Understanding Correlation: Exploring Relationships Between Variables

Have you ever wondered whether students who spend more hours in the library actually score higher marks? Or whether the number of years a person has been reading is linked to the size of their personal collection? Questions like these are everywhere in research, and they all point to a single statistical idea: correlation. Correlation helps us measure whether two things move together, in which direction they move, and how strongly. In this post, we will unpack what correlation really means, look at the main methods used to measure it, and work through a practical example, while keeping a sharp eye on what correlation can and cannot tell us.

What is correlation?
Direction of the relationship
Strength of the relationship
Methods of correlation
Scatter diagrams
Pearson’s product moment correlation
Spearman’s rank correlation
A real-life example
Understanding the limitations of correlation

What is correlation?

Correlation is a statistical technique that measures the relationship between two variables. In simple terms, it tells us whether a change in one variable is associated with a change in another. When two variables tend to change together, they are said to be correlated. Correlation is a bivariate analysis that captures both the strength of the association between two variables and the direction of that relationship.

The result of a correlation is expressed as a number called the correlation coefficient, usually written as r. This value always lies between -1 and +1. The two parts of this number carry different information.

Direction of the relationship

The sign of the coefficient tells us the direction. A positive sign means both variables move in the same direction, while a negative sign means they move in opposite directions. When two variables move in the same direction the correlation is positive, and when they move in opposite directions it is negative. For example, height and weight often show a positive correlation, while the price of a product and the quantity demanded usually show a negative one.

Strength of the relationship

The size of the number, ignoring its sign, tells us how strong the relationship is. A value of exactly +1 or -1 indicates a perfect relationship, where the variables move together in a completely consistent way. As the coefficient moves closer to zero, the association becomes weaker, and a value of zero means there is no linear or monotonic association between the two variables. So a coefficient of +0.9 signals a strong positive relationship, while +0.1 signals a very weak one.

Methods of correlation

There are several ways to study correlation, ranging from a quick visual check to precise mathematical formulas. The three methods covered in most research methodology courses are scatter diagrams, Pearson’s product moment correlation, and Spearman’s rank correlation. Each suits a different kind of data and purpose.

Scatter diagrams

A scatter diagram, also called a scatter plot, is the simplest way to study correlation. It is a graphical tool where each pair of values is plotted as a single point on a graph, with one variable on the X-axis and the other on the Y-axis. By observing how the points are distributed, we can identify the direction and strength of the relationship without performing any complex calculations.

The pattern of the dots reveals the type of correlation. When the points rise from the lower left to the upper right, both variables increase together, showing positive correlation; when they fall from the upper left to the lower right, the correlation is negative. If the points are scattered randomly with no clear pattern, there is little or no correlation. When all the points fall exactly on a straight line, the correlation is perfect.

The scatter diagram has clear advantages. It is easy to understand at a glance, and it is not distorted by extreme values the way some mathematical methods are. Its main limitation is that it only gives a rough idea of the relationship. It shows whether a correlation exists and its general nature, but it cannot give an exact numerical measure. For that, we need the calculated methods below.

Pearson’s product moment correlation

Pearson’s product moment correlation coefficient, symbolised by r, is the most widely used measure of correlation. It evaluates the linear relationship between two continuous variables, where a change in one variable is associated with a proportional change in the other. It works on the raw values of the data rather than on their order or rank.

Because Pearson’s method is based on the actual numbers, it comes with certain conditions. It is a parametric statistic and is best used for data that follows a normal or approximately normal distribution. It is the right choice when both variables are measured on a continuous scale, such as marks, height, income, or temperature, and when you expect a straight-line relationship between them. If these conditions are not met, the result can be misleading.

Spearman’s rank correlation

Spearman’s rank correlation, named after Charles Spearman and symbolised by r_s or the Greek letter rho, takes a different approach. Instead of using the raw data, it is based on the ranks of the values. It is a non-parametric statistic used for data that is not normally distributed or whose distribution is unknown, and it measures the strength and direction of a rank association between two variables.

This makes Spearman’s method especially useful in two common situations. The first is when the data is ordinal, meaning it can be ranked but not precisely measured, such as ratings, grades, or preference orders. Spearman correlation is often used to evaluate relationships involving ordinal variables and measures the monotonic relationship, where variables change together but not necessarily at a constant rate. The second is when the data is skewed or contains outliers. Spearman’s coefficient is appropriate when one or both variables are skewed or ordinal, and it is robust when extreme values are present.

The key difference is worth remembering. A perfect Spearman correlation results when two variables are related by any monotonic function, whereas a perfect Pearson correlation occurs only when they are related by a linear function. In short, choosing between Pearson and Spearman is not about preference. It is about matching the method to the nature of your data.

A real-life example

Let us apply this to a question a library or education researcher might genuinely ask: is a person’s age related to the number of books they have read? Suppose we survey ten readers and record each person’s age and the total number of books they say they have read. To study this, we first plot the pairs on a scatter diagram. If the points generally rise from left to right, we have visual evidence of a positive correlation, suggesting that older readers tend to have read more books, which makes intuitive sense since they have had more years to read.

To put a number on it, we could calculate Pearson’s coefficient if both age and book counts are reasonably distributed. Imagine we get an r of +0.75. This tells us there is a fairly strong positive relationship: as age increases, the number of books read tends to increase too. If instead we had simply asked readers to rank themselves as light, moderate, or heavy readers, the data would be ordinal, and Spearman’s rank correlation would be the appropriate method.

Understanding the limitations of correlation

Here is the most important caution in all of statistics: correlation does not imply causation. This phrase refers to the fact that we cannot legitimately conclude a cause-and-effect relationship between two variables solely from an observed association between them. Even a strong correlation between age and books read does not prove that growing older causes a person to read more.

One reason for this is the third-variable problem. Two variables can be statistically related not because one causes the other, but because some third hidden variable affects both; such relationships are called spurious correlations. In our example, education level or income might influence both age group and reading habits. A famous illustration is the correlation between ice cream sales and shark attacks, which rise together not because one causes the other, but because warmer weather drives both.

A second limitation is that the common correlation methods only detect linear or monotonic relationships. Correlation coefficients measure only linear (Pearson) or monotonic (Spearman) relationships, while other kinds of relationships are also possible. Two variables might have a strong curved relationship and still produce a correlation coefficient close to zero. This is exactly why drawing a scatter diagram first is good practice. It lets you see the shape of the relationship before trusting a single number.

Finally, a small correlation calculated from a very large sample can be statistically significant without being meaningful in practice. Numbers always need careful interpretation alongside the context of the research.

What do you think? If you found a strong positive correlation between the number of hours students spend in a library and their exam scores, what third variables might explain that link before you conclude one causes the other? And for a study comparing reader satisfaction ratings with the number of visits per month, would you choose Pearson’s or Spearman’s method, and why?

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PDF 📄

Table of Contents

What is correlation? 🔗

Direction of the relationship 🔗

Strength of the relationship 🔗

Methods of correlation 🔗

Scatter diagrams 🔗

Pearson’s product moment correlation 🔗

Spearman’s rank correlation 🔗

A real-life example 🔗

Understanding the limitations of correlation 🔗