Scatter Graphs
What you'll learn
- How scatter graphs show pairs of data.
- How to describe positive, negative, and no correlation.
- How to draw a sensible line of best fit.
- How to use the line of best fit to make estimates.
1. What a scatter graph shows
A scatter graph is used when you have two pieces of information about the same person, object, or event.
For example, for each student you might know:
- how many hours they revised
- what score they got in a test
Each student becomes one point on the graph.
The horizontal axis is called the xxx-axis. The vertical axis is called the yyy-axis.
Scatter graph
A scatter graph shows pairs of data as individual points. Each point has an xxx-value and a yyy-value.
Reading and plotting points
If a point is 6 across and 20 up, it means the xxx-value is 6 and the yyy-value is 20.
Reading and plotting points
-
Imagine a graph where the horizontal axis shows revision time in hours, and the vertical axis shows test score.
-
A point is plotted at 8 hours across and 24 marks up.
-
This means that student revised for 8 hours and scored 24 marks.
-
To plot a student who revised for 12 hours and scored 30 marks, go to 12 on the horizontal axis, then go up to 30 on the vertical axis.
-
Mark the point carefully with a small cross or dot.
Mixing up the axes
If the question says “hours revised” is on the horizontal axis, do not put it on the vertical axis. Always check the axis labels before reading or plotting.
2. Correlation: the pattern in the points
A variable is something that can change, such as time spent revising, test score, age, height, or price.
When the points on a scatter graph form a pattern, we say there is correlation.
Correlation
Correlation describes the relationship between two variables on a scatter graph.
There are three main types you need to recognise.
Positive correlation
Positive correlation means that as one variable increases, the other tends to increase too.
Example: more revision time and higher test score.
Negative correlation
Negative correlation means that as one variable increases, the other tends to decrease.
Example: as a car gets older, its value usually decreases.
No correlation
No correlation means there is no clear pattern between the two variables.
Example: shoe size and test score are unlikely to have a clear link.
Look from left to right
To decide the type of correlation, look at the overall direction of the points from left to right.
Naming the type of correlation
-
Suppose a scatter graph compares hours spent practising with score in a skills test.
-
The points generally go upwards from left to right.
-
As practice time increases, the score tends to increase.
-
This is positive correlation.
-
If a graph compared age of a phone with selling price, the points might go downwards from left to right, which would be negative correlation.
-
If the points are spread randomly with no clear direction, it is no correlation.
Correlation is not proof
Correlation shows that two variables are linked in the data, but it does not prove that one definitely causes the other.
3. Drawing a line of best fit
Sometimes the points make a rough straight-line pattern. To make estimates, you draw a line of best fit.
Line of best fit
A line of best fit is a straight line drawn through the middle of the points to show the overall trend.
Your line does not need to go through every point. It should follow the general pattern.
A good line of best fit usually has roughly the same number of points on each side.
Drawing a line of best fit
-
Look at the whole scatter graph before drawing anything.
-
Decide whether the points are generally going up, going down, or showing no clear pattern.
-
Place your ruler through the middle of the cluster of points, following the overall direction.
-
Adjust the ruler so there are roughly equal numbers of points above and below the line.
-
Draw one straight line through the points. Do not join the points one by one.
Joining the dots
A scatter graph is not a line graph. Do not connect each point to the next point. Use one smooth straight line of best fit instead.
Do not force it through zero
The line of best fit does not have to go through the origin unless the points suggest that it should.
4. Using a line of best fit to estimate
An estimate is an approximate answer. On a scatter graph, we use the line of best fit to predict a value.
The basic method is:
- start from the given value on one axis
- move to the line of best fit
- move across to the other axis
- read the estimate
Estimate
An estimate is a sensible approximate value, usually found from a graph or calculation.
Estimating a test score
-
A scatter graph compares practice time in hours with test score out of 40.
-
A line of best fit has already been drawn.
-
To estimate the score for someone who practised for 14 hours, find 14 on the horizontal axis.
-
Draw or imagine a vertical line up from x=14x=14x=14 until it reaches the line of best fit.
-
From that point, draw or imagine a horizontal line across to the vertical score axis.
-
If it meets the score axis near 26, write an answer such as “about 26 marks”.
Show your construction lines
In an exam, lightly draw the vertical and horizontal guide lines. They make your method clear, even if your estimate is slightly different from someone else’s.
5. How reliable is your estimate?
A line of best fit is most useful when you estimate within the range of the data already plotted.
The range of the data means from the smallest plotted value to the largest plotted value on an axis.
Interpolation and extrapolation
Interpolation means estimating within the range of the data. Extrapolation means estimating outside the range of the data.
Interpolation is usually more reliable. Extrapolation is risky because the trend may not continue forever.
Deciding if an estimate is reliable
-
Suppose the plotted revision times go from 2 hours to 18 hours.
-
Estimating the score for 10 hours is interpolation, because 10 hours is inside the range of the data.
-
Estimating the score for 30 hours is extrapolation, because 30 hours is outside the range of the data.
-
The 30-hour estimate is less reliable, because you are extending the line beyond the points you actually have.
No correlation means no useful prediction
If the scatter graph has no correlation, a line of best fit will not give a meaningful estimate.
In the exam
-
Check the axis labels first so you know what each point means.
-
Describe correlation by saying whether the points go up, go down, or show no clear pattern from left to right.
-
When estimating, use the line of best fit, draw guide lines, and write “about” before your answer.
Check yourself
-
Can you explain the difference between positive and negative correlation?
-
Can you draw a line of best fit without joining the dots?
-
Can you use a line of best fit to estimate a missing value from one axis?
