Scatter Plots
A scatter plot shows bivariate data, pairs of values for two numerical variables, as points on a number plane. The pattern shows the association: its direction, its form, such as linear, and its strength. This NSW Year 10 Advanced maths subtopic reads and draws scatter plots (NESA outcome MA5‑DAT‑C‑02).
You will draw scatter plots from tables, read values from them, and describe associations such as rainfall and crop yield, or engine size and fuel use. You will spot outliers and recording errors, and explain why an association does not prove that one variable causes the other. Data include river red gums, stopping distances and hot drinks sold at a Bathurst cafe. Harder questions use points on a perfect line to find unknowns. It is part of Year 10 Advanced Single Variable and Bivariant Statistics.
Every question with a fully worked solution.
- Scatter Plots - Video - Scatterplots Displaying bivariate numerical data Watch
Theory
A scatter plot shows pairs of values as points, so you can see whether two variables are related. Describe the association by its direction (positive or negative), strength (strong, moderate, weak) and form (linear or non-linear), and point out any outliers. This Year 10 Advanced maths page covers drawing scatter plots with sensible scales, describing them precisely, perfect linear relationships, and how one point can change the picture.
Bivariate data come in pairs, such as the height and arm span of each person. The independent variable (the one that might influence the other) goes on the horizontal axis; the dependent variable goes on the vertical axis.
The direction of an association is positive if \(y\) tends to increase as \(x\) increases, and negative if \(y\) tends to decrease. Its strength is how closely the points follow the pattern: strong, moderate, weak, or no association. Its form is linear (a straight band) or non-linear (a curve).
If every point lies exactly on a line, the relationship is perfect and has a rule \(y = mx + c\).
An association describes the overall trend, not every pair: some points can go against it. An outlier is a point well away from the pattern; a cluster is a group of points separate from the others, often a different group.
Gradient and rule of a perfect linear relationship through \((x_1, y_1)\) and \((x_2, y_2)\):
The mean point of \(n\) points is \((\bar{x}, \bar{y})\), so \(\sum x = n\bar{x}\) and \(\sum y = n\bar{y}\).
| Description | What the points look like |
|---|---|
| strong positive linear | close to a rising straight line |
| moderate negative linear | a falling band, noticeably spread |
| weak | a loose cloud with only a slight trend |
| no association | no trend at all |
| non-linear | a clear curve, e.g. levelling off or U-shaped |
Drawing a scatter plot
- Choose the axes: independent variable horizontal, dependent vertical, both labelled with units.
- Choose scales that spread the points out. If all values are far from zero, start the axis near the smallest value and mark the break.
- Plot each pair as a point. Do not join the points.
Describing and using it
- Direction, strength, form: e.g. "strong positive linear".
- Outliers: name the point, suggest a reason (a recording error, an unusual case) and describe the association with and without it.
- Clusters or groups: if points form separate groups (e.g. two kinds of car), describe each group; one trend through all of them can mislead.
- Read values: count points meeting a condition, or compare \(x\) and \(y\) for each point (e.g. the difference between two marks).
- Perfect relationships: find \(y = mx + c\) from two points, then test or complete the others.
| Training (hours per week) | 5 km time (minutes) |
|---|---|
| \(2\) | \(31\) |
| \(3\) | \(30\) |
| \(4\) | \(28\) |
| \(5\) | \(27\) |
| \(6\) | \(25\) |
| \(7\) | \(26\) |
| \(8\) | \(23\) |
| \(10\) | \(21\) |
(i) Which is the independent variable?
(ii) Draw a scatter plot.
(iii) Describe the association.
(i) Training could affect the time, so training is the independent variable, on the horizontal axis. Time is the dependent variable.
(ii) The times are all between \(21\) and \(31\) minutes, so start the vertical axis at \(18\) and show the break.
(iii) As training increases, the time tends to decrease, and the points lie close to a straight line: a strong negative linear association. (Not every pair fits: the runner on \(7\) hours was slower than the one on \(6\).)
(i) Describe the association.
(ii) Would a straight line of best fit be suitable?
(i) The temperature always falls (negative), and the points follow a smooth pattern closely (strong). Look at the drops each \(5\) minutes: \(18,\ 13,\ 9,\ 7,\ 5,\ 3\). They get smaller, so the points curve: a strong negative non-linear association.
(ii) No. A straight line would be too high in the middle and too low at both ends. The tea cools quickly at first and then levels off towards room temperature, which a curve describes better.
(i) Find the rule.
(ii) Find \(a\) and \(b\).
(iii) Is \((5, 15)\) on the line?
(i) Use the two complete points.
| \(m\) | \(=\) | \(\dfrac{10 - 4}{3 - 1} = 3\) |
| \(4\) | \(=\) | \(3(1) + c \quad\Rightarrow\quad c = 1\) |
| \(y\) | \(=\) | \(\boldsymbol{3x + 1}\) |
(ii) Substitute the known coordinate of each point.
| \(3a + 1\) | \(=\) | \(19 \quad\Rightarrow\quad a = \boldsymbol{6}\) |
| \(b\) | \(=\) | \(3(8) + 1 = \boldsymbol{25}\) |
(iii) \(3(5) + 1 = 16 \neq 15\), so \((5, 15)\) is not on the line; it would need \(y = 16\).
(i) Find \(p\) and \(q\).
(ii) Does removing it strengthen or weaken the association?
(i) The mean point is \((\bar{x}, \bar{y})\), so the totals are \(n\bar{x}\) and \(n\bar{y}\).
| \(p\) | \(=\) | \(8 \times 5 - 7 \times 4.5 = 40 - 31.5 = \boldsymbol{8.5}\) |
| \(q\) | \(=\) | \(8 \times 20 - 7 \times 21 = 160 - 147 = \boldsymbol{13}\) |
(ii) \((8.5, 13)\) has a large \(x\) but a small \(y\), below the mean \(y\) of \(20\) even though its \(x\) is well above average. It goes against the positive trend, so removing it makes the association stronger.
Common pitfalls
Frequently asked questions
How do you describe a scatter plot?
Give the direction (positive or negative), the strength (strong, moderate or weak) and the form (linear or non-linear), for example a strong positive linear association. Mention any outliers or clusters.
Which variable goes on the x-axis of a scatter plot?
The independent variable, the one that may influence the other, goes on the horizontal axis. The dependent variable goes on the vertical axis.
What is an outlier on a scatter plot?
A point that lies well away from the pattern of the other points. Look for a reason, such as a recording error, and describe the association with and without it.
What does a perfect linear relationship mean?
Every point lies exactly on one straight line, so the variables are connected by a rule y equals mx plus c.
Can a scatter plot show an association that is not a straight line?
Yes. Points can follow a curve, such as values that level off or a U shape. That is a non-linear association, and it can still be strong.