Correlation Coefficient Interpretation: How to Effectively Interpret the Correlation Coefficient

Data analysis is more relevant in today’s world than it ever was before. Data analysis techniques are an important part of all fields, from research and scientific study to business and marketing. Large companies often rely on data analysis techniques to get an edge over their competitors and sell more products or services. The main goal of data analysis is deriving useful conclusions from available data sources, which can then be used to make logical decisions. It is not an easy job- not everyone can do it- and consequently data analysts and statisticians are never short of work. To become a statistician, you need to be familiar with the major techniques of statistical analysis and be good with figures and numbers in general. You don’t need to want to be a statistician to learn statistics and math, however. Learning statistics will always be useful, whatever the field you’re in. If you’re interested in learning statistics, you can take our introductory course that helps you learn statistics in an easy manner. We cover all the things you need to know to acquire a thorough understanding of the topic- including the basics and the major techniques of data analysis.

In this tutorial, we’re going to take a look at how to interpret the correlation coefficient. It’ll be easier to understand this tutorial if you know how to calculate the correlation coefficient, as we show in this tutorial.

What is the Correlation Coefficient?

In statistics, the correlation coefficient indicates the strength of the relationship between two variables. When we say that two variables are correlated, it means that there exists a definable relationship between the two. If there is a positive correlation between the two variables, it means that when the value of one variable goes up, the value of the other goes up as well and conversely when the value of one variable goes down, the value of the other variable goes down as well. If there is a negative correlation between the two variables, this means that as the value of one variable goes up, the value of the other variable falls and conversely when the value of one variable falls, the value of the other variable goes up.

An example of positive correlation is the relationship between gas prices and food prices. When gas is expensive, food becomes more expensive too, and vice versa. An example of negative correlation is the relationship between the supply and demand of any product. As the supply of a product rises, its demand decreases and vice versa.

The correlation coefficient is used to measure the strength of the linear relationship between two variables on a graph. It also plots the direction of there relationship. The correlation coefficient is calculated by the following formula:

(r) =[ nΣxy - (Σx)(Σy) / Sqrt([nΣx2 - (Σx)2][nΣy2 - (Σy)2])]

What do all the letters stand for?

r: This is the correlation coefficient

n: n specifies the number of values we’re looking at. If we had five instances we were calculating the correlation coefficient for, the value of n would be 5.

x: This is the first data variable.

y: This is the second data variable.

Σ: The Sigma symbol (Greek) is used to calculate the sum of anything placed next to it.

Does the formula seem confusing? Let’s break it down to understand it better. If we were calculating the relationship between weight loss and exercise, for example, weight loss would be variable x and exercise would be variable y. If we were doing it for 10 people, the value of n would be 10. After calculating the results, we would get the value of r. Still confused? If you need more help understanding the formula (or if you want to learn more about correlation in general), we recommend you take our this course to help walk you through the basics of probability and statistics. 

Interpreting the Correlation Coefficient

Let’s continue using the example from above to help us interpret (understand and use) the correlation coefficient. You’re probably thinking that the more you exercise, the more the weight you lose right? That’s true in some cases, of course, but not all the time. Some people gain weight at the gym, others lose it. This muddles the relationship between the two variables we’re studying (exercise and weight loss).

It’s very rare to get a perfect positive (+1) or perfect negative (-1) relationship between two variables in the real world. Because of this, the value of the correlation coefficient will usually hover between 1 and -1, depending on the strength of the relationship between the two variables. To learn more about correlation and to get more examples that deal with the occurrence, you can take a look at our tutorials on the topic. Alternatively, you can just sign up for this introductory course on statistics and learn all the basics in one place.

The Exact Value of the Correlation Coefficient ‘r’

The closer the value of the correlation coefficient is to 1 or -1, the stronger the relationship between the two variables and the more the impact their fluctuations will have on each other. If the value of r is 1, this denotes a perfect positive relationship between the two and can be plotted on a graph as a line that goes upwards, with a high slope. If the value of r is 0.5, this will denote a positive relationship between the two variables and it can be plotted on a graph as a line that goes upward, with a moderate slope. If the value of r is 0, there is no relationship at all between the two variables. If the value of r is -0.5, this will denote a negative relationship between the two variables and it can be plotted on a graph as a line that goes downwards with a moderate slope. If the value of r is -1, it will denote a negative relationship between the two variables and it can be plotted on a graph as a line that goes downwards with a steep slope.

If the value of the correlation coefficient is between 0.1 to 0.5 or -0.1 and -0.5, the two variables in the relationship are said to be weakly related. If the value of the correlation coefficient is between 0.9 and 1 or -0.9 and -1, the two variables are extremely strongly related.

As we discussed earlier, a positive coefficient will show variables that rise at the same time. A negative coefficient, on the other hand, will show variables that move in opposite directions. It’s easy to tell the relationship between by checking the positive or negative value of the coefficient.

Statistical Probability Principle

The correlation coefficient can be further interpreted by performing additional calculations, like regression analysis, which we won’t discuss in detail in the current tutorial. The statistical probability principle can be employed to further understand the relationship between the two variables. If the correlation coefficient is high (.9), the statistical probability that the relationship between the two occurred by chance is very low. If the correlation coefficient is low (.1), the statistical probability that the relationship between the two variables occurred by chance is high. In some cases, when the variables under study are unclear or obscure, we may need to analyze them further with such techniques.

The correlation coefficient can be further interpreted or studied by forming a correlation coefficient matrix. To learn more about the correlation coefficient and the correlation matrix are used for everyday analysis, you can sign up for this course that delves into practical statistics for user experience.