Quick Summary

• Correlation describes the relationship between two quantities: positive, negative, or none (zero).
• Scatter graphs plot paired data as crosses (×) — never joined up — so you can see the pattern.
• "Correlation does not imply causation" — just because two things move together doesn't mean one causes the other.
• A line of best fit is drawn by eye through the mean point (x̄, ȳ) to summarise a linear trend.
• Interpolation (predicting inside your data range) is reliable; extrapolation (predicting outside it) is risky.
• A linear regression line (y = mx + b) is the calculator-generated "ideal" version of a line of best fit, and it always passes through the mean point too.

1. Correlation — What's the Relationship?

Imagine you're comparing two things about a group of people — say, their height and their shoe size. As one goes up, does the other tend to go up too? Go down? Or is there no obvious link at all? That relationship is what correlation measures.

There are three types, and you need to be able to spot each one just by looking at the shape a set of dots makes.

Positive Correlation

This is when both quantities increase together. As one goes up, so does the other. Think: as temperature rises, ice cream sales rise. On a graph, the dots drift from the bottom-left up to the top-right — like climbing a hill. This gives the line of best fit a positive gradient.

Negative Correlation

This is the opposite: as one quantity increases, the other decreases. Think: as a car gets older, its resale value drops. On the graph, dots run from top-left down to bottom-right, giving a negative gradient.

No (Zero) Correlation

Here, there's no pattern at all — the dots are scattered randomly, like a "cloud." Think: the mass of snails and people's exam scores — completely unrelated. There's no sensible straight line you could draw through it.

Strength matters too

Correlation isn't just "positive/negative/none" — it can also be weak or strong. The rule is simple: the closer the dots huddle to a straight line, the stronger the correlation. If they're tightly packed along an imaginary line, that's strong. If they're loosely scattered but still trending in a direction, that's weak.

"Correlation Does Not Imply Causation"

This is one of the most important ideas in this whole topic, and examiners love testing it. Here's the classic example from the textbook: if you record a sunflower's height and a puppy's weight every day for a few weeks, you'll probably find a strong positive correlation — as the sunflower gets taller, the puppy gets heavier.

But obviously, making your sunflower taller will not make your puppy heavier, and a heavier puppy won't make your sunflower grow! The real explanation is that both are increasing because of a third, hidden factor: time. They're both naturally growing as the days pass — they just happen to be measured over the same time period.

2. Scatter Graphs — How to Read & Draw Them

A scatter graph (or scatter diagram) is simply a way to visualise pairs of related data — for example, a set of students' Maths grades plotted against their Physics grades. Each student contributes one point to the graph, using their Maths score as the x-coordinate and their Physics score as the y-coordinate.

The rules for plotting

• The horizontal (x) axis and vertical (y) axis represent the two quantities you're comparing.
• Each data pair is plotted as a small cross, × — not a dot, not a circle — this is a standard convention so points are easy to see and line up precisely.
• Points are never joined up with lines to each other. Unlike a line graph, there's no "order" connecting one point to the next — each cross is an independent piece of data.

3. Line of Best Fit

Once you can see that a scatter graph shows a positive or negative correlation, you can draw a line of best fit — a single straight line that best represents the overall trend of the data. Think of it as "averaging out" all the scattered points into one clean line you can actually use for predictions.

Step 1: Find the Mean Point

Every line of best fit must pass through a special point called the mean point, written (x̄, ȳ) — pronounced "x-bar, y-bar." This point is simply the average of all the x-values and the average of all the y-values.

Step 2: Draw the Line

1. Calculate the mean point (x̄, ȳ).
2. Plot the mean point on the graph alongside all your other data crosses.
3. Draw a single, ruled straight line that passes through the mean point.
4. Extend it across the full width of your data set (not just a short segment in the middle).
5. Adjust the angle until you have roughly the same number of points on either side of the line along its whole length — this makes it a genuine "best fit," not just a line that looks close to a few points.

What about outliers?

If there's one point that clearly doesn't fit the general pattern (an outlier/extreme value), you should ignore it when drawing your line of best fit. It's still plotted on the graph, but it doesn't get to drag your line off course.

Using the Line to Make Predictions

Once your line of best fit is drawn, you can use it like a lookup tool. Say you want to estimate the y-value when x = 5: draw a vertical line up from x = 5 until it hits your line of best fit, then draw a horizontal line across to the y-axis to read off the estimated value.

Worked Example

Sophie wants to know if the price of a computer is related to how fast it runs a program. She tests 8 computers and records the price and the time (in seconds) each one takes:

(a) What type of correlation is shown, and what does it mean? The points run from top-left to bottom-right — a negative correlation. This means: the more a computer costs, the quicker it completes the task.

(b) Estimate the price of a computer that completes the task in 3.4 seconds.

Step 2 — Draw a horizontal line across from 3.4 seconds on the time axis until it meets your line of best fit, then drop straight down to read the price.

4. Linear Regression Lines

A line of best fit drawn "by eye" is a good estimate, but it's still a human guess — two students looking at the same scatter graph might draw slightly different lines. A linear regression line solves this by using statistical software (your graphic display calculator) to calculate the mathematically ideal line — the one that fits the data as closely as possible.

Finding the Equation on Your Calculator

You are not expected to calculate m and b by hand — this is a calculator skill. Here's the general process (exact button names vary by calculator model):

1. Step 1: Enter your raw data into your calculator's spreadsheet or statistics mode, as two columns (label them x and y if you can).
2. Step 2: From the statistics menu, select the linear regression option — usually written as (mx + b). On some calculators, you instead choose CALC → REG → X (for a linear/"X" model rather than quadratic or cubic), then select the ax+b format.
3. Step 3: Assign the correct columns to the x-list and y-list.
4. Step 4: Write down the values the calculator gives you for m and b, rounding to 3 significant figures (unless they come out as exact values).
5. Step 5: Substitute m and b into y = mx + b to state the full equation.

Worked Example

Rebecca, a regular jogger, records calories burned (y) against distance run in km (x):

(a) Find the equation of the regression line for y in terms of x.

Entering the data into the calculator's statistics mode and selecting linear regression gives:

(b) Draw the regression line on the scatter diagram.

To draw a straight line, you only need two points — pick any two convenient x-values and substitute them into the equation:

(c) Given that the mean distance run is 8 km, find the mean calories burned — without recalculating from the raw data.

Since the regression line always passes through the mean point, you can simply draw a vertical line up from x = 8 until it meets the regression line, then read across to the y-axis.

What to Memorise

Concepts Checklist

• I can identify positive, negative, and zero correlation from a scatter graph
• I can explain why correlation does not imply causation, with an example
• I know scatter graph points are plotted as crosses and never joined up
• I can calculate the mean point (x̄, ȳ) from a data set
• I can draw a line of best fit through the mean point, balancing points on either side
• I know to ignore outliers when drawing a line of best fit
• I can use a line of best fit to make predictions (interpolation)
• I understand why extrapolation is unreliable
• I can find the equation of a linear regression line using a calculator
• I know a regression line always passes through the mean point
• I can plot a regression line using two calculated points from its equation

Exam Tips & Common Mistakes