Understanding Negative Correlation Coefficient in Statistics

A negative correlation coefficient is used in statistics to indicate an inverse relationship between two variables. When one variable increases, the other tends to decrease, and the strength of this relationship is reflected in the value of the coefficient. Understanding how a negative correlation compares with positive or zero correlation coefficients helps clarify patterns within data. This is especially useful for analyzing real-world trends in fields like finance, economics, and science.

How to Calculate a Correlation Coefficient  

$\begin{aligned}&r=\frac{\sum(x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\sum(x_i-\bar{x})^2\sum(y_i-\bar{y})^2}}\\&\textbf{where:}\\&r=\text{Correlation coefficient}\\&x_i=\text{Values of the $x$-variable in a sample}\\&\bar{x}=\text{Mean of the values of the $x$-variable}\\&y_i=\text{Values of the $y$-variable in a sample}\\&\bar{y}=\text{Mean of the values of the $y$-variable}\end{aligned}$​r=∑(xi​−xˉ)2∑(yi​−yˉ​)2​∑(xi​−xˉ)(yi​−yˉ​)​where:r=Correlation coefficientxi​=Values of the x-variable in a samplexˉ=Mean of the values of the x-variableyi​=Values of the y-variable in a sampleyˉ​=Mean of the values of the y-variable​

Comparing Negative and Positive Correlations 

A negative correlation demonstrates a connection between two variables in the same way as a positive correlation coefficient and the relative strengths are the same. A correlation coefficient of 0.85 shows the same strength as a correlation coefficient of -0.85.

Correlation coefficients are always values between -1 and 1 where -1 shows a perfect, linear negative correlation and 1 shows a perfect, linear positive correlation. This list shows what some correlation coefficient values indicate:

Exactly –1. A perfect negative, downward-sloping linear relationship

–0.70. A strong negative, downward-sloping linear relationship

–0.50. A moderate negative, downhill-sloping relationship

–0.30. A weak negative, downhill-sloping linear relationship

0. No linear relationship

+0.30. A weak positive, upward-sloping linear relationship

+0.50. A moderate positive, upward-sloping linear relationship

+0.70. A strong positive, upward-sloping linear relationship

Exactly +1. A perfect positive, upward-sloping linear relationship

Thinking about the numeric value of a correlation coefficient as a percentage. A 20% move higher for variable X would equate to a 20% move lower for variable Y.

Understanding Extreme Correlation Coefficients 

A correlation coefficient of zero or close to zero shows no meaningful relationship between variables. A coefficient of -1.0 or +1.0 indicates a perfect correlation. A change in one variable perfectly predicts the changes in the other. These numbers are rarely seen in reality because perfectly linear relationships are rare.

An example of a strong negative correlation would be -0.97. The variables would move in opposite directions in a nearly identical move. The values demonstrate the strength of a relationship as the numbers approach 1 or -1. Numbers of 0.92 or -0.97 would show a strong positive and negative correlation respectively.

Real-World Examples of Correlation Coefficients 

The amount of snowfall decreases as the temperature increases outside. This shows a negative correlation and would have a negative correlation coefficient.

A positive correlation coefficient would be the relationship between temperature and ice cream sales. Ice cream sales increase as temperature increases. This relationship would have a positive correlation coefficient.

A relationship with a correlation coefficient of zero or very close to zero might be temperature and fast food sales assuming there's zero correlation for illustrative purposes. Temperature typically has no bearing on whether people consume fast food.

The Bottom Line

The strength of a negative correlation can vary, but a correlation of –1 represents the strongest negative linear relationship between two variables. Contrary to a common misconception, –1 doesn't indicate the absence of a relationship. Rather, it signals a perfect inverse relationship along a straight line. The minus sign simply reflects direction, not the weakness of the relationship.