Scatter Graphs And Correlation

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Notes

Scatter Graphs

Scatter graphs (or scatter diagrams) plot pairs of data using crosses (×).
The vertical and horizontal axes represent the two quantities being measured.
Points are not joined up; the overall shape shows the type of correlation.
**Positive correlation**: points go from bottom left to top right (positive gradient).
**Negative correlation**: points go from top left to bottom right (negative gradient).
**No correlation**: points form a random cloud with no apparent pattern.
Correlation can be **weak** or **strong** depending on how close points are to a straight line.

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., car age and value).
**Zero correlation**: no relationship between the two quantities (e.g., snail mass and exam score).
**Correlation does not imply causation** – a relationship may be due to a third factor (e.g., time).

Lines of Best Fit

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

Using Lines of Best Fit for Predictions

To estimate a value, read off the line at the given x or y coordinate.
**Interpolation**: predicting within the range of the data – reliable.
**Extrapolation**: predicting beyond the data range – unreliable and should be avoided.
Always use the line of best fit, not individual points, for predictions.

Positive Correlation

Negative Correlation

No Correlation

Line of Best Fit

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
- DZero correlation
2.A scatter diagram shows points going from top left to bottom right. What type of correlation is this?
Easy
- APositive correlation
- BNegative correlation
- CNo correlation
- DWeak correlation
3.Which of the following best describes 'no correlation' in a scatter graph?
Easy
- APoints form a straight line
- BPoints form a cloud with no pattern
- CPoints go from bottom left to top right
- DPoints go from top left to bottom right
4.The average speed of a train and the time taken for a journey: what type of correlation would you expect?
Easy
- APositive correlation
- BNegative correlation
- CNo correlation
- DWeak correlation
5.The distance travelled by a car and the amount of fuel used: what type of correlation would you expect?
Easy
- APositive correlation
- BNegative correlation
- CNo correlation
- DZero correlation
6.A scatter diagram shows points with a clear upward trend from left to right. Which statement is correct?
Medium
- AThere is a negative correlation
- BThere is a positive correlation
- CThere is no correlation
- DThe line of best fit has a negative gradient
7.In a scatter diagram, a line of best fit should be:
Medium
- AA curved line passing through all points
- BA single ruled straight line with roughly equal points on each side
- CA line connecting the first and last points
- DA line that passes through the origin
8.When using a line of best fit to estimate a value within the range of the data, this is called:
Medium
- AExtrapolation
- BInterpolation
- CCorrelation
- DCausation

Summary

## Scatter Graphs & Correlation ### Correlation - **Positive correlation**: as one quantity increases, the other increases (e.g., temperature and ice cream sales). Points go from bottom left to top right. - **Negative correlation**: as one quantity increases, the other decreases (e.g., car age and value). Points go from top left to bottom right. - **No correlation**: no apparent relationship; points form a cloud. - **Strength**: strong if points lie close to a straight line; weak if scattered. - **Correlation ≠ causation**: a relationship does not mean one causes the other (e.g., sunflower height and puppy weight both increase with time). ### Scatter Graphs - Plot pairs of data as crosses (×). - Axes represent the two quantities. - Do not join points. ### Lines of Best Fit - Draw a single straight line by eye through the data. - Extend across the full data set. - Roughly equal numbers of points on each side of the line. - Ignore outliers (extreme values that don't fit the pattern). ### Using the Line of Best Fit - **Interpolation**: estimate values within the range of data (reliable). - **Extrapolation**: estimating beyond the data (unreliable). - To estimate: read off the y-value for a given x (or vice versa) using the line. ### Exam Tips - Use a transparent ruler to position the line of best fit. - Always draw a line of best fit when asked. - State correlation type (positive, negative, or none).

Key terms

Correlation: A relationship between two quantities; can be positive, negative, or zero.
Positive correlation: As one quantity increases, the other increases.
Negative correlation: As one quantity increases, the other decreases.
No correlation: No apparent relationship between two quantities.
Line of best fit: A straight line drawn on a scatter graph to show the trend; used for predictions.
Interpolation: Estimating a value within the range of known data.
Extrapolation: Estimating a value outside the range of known data; often unreliable.
Outlier: A data point that does not fit the general pattern; may be ignored when drawing a line of best fit.

Slides

What are 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 general shape formed by the points shows the type of correlation.

How are points plotted on a scatter graph?

