Question

Association. A researcher investigating the association between two variables collected some data and was surprised when he calculated the correlation. He had expected to find a fairly strong association, yet the correlation was near 0 . Discouraged, he didn't bother making a scatterplot. Explain to him how the scatterplot could still reveal the strong association he anticipated.

Short answer

A scatterplot might reveal a strong non-linear association, even if the correlation is near 0.

Step-by-step solution

Understand Correlation Coefficient

The correlation coefficient, denoted as \(r\), measures the strength and direction of a linear relationship between two variables. A correlation near 0 indicates a very weak linear relationship, implying that the data points do not follow a linear trend.

Recognize Non-linear Associations

If the relationship between the two variables is non-linear (for example, quadratic, exponential, or another form of curve), the correlation coefficient might be near 0, even though there is a strong association of a non-linear nature.

Create the Scatterplot

Plot the data on a scatterplot to visually inspect the relationship between the variables. By doing this, you can directly observe patterns, clusters, or any type of non-linear relationship that won't be reflected by the correlation coefficient alone.

Analyze the Scatterplot

Look for shapes or trends in the data points on the scatterplot. For example, a parabolic shape (U or inverted U), exponential growth or decay, or any circular or spiral pattern could suggest a non-linear relationship not captured by the correlation coefficient.

Conclude with a Visual Analysis

Based on the scatterplot's visual representation, determine if there is indeed a strong association between the variables, just not in a linear form. This emphasizes why scatterplots are crucial for identifying relationships that the correlation coefficient may miss.

Key concepts

Non-linear relationships

Continuous data can often be complex and not follow a straight line. Non-linear relationships occur when two variables are related, but their relationship doesn't create a straight line when graphed. This could be a curve, like a parabola or an exponential growth pattern. These types of relationships are not appropriately measured by the correlation coefficient, which only assesses linear connections.
A common example of a non-linear relationship is a quadratic relationship, where one variable might increase as another increases, but only up to a point, after which it decreases,— creating an inverted U-shape. Recognizing the specifics of these associations can help in understanding the real dynamics underlying the datasets.

Scatterplot analysis

A scatterplot is a type of graph used to visually represent the relationship between two variables. Each point on the graph corresponds to one observation (or data point) comprising a pair of numbers.
By examining the distribution and arrangement of these points, patterns such as clusters or outliers can be distinguished. This visual representation can uncover trends and relationships, whether they are linear or non-linear.
Creating a scatterplot is an essential step in data analysis because it provides a visual understanding that numerical statistics, such as the correlation coefficient, may not reveal. Being a simple yet powerful tool, it allows researchers to observe directly the nature of the relationships within their data, making it clear if relationships are present—even in non-linear forms.

Correlation coefficient

The correlation coefficient, denoted by 'r', is a numerical measure that assesses the strength and direction of a linear relationship between two variables. It ranges between -1 and 1.

• If 'r' is close to 1, it indicates a strong positive linear relationship.
• If 'r' is close to -1, it suggests a strong negative linear relationship.
• A value near 0 implies little to no linear relationship.

However, this measure has its limitations. When the underlying relationship is non-linear, the correlation coefficient can be very close to zero—even when a strong association exists.
Understanding the limitation of 'r' is crucial, as it reaffirms the importance of using additional analysis methods, such as scatterplot inspections, to fully comprehend data relationships.

Linear vs non-linear associations

It's important to distinguish between linear and non-linear associations when studying relationships between variables. Linear associations involve a straight-line relationship where changes in one variable predict changes in another at a constant rate. These associations are easily captured by the correlation coefficient.
Non-linear associations, on the other hand, entail relationships that are not constant—they could involve curvature, peaks, or troughs. Because these patterns deviate from a straight-line form, the correlation coefficient might fail to capture the essence of these associations. This is why non-linear associations can appear inconsequential if the analysis relies solely on a correlation coefficient, emphasizing the necessity for visual tools like scatterplots to explore the true nature of the connections. Understanding both types helps provide a comprehensive picture and avoid misinterpretations in data analysis.