Bivariate Data and Scatter Plots
Theory
A scatter plot shows whether two numerical variables are related. This Year 10A maths page shows how to plot bivariate data, describe an association by its form, direction and strength, spot an outlier, draw a line of best fit by eye, and use it to make predictions by interpolation and extrapolation.
Bivariate data has two numerical values for each person or item, such as height and arm span. A scatter plot shows each pair as one point. The independent variable goes on the horizontal axis.
The relationship is described by its form (linear if the points lie near a straight line), its direction (positive: as one variable increases, so does the other; negative: as one increases, the other decreases) and its strength (strong, moderate or weak, by how closely the points follow a line).
If all the points lie exactly on a line, the relationship is perfect. If the points show no pattern, there is no relationship.
An outlier is a point that is far away from the pattern of the others.
A line of best fit is a straight line drawn by eye through the middle of the points, following their direction, with about as many points above it as below.
A line of best fit through \((x_1, y_1)\) and \((x_2, y_2)\):
\(c\) is the value where the line crosses the vertical axis (when \(x = 0\)).
How to draw and describe a scatter plot
- Choose axes: the independent variable horizontally, the dependent variable vertically. Use even scales.
- Plot each pair as one point.
- Describe the form (linear or not), the direction (positive or negative) and the strength (strong, moderate, weak), and mention any outliers.
How to use a line of best fit
- Draw the line by eye, following the trend, with roughly equal numbers of points on each side. Ignore an obvious outlier.
- To predict \(y\): go up from the \(x\)-value to the line, then across to the vertical axis.
- To find \(x\): go across from the \(y\)-value to the line, then down.
- Or use the equation: find the gradient from two points on the line, write \(y = c + mx\) and substitute.
- Comment: inside the data range is interpolation (more reliable); outside is extrapolation (less reliable).
Check form, direction and strength in turn.
| \(\text{form}\) | \(\) | the points lie close to a straight line: linear |
| \(\text{direction}\) | \(\) | times fall as distance rises: negative |
| \(\text{strength}\) | \(\) | very little scatter: strong |
A strong negative linear relationship: runners who run further each week tend to have faster (lower) times.
(i) Predict the goals for \(6\) hours of training.
(ii) About how many hours of training go with \(20\) goals?
The line rises \(24 - 4 = 20\) goals over \(10\) hours, so its gradient is \(2\) goals per hour. Its equation is \(y = 4 + 2x\).
| \(\text{(i) } y\) | \(=\) | \(4 + 2 \times 6 = \mathbf{16}\textbf{ goals}\) |
| \(\text{(ii) } 20\) | \(=\) | \(4 + 2x\) |
| \(x\) | \(=\) | \(\mathbf{8}\textbf{ hours}\) |
On a graph, (i) reads up from \(6\) to the line, then across; (ii) reads across from \(20\), then down. Both are inside \(1\) to \(10\) hours, so both are interpolation.
(i) Which point is an outlier?
(ii) Which line, A or B, is the better line of best fit? Why?
(i) An outlier is far away from the pattern of the other points. Point P at \((3, 9)\) is far above the trend.
(ii) Line A. It follows the direction of the points, with about the same number of points above and below it along its whole length. Line B is too flat: it is above the points on the left and below them on the right.
(i) Find the gradient and explain its meaning.
(ii) Predict the price at \(3\) years and at \(8\) years. Comment.
(i) Gradient \(=\) rise \(\div\) run.
| \(m\) | \(=\) | \(\dfrac{160 - 400}{5 - 1} = \dfrac{-240}{4} = \mathbf{-60}\) |
The price drops by about \(\$60\) for each extra year of age.
(ii) Start at \((1, 400)\) and drop \(\$60\) a year.
| \(3\text{ years}\) | \(:\) | \(400 - 60 \times 2 = \mathbf{\$280}\) |
| \(8\text{ years}\) | \(:\) | \(400 - 60 \times 7 = -\$20\) |
\(3\) years is interpolation and is reasonable. \(8\) years is extrapolation, far outside the data, and gives an impossible negative price.
Common pitfalls
Frequently asked questions
What does a scatter plot show?
It shows pairs of numerical data as points, so you can see whether the two variables are related, how strongly, and in which direction.
What is the difference between a positive and a negative relationship?
In a positive relationship both variables tend to increase together, so the points rise to the right. In a negative relationship one decreases as the other increases, so the points fall to the right.
How do you draw a line of best fit?
Draw a straight line by eye that follows the direction of the points, through the middle of them, with about the same number of points above and below. It does not have to pass through any particular point.
What is the difference between interpolation and extrapolation?
Interpolation predicts inside the range of the data and is usually reliable. Extrapolation predicts outside the range and can be unreliable, because the pattern may change.
What is an outlier?
A point that lies well away from the pattern of the other points. Check whether it is a mistake, and leave it out when drawing the line of best fit.