Scatter Graphs And Correlation

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Notes

Scatter Graphs

**Scatter graphs** (or scatter diagrams) plot pairs of data, e.g. Maths grade vs Physics grade.
Points are plotted as crosses (×) and are **not joined up**.
The vertical and horizontal axes represent the two quantities being measured.
The general shape formed by the points shows the type of correlation.

Types of Correlation

**Positive correlation**: as one quantity increases, the other also increases (points go from bottom left to top right).
**Negative correlation**: as one quantity increases, the other decreases (points go from top left to bottom right).
**No (zero) correlation**: no apparent relationship (points form a cloud).
Correlations can be **weak** or **strong** – the stronger the correlation, the closer the points lie to a straight line.

Correlation vs Causation

**Correlation does not imply causation** – a relationship does not mean one variable causes the other.
Example: sunflower height and puppy weight both increase over time, but neither causes the other.
Both quantities may be increasing due to a third factor (e.g. time).

Line of Best Fit

A **line of best fit** is a single ruled straight line drawn by eye on a scatter graph when correlation exists.
It should extend across the full data set, with roughly equal numbers of points on either side.
Ignore **outliers** (extreme values that don't fit the pattern) when drawing the line.
Use a transparent ruler to help position the line.

Using the Line of Best Fit

The line can be used to **predict** values (e.g. estimate y when x is given).
**Interpolation**: predicting within the range of the data – reliable.
**Extrapolation**: predicting beyond the data range – unreliable and should be avoided.
Example: from a line of best fit, estimate the price of a computer that takes 3.4 seconds to run a program.

Common Exam Questions

Read values from a scatter graph (e.g. number of phones when 42 people are in a building).
Identify the type of correlation from a scatter diagram.
Plot additional points from a table onto an existing scatter graph.
Draw a line of best fit and use it to estimate a value.
Describe the relationship between two variables (e.g. age and value of cars).

Types of Correlation

Line of Best Fit Example

Interpolation vs Extrapolation

Scatter Graph with Outlier

Practice questions

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1.What type of correlation is shown when one quantity increases and the other quantity also increases?
Easy
- APositive correlation
- BNegative correlation
- CNo correlation
- DWeak correlation
2.A scatter diagram shows points that go from top left to bottom right. What type of correlation does this show?
Easy
- APositive correlation
- BNegative correlation
- CNo correlation
- DStrong correlation
3.The scatter diagram shows the number of people and the number of phones in each of 8 buildings. One building contains 42 people. How many phones are in that building? (Assume you can read from the graph.)
Easy
- A30
- B35
- C40
- D45
4.What type of correlation is described by the statement: 'We eat more ice cream as the temperature rises'?
Easy
- APositive correlation
- BNegative correlation
- CNo correlation
- DCausation
5.In a scatter graph showing the age and value of cars, the points go from top left to bottom right. Describe the relationship between age and value.
Easy
- AAs age increases, value increases
- BAs age increases, value decreases
- CNo relationship
- DAs age decreases, value decreases
6.A scatter graph shows the distance from the city centre and monthly rent of apartments. Points go from top left to bottom right. What does this tell you?
Easy
- AThe further from the centre, the higher the rent
- BThe further from the centre, the lower the rent
- CNo relationship
- DRent is constant
7.In a scatter diagram of physics marks and chemistry marks, the points show a positive correlation. Which statement is correct?
Medium
- AHigher physics marks tend to have higher chemistry marks
- BHigher physics marks tend to have lower chemistry marks
- CPhysics marks cause chemistry marks
- DThere is no relationship
8.A line of best fit is drawn on a scatter diagram. For a given x-value, how do you estimate the corresponding y-value?
Medium
- ARead up from x to the line, then across to y
- BRead across from y to the line, then down to x
- CFind the mean of all y-values
- DUse the closest data point

