LearnYMA01WST03 · Statistics 3Regression and correlationThe least squares regression line

Regression and correlation · Subtopic 2 of 2

The Least Squares Regression Line

The regression line of $y$ on $x$ , $y=a+bx$ , is the straight line that makes the squared vertical distances from the points as small as possible. Its gradient is $b=\dfrac{S_{xy}}{S_{xx}}$ and it always passes through $(\bar x,\bar y)$ , so $a=\bar y-b\bar x$ .

Edexcel YMA01 WST01Edexcel YMA01 WST03Edexcel 9MA0 Paper 3Edexcel 9FM0 OptionAQA 7357 Paper 3≈ 35 minRevision 6 · updated 2026-10-27

Section 1 of 14

Learning objectives

Identify the explanatory and response variables.
Calculate $b$ and $a$ for the regression line of $y$ on $x$ .
Interpret $a$ and $b$ in context and make predictions.
Know when predictions are unreliable, and convert coded regression lines.

Pause and think

Draw a line through a scatter of points "by eye". How could you decide which of two lines fits better?

Section 2 of 14

Core idea

Start hereThe best straight line

Points on a scatter graph rarely lie on a line. Many lines could go "through the middle", but one is special: it makes the total of the squared vertical misses as small as possible. That's the least squares regression line.

The big idea

$y=a+bx$ with $b=\frac{S_{xy}}{S_{xx}}$ , $a=\bar y-b\bar x$ . It passes through $(\bar x,\bar y)$ . Interpolate safely; extrapolate with caution.

1. The line

Regression line y = 2.42 + 1.32x with residuals

Each dashed line is a residual: actual

y

minus predicted

y

. The line makes the total of their squares as small as it can be. It always passes through

(\bar x,\bar y)=(7,11.6)

b=\frac{S_{xy}}{S_{xx}}\qquad a=\bar y-b\bar x

$b$: gradient: change in $y$ per unit of $x$ .
$a$: intercept.
residual: observed − predicted.

Worked example $n=8$ , $\sum x=40$ , $\sum y=160$ , $S_{xx}=42$ , $S_{xy}=63$

Find the regression line of $y$ on $x$ .

Goal: $a$ , $b$ .
Strategy: Means, then formulas.

$\bar x=5,\ \bar y=20$
Divide by 8.
$b=\tfrac{63}{42}=1.5$
Gradient.
$a=20-1.5(5)=12.5$
Through the means.

Answer

y=12.5+1.5x

✓ CheckAt

x=5

20=\bar y

✓.

2. Using it

Worked examplePredicting

Predict $y$ at $x=6$ and $x=30$ (data range $1$ – $10$ ). Residual at $(4,17)$ ?

Goal: Predictions; comment; residual.
Strategy: Substitute.

$x=6:\ 21.5$
Interpolation: reliable.
$x=30:\ 57.5$
Extrapolation: unreliable.
$17-(12.5+6)=-1.5$
Residual: point lies below the line.

Answer

21.5

reliable;

57.5

unreliable; residual

-1.5

✓ CheckPredicted at

x=4

12.5+1.5(4)=18.5

;

17<18.5

so negative ✓.

Common misconceptionWrong direction Use

y

x

to predict

y

from

x

; predicting

x

from

y

needs the other line.

Pause and think

Interpret $b=1.5$ .

Summary

$b=S_{xy}/S_{xx}$ ; $a=\bar y-b\bar x$ .
Interpolate, don't extrapolate.

Residual: Observed − predicted.
Extrapolation: Predicting outside the data.

HistoryLegendre and Gauss. Adrien-Marie Legendre published the method of least squares in 1805 to fit comet orbits. Carl Friedrich Gauss claimed he had used it since 1795, sparking a priority dispute. The word "regression" came later, from Galton's studies of height.

Fun factRegression to the mean. Galton noticed that very tall parents tended to have children who were tall, but less extremely so. He called this "regression towards mediocrity", and the name "regression" stuck to the whole method.

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