Linear Regression

Revise Linear Regression for Further Statistics 2 WFS02 (A2 Level) — revision notes and instant AI marking.

📖 Revision notes · preview

Edexcel Further Statistics 2

Linear Regression

The Big Idea When two things are linked, you can draw one "line of best fit" through the data (using a formula, not a ruler) that lets you predict one value from the other — and then check with numbers, not just a gut feeling, whether that line is actually any good.

Summary — What This Chapter Covers

Least squares regression finds the line y = a + bx that minimises the total squared distance between the line and every data point.
S_xx, S_xy, S_yy are three "helper sums" you calculate first — everything else (the gradient, the intercept, how good the fit is) is built from them.
Interpolation vs extrapolation — predicting inside your data range is safe; predicting outside it is a guess dressed up as maths.
Coded data lets you work with small, friendly numbers instead of huge ones, then convert your answer back using substitution.
Residuals are the leftover gaps between what actually happened and what your line predicted — plotting them tells you if a straight line was ever the right shape to use.
RSS (Residual Sum of Squares) turns "how good is my fit" into a single number — smaller is better.

1. Least Squares Linear Regression

Imagine you've got a scatter of points on a graph — say, temperature against reaction time. You could draw a line through them freehand, but everyone would draw it slightly differently. Least squares regression removes the guesswork: it's the one specific line that makes the total "error" between the line and the real points as small as mathematically possible.

Here's the trick to understanding *why* it's called "least squares": for every point, there's a vertical gap between where the point actually is and where the line says it should be. That gap is called a residual. Some residuals are positive (point is above the line), some are negative (point is below). If you just added them up, positives and negatives would cancel out and hide how good or bad the fit really is. So instead, you square every residual first (which also makes them all positive) and add those up. The regression line is the one specific line that makes this sum — the sum of squared residuals — as small as it can possibly be. That's the whole idea behind the name "least squares."

The Regression Line

y = a + bx

where y is the dependent variable (the thing being predicted), x is the independent variable (the thing you already know), b is the gradient of the line, and a is the y-intercept.

The Three Helper Sums

S_xx = Σx² − (Σx)²/n
S_xy = Σxy − (Σx)(Σy)/n
S_yy = Σy² − (Σy)²/n

These aren't random — they're standardised measures of "spread" and "shared spread." S_xx measures how spread out the x-values are. S_yy measures how spread out the y-values are. S_xy measures how x and y vary together — this is what tells you the relationship exists at all.

Gradient and Intercept

b = S_xy / S_xx a = ȳ − bx̄

You always find b first (it only needs the helper sums), then use it to find a. Note that ȳ = Σy/n and x̄ = Σx/n are just the means of the data — and a neat fact worth remembering: the regression line always passes through the point (x̄, ȳ), the "centre of mass" of your data. That's exactly why the formula for a works — you're finding the intercept that forces the line through that centre point.

a b mean

🔓 Read the full Linear Regression note → You're seeing the preview · sign in to read it all

Also in the full note

2. Interpolation & Extrapolation
3. Coded Data
4. Residuals
5. Residual Sum of Squares (RSS)
What to Memorise
Concepts Checklist
Exam Tips — Common Mistakes & Mark-Scheme Traps
Worked Example

What's inside

📖 Revision notes 🎯 Learn mode ✦ AI flashcards ✓ Instant AI marking 🧊 3D explorers 🧪 Experiments & simulations 📈 Progress tracking

📄 Practise Linear Regression with Further Statistics 2 WFS02 past papers Every paper with its mark scheme — answer online, marked instantly. Open →

Read the full Linear Regression notes free

That's the preview — create a free account to read the rest, plus flashcards and practice questions with instant AI marking. No credit card.

Unlock the full notes free →

Linear Regression