A correlation coefficient is used in statistics to describe a pattern or relationship between two variables. A negative correlation describes the extent to which two variables move in opposite directions. For example, for two variables, X and Y, an increase in X is associated with a decrease in Y. A negative correlation coefficient is also referred to as an inverse correlation. Correlation relationships are graphed in scatterplots.

### Key Takeaways

- A correlation coefficient measures the strength of the relationship between two variables.
- The most commonly used correlation coefficient is the Pearson coefficient, which ranges from -1.0 to +1.0.
- A positive correlation indicates two variables that tend to move in the same direction.
- A negative correlation indicates two variables that tend to move in opposite directions.
- A correlation coefficient of -0.8 or lower indicates a strong negative relationship, while a coefficient of -0.3 or lower indicates a very weak one.

$\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}$

## Negative Versus Positive Correlation

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. In other words, 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. The list below shows what different 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

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

A strong correlation does not indicate a causal relationship.

## 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, where a change in one variable perfectly predicts the changes in the other. In reality, these numbers are rarely seen, as perfectly linear relationships are rare.

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

## Examples of Positive and Negative Correlation Coefficients

For example, as the temperature increases outside, the amount of snowfall decreases; this shows a negative correlation and would, by extension, have a negative correlation coefficient.

A positive correlation coefficient would be the relationship between temperature and ice cream sales; as temperature increases, so too do ice cream sales. 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) because temperature typically has no bearing on whether people consume fast food.

## What Does a Correlation Coefficient of Zero Mean?

A correlation coefficient of zero indicates the absence of a relationship between the two variables being studied. If two variables have a correlation coefficient of zero, then it is impossible to predict if or how one variable will change in response to changes in the other variable.

## Does a Correlation Coefficient of -0.8 Indicate a Strong or Weak Negative Correlation?

A correlation coefficient of -0.8 indicates an exceptionally strong negative correlation, meaning that the two variables tend to move in opposite directions. The closer the coefficient is to -1.0, the stronger the negative relationship will be.

## What Is the Difference Between a Negative Correlation and a Positive Correlation?

A negative correlation indicates two variables that tend to move in opposite directions: a positive change in one variable will be accompanied by a negative change in the other variable. A positive correlation indicates that the variables move in the same direction: a positive change in one variable will tend to accompany a positive change in the other variable.

## The Bottom Line

A negative correlation can indicate a strong relationship or a weak relationship. Many people think that a correlation of –1 indicates no relationship. But the opposite is true. A correlation of -1 indicates a near-perfect relationship along a straight line, which is the strongest relationship possible. The minus sign simply indicates that the line slopes downwards, and it is a negative relationship.