Scatter Graphs and Correlation

This lesson covers scatter graphs — how to plot them, identify the type of correlation, draw a line of best fit, and use it to make predictions. Scatter graphs are tested frequently on the AQA GCSE Mathematics paper and are a key topic for understanding the relationship between two variables.

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What Is a Scatter Graph?

A scatter graph (also called a scatter diagram or scatter plot) is used to show the relationship between two variables. Each point on the graph represents one piece of data with two values — one plotted on the x-axis and one on the y-axis.

When to Use a Scatter Graph

Scatter graphs are used when you want to investigate whether there is a relationship (correlation) between two quantitative variables. For example:

• Temperature and ice cream sales
• Hours of revision and test score
• Age of a car and its value
• Height and shoe size

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Plotting a Scatter Graph

Worked Example

A teacher recorded the number of hours of revision and the test score for 10 students.

Hours of Revision	Test Score (%)
2	35
3	42
4	50
5	55
5	60
6	62
7	70
8	75
9	82
10	90

To plot the scatter graph:

1. Draw two axes with appropriate scales. Label them clearly (e.g. "Hours of Revision" on the x-axis, "Test Score (%)" on the y-axis).
2. Plot each pair of values as a single cross (x) or dot.
3. Give the graph a title.

Exam Tip: Always use a cross (x) or a small dot to plot points — not large circles. Make sure each point is plotted accurately. Examiners check individual points and can deduct marks for inaccurate plotting.

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Types of Correlation

Correlation describes the relationship between two variables on a scatter graph.

Type	Description	What It Looks Like
Positive correlation	As one variable increases, the other also increases	Points slope upwards from left to right
Negative correlation	As one variable increases, the other decreases	Points slope downwards from left to right
No correlation	There is no clear relationship between the variables	Points are scattered randomly

graph LR
    A[Correlation Types] --> B[Positive]
    A --> C[Negative]
    A --> D[No Correlation]
    B --> B1[Both variables increase together]
    C --> C1[One increases while other decreases]
    D --> D1[No clear pattern or relationship]

Strength of Correlation

Correlation can also be described by its strength:

• Strong correlation — the points lie close to a straight line.
• Weak correlation — the points are more spread out but still show a general trend.

Examples

• Strong positive correlation: Hours of sunshine and ice cream sales.
• Weak positive correlation: Height and shoe size (general trend but lots of variation).
• Strong negative correlation: Age of a car and its resale value.
• No correlation: Shoe size and IQ score.

Exam Tip: When describing correlation, you must state both the type (positive, negative, or none) and the strength (strong or weak). Writing "positive correlation" is good, but "strong positive correlation" is better and shows fuller understanding.

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The Line of Best Fit

A line of best fit is a straight line drawn through the data points on a scatter graph that best represents the overall trend. It does not have to pass through every point.

How to Draw a Line of Best Fit

1. The line should follow the general direction of the points.
2. There should be roughly the same number of points above and below the line.
3. The line should pass through or very close to the mean point (the mean of all x-values and the mean of all y-values).
4. The line should extend across the full range of the data.
5. Do not force the line through the origin (0, 0) unless the data suggests it should.

Finding the Mean Point

For the revision data above:

• Mean hours = (2 + 3 + 4 + 5 + 5 + 6 + 7 + 8 + 9 + 10) / 10 = 59/10 = 5.9
• Mean score = (35 + 42 + 50 + 55 + 60 + 62 + 70 + 75 + 82 + 90) / 10 = 621/10 = 62.1

The mean point is (5.9, 62.1). Your line of best fit should pass through or very close to this point.

Exam Tip: In the exam, you may be asked to calculate and plot the mean point. Always mark it clearly with a labelled cross. Then draw your line of best fit so it passes through this point. Use a ruler for a straight line.

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Using the Line of Best Fit to Make Predictions

Once you have a line of best fit, you can use it to estimate the value of one variable given the other.

Interpolation

Interpolation means using the line of best fit to estimate a value within the range of the data. This is generally reliable because you are predicting within known data.

Worked Example

Using the revision data: How many marks would a student who revised for 6.5 hours expect to score?

Find 6.5 on the x-axis, draw a vertical line up to the line of best fit, then draw a horizontal line across to the y-axis. Read off the value (approximately 67%).

Extrapolation

Extrapolation means using the line of best fit to estimate a value outside the range of the data. This is generally unreliable because the trend may not continue beyond the data range.

Worked Example

Using the revision data: How many marks would a student who revised for 15 hours expect to score?

This is extrapolation because 15 hours is outside the range of the data (2–10 hours). The estimate would be unreliable because we do not know if the pattern continues — it is unlikely that test scores keep rising indefinitely (a score above 100% is impossible).

graph TD
    A[Using Line of Best Fit] --> B{Is the value within the data range?}
    B -- Yes --> C[Interpolation]
    B -- No --> D[Extrapolation]
    C --> E[Generally reliable]
    D --> F[Generally unreliable]
    F --> G[The trend may not continue beyond the data]

Exam Tip: The difference between interpolation and extrapolation is a common exam question. If asked to comment on the reliability of an estimate, check whether it is within the data range. If it is outside the range, say the estimate is "unreliable because it involves extrapolation and the trend may not continue beyond the given data".

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Outliers on Scatter Graphs