What Is the Correlation Coefficient?

The correlation coefficient is a statistical measure of the strength of a linear relationship between two variables. Its values can range from -1 to 1. A correlation coefficient of -1 describes a perfect negative, or inverse, correlation, with values in one series rising as those in the other decline, and vice versa. A coefficient of 1 shows a perfect 澳洲幸运5官方开奖结果体彩网:positive correlation, or a direct relationship. A correlation🀅 coefficient of 0 means there is no linear relationship.

Correlation coefficients are used in science and finance to assess the degree of association between two variables, factors, or data sets. For example, as high oil prices are favorable for crude producers, one might assume that the correlation between oil prices and forward returns on oil stocks is strongly positive. Calculating the 澳洲幸运5官方开奖结果体彩网:correlation coefficient for these variables based on market data reveals a moderate and inconsistent correlation over lengthy periods.

Understanding the Correlation Coefficient

Different types of correlation coefficients are used to assess correlation based on the properties of the compared data. By far the most common is the 澳洲幸运5官方开奖结果体彩网:Pearson coefficient, known as “Pearson’s R,” which measures the st𒊎rength and direction of a linear relationship betwee♉n two variables.

The Pearson coefficient uses a mathematical statistics formula to measure how closely the data points combining the two variables (with the values of one data series plotted on the x-axis and the corresponding values of the other series on the y-axis) approximate the 澳洲幸运5官方开奖结果体彩网:line of best fit. The line of best fit can be determined through regression analysis.

The further the coefficient is from zero, whether it is positive 🍰or negative, the better the fit and the greater the correlation. The values of -1 (for a negative correlation) and 1 (for a positive one) describe perfect fits in which all data poi❀nts align in a straight line, indicating that the variables are perfectly correlated.

In other words, the relationship isꦕ so predictable that the value of one variable can be determined from the matched value of the other. The closer the co🤡rrelation coefficient is to zero, the weaker the correlation, until at zero no linear relationship exists at all.

Assessments of correlation streng♎th based on the correlation coefficient value vary by application. In physics and chemistry, a correlation coefficient should be lower than -0.9 or higher than 0.9 for the correlation to be considered meaningful, while in social sciences the threshold could be as high as -0.5 and as low as 0.5.

For correlation coefficients derived from sampling, the determination of statistical significance depends on the p-value, which is calculated from the data sample’s size as well as the value of the coefficient.

Formula for the Correlation Coefficient

To calculate the Pearson correlation, start by determining each variable’s 澳洲幸运5官方开奖结果体彩网:standard deviation as well as the 澳洲幸运5官方开奖结果体彩网:covariance between them. The correꦺlation coefficient is covarian♊ce divided by the product of the two variables’ standard deviations.

$\begin{aligned} &\rho_{xy} = \frac { \text{Cov} ( x, y ) }{ \sigma_x \sigma_y } \\ &\textbf{where:} \\ &\rho_{xy} = \text{Pearson product-moment correlation coefficient} \\ &\text{Cov} ( x, y ) = \text{covariance of variables } x \text{ and } y \\ &\sigma_x = \text{standard deviation of } x \\ &\sigma_y = \text{standard deviation of } y \\ \end{aligned}$​ρxy​=σx​σy​Cov(x,y)​where:ρxy​=Pearson product-moment correlation coefficꦛientCov(x,y)=covariance of variables x and yσx​=standard deviation of xσy​=standard deviation of y​

Standard deviation is a measure of the dis𓄧persion of data from its average. Covariance shows whether the two variables tend to 🔯move in the same direction, while the correlation coefficient measures the strength of that relationship on a normalized scale, from -1 to 1.

The formula above can be elaborated as

$\begin{aligned}&r = \frac { n \times ( \sum (X, Y) - ( \sum (X) \times \sum (Y) ) ) }{ \sqrt { ( n \times \sum (X ^ 2) - \sum (X) ^ 2 ) \times ( n \times \sum( Y ^ 2 ) - \sum (Y) ^ 2 ) } } \\&\textbf{where:}\\&r=\text{Correlation coefficient}\\&n=\text{Number of observations}\end{aligned}$​r=(n×∑(X2)−∑(X)2)×(n×∑(Y2)−∑(Y)2)​n×(∑(X,Y)−(∑(X)×∑(Y)))​where:r=Correlation coefficientn=Number of observations​​

Correlation Statistics and Investing

The correlation coefficient is particularly helpful in ✅澳洲幸运5官方开奖结果体彩网:assessing and managing investment ris🍸ks. For example, 澳洲幸运5官方开奖结果体彩网:modern portfolio theory suggests diversification can reduce the volatility of a portfolio’s returns, curbing risk. The correlation coefficient between historical returns 澳洲幸运5官方开奖结果体彩网:can indicate wheth𝓀er adding an investment to a portfolio will improve its diversification.

Correlation calculations are also a staple of 澳洲幸运5官方开奖结果体彩网:factor investing, a strategy for constructing a portfolio based on factors associated with excess returns. Meanwhile, quantitative traders use historical correlations and correlation coefficients to anticipate near-term changes in securities prices.

Limitations of the Pearson Corಞrelation Coefficient

Correlation does not imply causation, as the saying goes, and the Pearson coe🃏fficient cannot determine whether one of the correlated variables is dependent on the other.

Nor does the correlation coefficient show what proportion of the variation in the dependent variable is attributable to the independent variable. That's shown by the 澳洲幸运5官方开奖结果体彩网:coefficient of determination, also known as “澳洲幸运5官方开奖结果体彩网:R-squared,💖” which is simply the correlation coefficient squared💜.

The correlation coefficient also does not describe the slope of the line of best fit; the slope can be determined with the 澳洲幸运5官方开奖结果体彩网:least squares method in regression analysis.

The Pearson correlation coefficient can’t be used to assess nonlinear associations or those arising frꩲom sampled data not subject to a normal distribution. It can also be distorted by outliers—data points far outside the scatterplot of a distribution.

Those relationships can be analyzed using nonparametric methods, such as Spearman’s correlation coefficient, the Kendall rank correlation coefficient, or a polychoric correlation coefficient.

Finding Correlation Coefficients in Excel

There are a few ways to 澳洲幸运5官方开奖结果体彩网:calculate correlation in Excel. The simplest way is to input two data series in adjacent columns and use the built-in correlation formula:

If you want to create a correlation matrix across a range of data sets, Excel has a data analysis plugin.  To use it, you must first enable the data analysis ToolPak. This can be done by clicking on "file," and then "options," which should open the Excel options dialogue box. In the box, click on "add-ins" and then on the "manage" dropdown select "Excel add-ins" and click on "go." This will cause the add-ins box to appear. Check the checkbox for "analysis TookPak," then click "ok." The enable process should now be complete.

To use the data analysis plugin, click on the "data" ribbon and then select "data analysis," which should open a box. In the box, click on "correlation" and then "ok." The correlation box will now open and you can enter the input ranges, either manually or by selecting the relevant cells.

In this case, our columns are titled, so we want to check the box “labels in first row,” so Excel knows to treat these as titles. Then you can choose to output on the same sheet or on a new sheet.

Hitting enter will produce the correlation matrix. You can add so♏me t🐭ext and conditional formatting to clean up the result.

The Bottom Line

The correlation coefficient describes how one variable moves in relation to another. A posไitive correlation indicates that the two move in the same direction, with a value of 1 denoting a perfect positive correlation. A value of -1 shows a perfect negative, or inverse, correlation, while 0 means no linear correlation exists.