Most of the time, we want to scatterplots of bivariate data to find patterns in data and make inferences about a possible relationship between the two variables. To do this, we look at both the form (or shape) and the strength of the association.
Linear and non-linear relationships
The first way to analyse scatterplots is to describe the shape that the bivariate data takes. Sometimes the data clusters around some kind of curve, so the relationship is:
- linear (a straight line), or
- non-linear (not a straight line) and the two variables have a non-linear relationship. Non-linear data could have a quadratic (parabolic), exponential or hyperbolic shape.
Linear relationship
Quadratic relationship
Exponential relationship
Hyperbolic relationship
Positive and negative relationships
We can further describe linear relationships by whether they are increasing (positive gradient) or decreasing (negative gradient).
- A linear relationship where the dependent variable increases as the independent variable increases is called a positive linear relationship.
- A linear relationship where the dependent variable decreases as the independent variable increases is called a negative linear relationship.
Positive relationship
Negative relationship
What does it mean if the line is close to horizontal (0 gradient)? In terms of data, the dependent variable does not change as the independent variable increases. In other words, the dependent variable doesn't actually depend on the other variable, so we say there is likely no relationship.
The words positive and negative only apply when describing linear relationships. For non-linear relationships (like a quadratic relationship) there can be a mix of gradients - positive in one part and negative in the other.
Strong and weak relationships
The second way to analyse scatterplots is to describe the strength of the relationship. If the data points cluster very closely around a curve, we say that there is evidence of a strong relationship. If the data points are very spread out but there is still an overall curve we say there is evidence of a weak relationship. If the data points are somewhere in the middle we say that there is evidence of a moderate relationship.
For example, almost all data points of a strong linear relationship will lie on or very close to a straight line. If the data points are arbitrarily spread out, then there is probably no linear relationship at all. This could mean that there is a non-linear relationship, or that the two variables are completely unrelated.
Strong relationship
Moderate relationship
Weak relationship
No relationship
We can never be completely certain that there's a linear relationship between any two variables from a scatterplot. This is why we say that that "there is evidence of a linear relationship" or that "there is probably no relationship". To be brief with our words, we often say "there is a linear relationship" and "there is no relationship", but is important to keep in mind what is meant by this.
Practice questions
Question 1
Describe the relationship between the variables observed in the scatterplot below.
Strong positive linear relationship
AStrong negative linear relationship
BNo linear relationship
CWeak positive linear relationship
DWeak negative linear relationship
E
Question 2
Describe the relationship between the variables observed in the scatterplot below.
Weak parabolic relationship
AStrong parabolic relationship
BNo relationship
CWeak linear relationship
DStrong linear relationship
E
Question 3
Which of the following scatterplots demonstrates no relationship?
- Loading Graph...ALoading Graph...BLoading Graph...CLoading Graph...DLoading Graph...E