Scatter plots and lines of best fit

Interpret scatter plots, correlation and lines of best fit, and use models to make predictions.

Also known as: correlation, line of best fit, two-variable data, linear models.

SAT · Problem-Solving and Data Analysis GCSE Maths · Statistics IGCSE Maths · Statistics

Key ideas

Positive correlation: as one variable increases, the other tends to increase.
A line of best fit's slope is the predicted change in y for each 1-unit increase in x.
Predictions inside the data range (interpolation) are more reliable than outside it (extrapolation).
Correlation does not prove causation.

Reading scatter plots

Each point is one observation of two variables. Describe the pattern by direction (positive or negative), strength (how close points lie to a line) and any outliers.

Lines of best fit

A model such as $y = 2.5x + 40$ predicts $y$ from $x$ . Here the slope $2.5$ means each extra unit of $x$ is associated with an increase of about $2.5$ in $y$ ; the intercept $40$ is the predicted $y$ when $x = 0$ (only meaningful if $x = 0$ makes sense).

The residual for a point is actual $-$ predicted.

Using models carefully

Predict within the range of the data where possible. A strong correlation between ice-cream sales and sunburn doesn't mean one causes the other; both rise with sunny weather.

Worked examples

Example 1 · Interpreting a slope

A model for a plant's height is $h = 1.8w + 12$ , where $h$ is height in cm and $w$ is weeks since planting. Interpret $1.8$ .

Step 1.1.8 is the slope: the predicted increase in height for each additional week.

Answer: The plant is predicted to grow about 1.8 cm per week.

Example 2 · Residual

Using $y = 3x + 5$ , a point has $x = 4$ and $y = 20$ . Find the residual.

Step 1.Predicted $y = 3(4) + 5 = 17$ .
Step 2.Residual $= 20 - 17 = 3$ .

Answer: 3

Common mistakes

Interpreting the intercept when x = 0 is outside the data or meaningless.
Calling a correlation "causation".
Computing residual as predicted − actual (the sign matters).

Try it yourself

Original practice questions. Your answer is checked instantly.

Question 1Scatter plots and lines of best fitEasier

In a scatter plot, students who spent more hours revising tended to score higher on a test. What kind of correlation does this show?

Question 2Scatter plots and lines of best fitEasier

A line of best fit is $y = 2.5x + 40$ . What value of $y$ does it predict when $x = 12$ ?