S4.2StatisticsStretch

Linear Regression

The regression line is the “line of best fit” through bivariate data, written $y = a + b x$ . It lets you predict the response from the explanatory variable — but only safely within the range of the data.

25 min Video by Zeeshan Zamurred Correlation

Edexcel AS Level Maths: Chapter 4 Correlation (Part 2)Watch the full walkthrough before the notes below.

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What you'll be able to do

Understand the regression line y = a + bx
Interpret the gradient and intercept in context
Use the line to make predictions
Distinguish interpolation from extrapolation

The regression line

The least-squares regression line of $y$ on $x$ has the form $y = a + b x$ , where $b$ is the gradient and $a$ the intercept. It is the line that best fits the data by minimising the vertical distances to the points.

y = a + b x

b

= gradient (change in

y

per unit

x

a

= intercept.

Interpreting the coefficients

In context, $b$ is the predicted $change in the response$ for each one-unit increase in the explanatory variable, and $a$ is the predicted response when $x = 0$ . Always state units.

b = 2

is the gradient.

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Interpolation vs extrapolation

Predicting $within$ the data range ( $interpolation$ ) is reliable. Predicting $outside$ it ( $extrapolation$ ) is unreliable, because the linear pattern may not continue.

Tip — Trust predictions inside the data range; be wary of extrapolating beyond it.

Formula recap

y = a + b x

Regression line of y on x.

b = change in y per unit x

Gradient meaning.

interpolation (safe) vs extrapolation (risky)

Prediction validity.

Common mistakes to avoid

Trusting predictions far outside the data range.

Extrapolation is unreliable — the relationship may not hold there.

Using the line to predict x from y.

The y-on-x regression line is for predicting y from x, not the reverse.

Key takeaways

Regression line: y = a + bx (best fit).
b = change in y per unit x; a = y when x = 0.
Interpolation (within data) is reliable; extrapolation (beyond) is not.

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