Line of best fit by eye through the double mean point; the regression line y = a + bx; interpreting gradient and intercept; using the line for prediction with awareness of interpolation and extrapolation.

What this dot point is asking

Edexcel code 2e.04 requires you to determine a line of best fit by eye, drawn through the calculated double mean point $(\bar{x}, \bar{y})$, and (Higher tier) to use the regression line of the form $y = a + bx$. You must interpret the gradient and intercept in context and use the line to make predictions, while being aware of the dangers of interpolation and extrapolation. Non-linear models are not tested.

Drawing the line of best fit

A line of best fit is a single straight line that passes as close as possible to all the points on a scatter diagram, showing the overall trend. The key technique Edexcel requires is to use the double mean point.

Plotting $(\bar{x}, \bar{y})$ first gives you a fixed, accurate anchor, so your line is no longer drawn purely by eye. You then rotate the line about this point until it balances the points above and below.

The regression line y = a + bx

At Higher tier the line of best fit is treated as a regression line written in the form

This is the same straight-line idea as $y = mx + c$ from mathematics, just with the constant written first. You can find the equation from two points on the line, or from the gradient and the double mean point, by substituting into $y = a + bx$ and solving for $a$.

Interpreting gradient and intercept

Marks are won by interpreting the line in context, not just stating numbers:

• The gradient $b$ is the rate of change: "for each extra year of age, the value falls by GBP $800$" for a gradient of $-0.8$ (thousand pounds per year).
• The intercept $a$ is the predicted $y$ when $x = 0$: "a brand new car (age $0$) is predicted to be worth GBP $12{,}000$". Be careful, because the intercept is only meaningful if $x = 0$ is sensible for the context.

Using the line for prediction

To predict a value, substitute the known $x$ into the equation (or read off the line). This is reliable for interpolation (an $x$ inside the data range) but unreliable for extrapolation (an $x$ beyond the data), because the linear trend may not continue. Always check whether the prediction point lies within the range of the original data before trusting it.

The strength of the correlation also affects how much you should trust a prediction. If the points lie close to the line (strong correlation), predictions made by interpolation are reasonably reliable; if the points are widely scattered (weak correlation), even an interpolated prediction carries a large uncertainty. So when judging a prediction, consider both whether it is an interpolation or extrapolation and how strong the underlying correlation is.

When a line of best fit is appropriate

Edexcel only tests linear models, so a line of best fit should be used when the scatter diagram shows a roughly straight-line trend. If the points clearly curve, a straight line is a poor model and any prediction from it is unreliable. You should also ignore a single outlier when positioning the line by eye, since one stray point can distort it; mention the outlier rather than letting it drag the line. Recognising that a straight line does not suit curved data is part of choosing an appropriate model.