Question

Explain how to decide whether a data set can be represented by a linear model.

Short answer

To decide whether a data set can be represented by a linear model, evaluate the correlation between variables, consistency in rate of change, and proximity of data points to a line of best fit. If these indicators align with a linear relationship, a linear model is likely appropriate.

Step-by-step solution

Understand the characteristics of a linear model

A linear model characterizes a relationship between two variables in which a change in one variable results in a proportional change in the other variable. In mathematical terms, a linear model represents a straight-line graph, and is usually represented by the equation \(y = mx + c\), where \(m\) is the slope of the line, \(c\) is the y-intercept and \(y\) and \(x\) are the variables being studied.

Examine the correlation between the variables

Analyze your data for correlation between the variables. This could involve creating a scatter plot of your data points to see if they tend to form a straight line when graphed. Calculation of a correlation coefficient, a statistical measure that describes the degree of relationship between two variables, can also help (values close to -1 or 1 imply a strong linear relationship).

Look for consistent rate of change

In a linear model, there's usually a constant rate of change. This means that for every unit increase or decrease in one variable, the other variable increases or decreases by a consistent amount. You can often detect this by looking at the numbers in the data set or by calculating the slopes between various points to see if they're approximately equal.

Evaluate fit to a best fit line

Using a line of best fit, or a trendline, on your scatter plot can help you decide whether a linear model is appropriate. If the majority of data points are close to this line, it suggests the data set can be represented well by a linear model. If the points are widely spread from the line, a non-linear model may better represent the data.