Summary

## Scatter Graphs & Correlation ### Correlation - **Positive correlation**: as one quantity increases, the other increases (e.g., temperature → ice cream sales). Points go from bottom left to top right. - **Negative correlation**: as one quantity increases, the other decreases (e.g., age of car → value). Points go from top left to bottom right. - **No (zero) correlation**: no apparent relationship; points form a cloud. - Correlation can be **weak** or **strong** (closer to a straight line = stronger). - **Correlation does not imply causation**: two variables may correlate due to a third factor (e.g., time). ### Scatter Graphs - Plot pairs of data as **crosses (×)**; do not join points. - Axes represent the two quantities. ### Lines of Best Fit - Drawn **by eye** as a single straight line using a ruler. - Extend across the full data set; roughly equal points on each side. - Ignore **outliers** (extreme values not fitting the pattern). - Use to **predict** values: - **Interpolation**: predict within the data range (reliable). - **Extrapolation**: predict beyond the data range (unreliable). - To estimate: read off from the line at a given x (or y). ### Key Tips - Use a transparent ruler to see all points. - Always describe correlation with context (e.g., "negative correlation: as distance increases, rent decreases"). - When asked "estimate", use your drawn line of best fit. - Do not use line of best fit to predict far outside the data range (extrapolation).

Key terms

Positive correlation: When one quantity increases, the other also increases; points slope upward from left to right.
Negative correlation: When one quantity increases, the other decreases; points slope downward from left to right.
No (zero) correlation: No apparent relationship between the two quantities; points are scattered randomly.
Line of best fit: A straight line drawn on a scatter graph to represent the trend, used for predictions.
Interpolation: Predicting a value within the range of the given data; reliable.
Extrapolation: Predicting a value outside the range of the given data; unreliable.
Outlier: An extreme data point that does not fit the general pattern; ignored when drawing a line of best fit.

Slides

What is a Scatter Graph?

A scatter graph plots pairs of data (e.g., Maths vs Physics marks).
Points are shown as crosses (×) and are not joined up.
The pattern of points shows the relationship between two quantities.

How are points plotted on a scatter graph?

As crosses (×)
As dots (•)
Joined with lines
As bars

Types of Correlation

Positive correlation: as one quantity increases, the other also increases (e.g., temperature and ice cream sales).
Negative correlation: as one quantity increases, the other decreases (e.g., age and value of a car).
No correlation: no apparent relationship between the quantities.

Which of these are true?

Positive correlation means both quantities increase together.
Negative correlation means both quantities decrease together.
No correlation means there is no clear relationship.

Correlation Does Not Imply Causation

Just because two quantities correlate does not mean one causes the other.
Example: sunflower height and puppy weight both increase over time, but one does not cause the other.
Both may be influenced by a third factor (e.g., time).

Which statement is correct?

Correlation always means causation.
Correlation does not imply causation.
Positive correlation always means one causes the other.
Negative correlation means no relationship.

Drawing a Line of Best Fit

A line of best fit is a single straight line drawn by eye.
It should extend across the full data set.
Roughly equal numbers of points should be on each side of the line.
Ignore any outliers (extreme values) when drawing the line.

A line of best fit should be $a ____$ line drawn by eye.

Using a Line of Best Fit to Predict

Use the line to estimate values (e.g., find y when x is given).
Interpolation: predicting within the range of data points – reliable.
Extrapolation: predicting outside the range – unreliable.

What is it called when you predict a value within the range of data points?

Extrapolation
Interpolation
Correlation
Causation

Identifying Correlation from a Scatter Graph

Positive correlation: points go from bottom left to top right.
Negative correlation: points go from top left to bottom right.
No correlation: points form a random cloud.
Correlation can be weak (points spread out) or strong (points close to a line).

Which of these describe positive correlation?

Points go from bottom left to top right.
Points go from top left to bottom right.
As one quantity increases, the other increases.

Worked Example: Scatter Graph and Line of Best Fit

Sophie recorded price and time for 8 computers.
She plotted points and saw a negative correlation: higher price, shorter time.
She drew a line of best fit and estimated that a computer taking 3.4 seconds costs about £620.

In Sophie's example, the correlation $was ____$ because as price increased, time decreased.

Exam Tips

Use a transparent ruler to slide around and find the best position for the line of best fit.
Always plot points as crosses (×).
When estimating, draw lines from the axis to the line of best fit and read off carefully.

What tool is recommended for drawing a line of best fit?

A compass
A transparent ruler
A protractor
A set square

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