As dots
As crosses
As circles
Joined with lines

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).

Which type of correlation is shown when points go from top left to bottom right?

Positive
Negative
No correlation
Weak correlation

Correlation Does Not Imply Causation

If two quantities correlate, it 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 increasing due to another factor (e.g. time).

Which of these are true about correlation?

Positive correlation means both quantities increase together.
If two quantities correlate, one must cause the other.
No correlation means there is no relationship.

Line of Best Fit

If a scatter graph shows positive or negative correlation, a line of best fit can be drawn.
It is a single ruled straight line drawn by eye, extending across the full data set.
Roughly as many points should be on either side of the line; ignore outliers.

What should you do with an outlier when drawing a line of best fit?

Include it exactly on the line
Ignore it
Make the line pass through it
Remove it from the graph

Using a Line of Best Fit

Use the line to predict values (interpolation) within the data range.
To estimate y for a given x, draw a vertical line from x to the line of best fit, then horizontal to the y-axis.
Extrapolation (predicting beyond the data) is unreliable.

Predicting values within the range of data is called ____. Predicting values outside is called ____.

Worked Example: Drawing a Scatter Graph

Plot each point carefully using crosses.
Example: Price (£) vs Time (secs) for 8 computers.
After plotting, identify the type of correlation.

In the worked example, what type of correlation was shown between price and time?

Positive
Negative
No correlation
Weak positive

Worked Example: Using Line of Best Fit

Draw a line of best fit by eye.
To estimate price for a given time, draw horizontal line from time to line of best fit, then vertical to price axis.
Example: a computer taking 3.4 seconds costs about £620.

To estimate the price for a time of 3.4 seconds, draw $a ____$ line from 3.4 to the line of best fit, then $a ____$ line down to the price axis.

Summary

Scatter graphs show relationships between two quantities.
Correlation can be positive, negative, or none.
A line of best fit helps predict values (interpolation) but beware of extrapolation.

Which statements are correct?

A line of best fit should have roughly equal points on each side.
Extrapolation is more reliable than interpolation.
Negative correlation means points go from bottom left to top right.

Flashcards

What is correlation?

Correlation describes how two quantities are related to each other.

What is positive correlation?

Positive correlation is when one quantity increases and the other quantity also increases.

What is negative correlation?

Negative correlation is when one quantity decreases while the other quantity increases.

What is no (zero) correlation?

No correlation is when there is no apparent relationship between the two quantities.

What does 'correlation does not imply causation' mean?

If two quantities correlate, it does not mean that the first causes the second. Both may be influenced by another factor.

What are scatter graphs used for?

Scatter graphs (or scatter diagrams) are used to plot pairs of data, e.g., students' Maths grades against their Physics grades.

How are points plotted on a scatter graph?

Points are plotted as crosses (×) and are not joined up.

What does a positive correlation look like on a scatter graph?

Points go from bottom left to top right (positive gradient).

What does a negative correlation look like on a scatter graph?

Points go from top left to bottom right (negative gradient).

What does no correlation look like on a scatter graph?

Points appear as a cloud with no clear pattern.

What is a line of best fit?

A line of best fit is a straight line drawn on a scatter graph to show the trend when there is positive or negative correlation.

How do you draw a line of best fit?

Draw a single ruled straight line by eye that extends across the full data set, with roughly as many points on either side.

How do you handle an outlier when drawing a line of best fit?

Ignore the outlier when drawing the line of best fit.

What is interpolation?

Interpolation is using a line of best fit to predict values within the region covered by the data points.

What is extrapolation and why is it unreliable?

Extrapolation is predicting values outside the data range; it is unreliable because the trend may not continue.

How do you use a line of best fit to estimate a value?

Read off the y-value from the line at a given x-value, or vice versa, by drawing horizontal/vertical lines to the axes.

What type of correlation is shown if points go from top left to bottom right?

Negative correlation.

What type of correlation is expected between average speed of a train and time taken for a journey?

Negative correlation (higher speed means less time).

What type of correlation is expected between distance travelled by a car and amount of fuel used?

Positive correlation (more distance uses more fuel).

What type of correlation is 'we eat more ice cream as temperature rises'?

Positive correlation.

What is the range of a data set?

The range is the difference between the highest and lowest values.

How do you find the median?

Arrange values in order and find the middle value; if even number, average the two middle values.

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