Response and explanatory variables and association

Understanding the association between two variables is essential for effective data analysis and decision-making. Distinguishing between explanatory and response variables and identifying whether data is categorical or numerical helps determine appropriate analytical methods. For two numerical data, scatterplots provide a visual display, with key features such as direction, form, and strength of association aiding interpretation. These concepts support trend identification, reliable predictions, and practical applications in fields like science, economics, and social research.

Use this page to revise the following concepts within response and explanatory variables and association:

Bivariate Data and Types of Data

Bivariate data consists of two variables recorded from the same subject, often used to examine associations between them (e.g., “Is there a relationship between study time and exam scores?”). Categorical data represents distinct groups that cannot be measured numerically (e.g., eye colour; preferred mode of transport), while numerical data involves measurable or countable values (for example, height, or number of siblings).

For a more detailed explanation of bivariate, categorical, and numerical data, refer to Types of Data.

Explanatory and Response Variables

Before exploring the association between two variables, it is important to identify the explanatory and response variables. The explanatory variable is the variable that is expected to influence or predict changes in the response variable.

Summary Table with Explanatory and Response Variables Examples

Question Explanatory Variable Response Variable
Do older people watch TV for a longer duration? Age Hours of TV watched
Can we predict a person’s holiday expenditure based on their annual income? Annual Income Holiday expenditure
Does a student’s test result depend on the number of hours they sleep? Hours of sleep Test result

Scatterplot

A scatterplot is a graphical representation of bivariate data when both variables are numerical. Each data point on the plot represents an individual observation. The response variable is plotted on the vertical \(y\)-axis, while the explanatory variable is plotted on the horizontal \(x\)-axis. Scatterplots are useful for visualising relationships and identifying trends between two numerical variables.

Clear Pattern and Association

Scatterplots can be used to determine whether there is an association between the two variables. If the data points are randomly scattered across the plots, no clear pattern is present, indicating there is no association between the two variables. However, if the data points follow a noticeable trend rather than being randomly distributed, a clear pattern exists.

When a clear pattern is observed, the association can be described using three key features: direction, form, and strength.

No clear pattern Clear pattern

The points in the scatterplot are randomly scattered across the plot.

First graph: The data points in the scatterplot following a clear downward trend. The points are closely clustered around an invisible diagonal path from the upper left to the lower right. Second graph: The scatterplot displays a set of data points spread across the graph, following a general downward trend. The spacing between points is moderate, with some points lying closer together and others slightly father apart. Third graph: The scatterplot displays data points forming a very tight upward trend. The points are closely clustered around an invisible diagonal line from the lower left to the upper right. The pattern is nearly linear.

Direction and Outliers

The direction of a scatterplot describes the overall trend of the data points.

  • Positive association between variables: the points in the scatterplot trend upwards from left to right, indicating that as the value of the explanatory variable increases, the value of the response variable tends to increase.
  • Negative association between variables: the points in the scatterplot trend downwards from left to right, indicating that as the value of the explanatory variable increases, the value of the response variable tends to decrease.
  • No association between variables: the points are randomly scattered across the plot, showing no consistent change in the value of the response variable when the value of the explanatory variable changes.

Outliers are the points that do not follow the overall pattern of the scatterplot and are clearly separated from the other data points. Identifying outliers is important, as outliers may indicate potential errors in data collections and unusual observations that require further investigation.

Positive association Negative association No associationOutliers

The points in the scatterplot trend upwards from left to right. The points are generally close to each other.

The points in the scatterplot trend downwards from left to right. The spacing between points is moderate, with some points lying closer together and others slightly father apart.

The points in the scatterplot are randomly scattered across the plot.

The scatterplot shows data points generally trending upward from left to right. However, one point is positioned high above the main cluster, while another is far below it, both separated from the rest of the data.

Form

The form of a scatterplot describes the overall pattern of the data points and can be classified as linear or non-linear.

  • Linear form: the points are scattered around a straight line, following a linear pattern.
  • Non-linear form: the points scattered around a curved line, following a non-linear pattern.

Dotted lines are sometimes added to highlight the pattern of data points.

Linear Form Non-linear Form

Data points trend upward from left to right. A red dotted line is added to highlight the pattern, which follows a straight line.

Data points trend upward from left to right. A red dotted line is added to highlight the pattern, which follows a curved line.

Strength

The strength of an association measures how closely the data points follow a specific pattern and can be classified as strong, moderate, and weak.

  • Strong association: data points are tightly clustered around a trend line, forming a clear pattern.
  • Moderate association: data points are more broadly spread around a trend line.
  • Weak association: data points are loosely scattered around a trend line.
Strong Positive Association Moderate Negative Association Weak Positive Association

Data points in the scatterplot are tightly clustered around an increasing trend from left to right.

Data points in the scatterplot are moderately clustered around a decreasing trend from left to right.

Data points in the scatterplot are loosely clustered around an increasing trend from left to right.