,926(409×485))
  • r=9,883÷23,414r = 9,883 \div 23,414r=9,883÷23,414
  • r=0.42r = -0.42r=0.42

The two data sets have an inverse correlation of -0.42.

What Does Inverse Correlation Tell You?

Inverse correlation tells you that when one variable rises, the other falls. In financial markets, the best example of an inverse correlation is probably the one between the U.S. dollar and gold. As the U.S. dollar depreciates against major currencies, gold is generally perceived to rise, and as the U.S. dollar appreciates, gold declines in price.

Two points need to be kept in mind with regard to a negative correlation. First, the existence of a negative correlation, or positive correlation for that matter, does not necessarily imply a causal relationship. Second, the relationship between two variables is not static and fluctuates over time, which means the variables may display an inverse correlation during some periods and a positive correlation during others.

Key Takeaways

  • Even though two sets of data may have a strong negative correlation, this does not imply that the behavior of one has any influence on or causation relationship with the other.
  • The relationship between two variables can change over time and may have periods of positive correlation as well.

Limitations of Using Inverse Correlation

Correlation analyses can reveal useful information about the relationship between two variables, such as how the stock and bond markets often move in opposite directions. However, the analysis does not fully consider outliers or unusual behavior of a few data points within a given set of data points, which could skew the results.

Also, when two variables show a negative correlation, there might be several other variables that, while not included in the correlation study, do in fact influence the variable in question. Even though two variables have a very strong inverse correlation, this result never implies a cause and effect relationship between the two. Finally, using the results of a correlation analysis to extrapolate the same conclusion to new data carries a high degree of risk.