Coaching Centre
Scatterplots and describing association
Bivariate data, explanatory and response variables, drawing a scatterplot, and describing an association by its direction, form and strength, including outliers.
A dataset is bivariate when every item has two measurements. Twenty students each give you their hours of sleep and their reaction time. Fifteen used cars each give you an age and a price. One number on its own tells you nothing about the link. You need to see all the pairs at once, and a scatterplot does exactly that.
Which variable goes where?
Before you plot anything, decide which variable might explain or predict the other.
- The explanatory variable (also called the independent variable) is the one you think does the influencing. It goes on the horizontal axis, $x$.
- The response variable (dependent variable) is the one that responds. It goes on the vertical axis, $y$.
Ask yourself: "Does $A$ affect $B$, or does $B$ affect $A$?" A car's age affects its price, not the other way round, so age is explanatory. Sometimes neither obviously drives the other (arm span and height). Then either choice is fine, as long as you are consistent and the question doesn't tell you otherwise.
Describing an association
When you describe a scatterplot, HSC markers want three things, in this order. A handy memory aid is D-F-S.
| Feature | Question to ask | Words to use |
|---|---|---|
| Direction | As $x$ increases, what does $y$ tend to do? | positive (up), negative (down), or no association |
| Form | Do the points follow a straight line or a curve? | linear or non-linear |
| Strength | How tightly do the points cluster around that line or curve? | strong, moderate, weak |
A complete description sounds like this: "There is a strong, positive, linear association between hours of study and exam mark." Then add context: "Students who study more tend to score higher."
Ten used hatchbacks on a Sydney car-sales site were recorded by age (years) and price (in thousands of dollars). The scatterplot shows points falling from about $(1, 24)$ down to $(10, 6)$, packed closely around a straight line. Describe the association.
Outliers and non-linear patterns
An outlier in a scatterplot is a point that sits well away from the overall pattern. It may be a recording mistake, or something genuinely unusual, like a 10-year-old car that's a collector's item. Always mention an outlier if you see one, and say what it might mean.
Not every pattern is a straight line. Car prices often drop quickly in the first few years, then level off: that is a non-linear (curved) association. Saying "strong negative linear" about a clear curve loses the form mark.
For each new scatterplot, choose a direction, a form and a strength, then press Check. Aim for five in a row.
When you get one wrong, look at the feedback. Is it usually the strength you misjudge? Moderate is the hardest to call.
Your turn
A teacher records, for each student, the number of hours spent gaming per week and their average sleep per night. The points fall from left to right, but they're spread quite widely around a line. Which variable should go on the $x$-axis, and how would you describe the association?
Show the answer
Gaming hours is the explanatory variable (it might reduce sleep), so it goes on the $x$-axis and sleep goes on the $y$-axis. The points fall, follow a rough line and are widely spread, so this is a moderate (or weak), negative, linear association: students who game more tend to sleep less.
A study looks at the number of hours a Year 12 student works at a casual job each week and their HSC trial mark. The explanatory variable is:
Hours worked might influence the trial mark, not the reverse, so hours worked is explanatory and goes on the $x$-axis.
On a scatterplot, the response variable is drawn on:
The response variable is $y$, so it goes on the vertical axis.
As the price of a concert ticket increases, fewer tickets tend to be sold. This is a:
As $x$ (price) increases, $y$ (tickets sold) tends to decrease. That is a negative direction.
Points on a scatterplot curve upwards more and more steeply as $x$ increases, with very little scatter. The best description of the form is:
Form is about the shape the points follow. A curve that gets steeper is non-linear, even though it is strong and positive.
Which description is complete for a 2-mark "describe the association" question?
Only this one gives direction, form and strength and links them to the variables in context.
A point on a scatterplot of age against height for a junior soccer team shows a 9-year-old who is 172 cm tall, far above the other points. This point is best described as:
It sits well away from the overall pattern, so it is an outlier. It may be a recording error (perhaps the coach's height), so it should be checked.
The points on a scatterplot are spread widely, with only a slight upward trend. The strength is best described as:
Strength is about how tightly the points cluster. Wide spread with a slight trend means weak.
Which pair of variables is likely to show no association?
There is no reason for shoe size to be linked to a maths mark, so we expect no association. The other three pairs have an obvious link.
Exam style: A scatterplot of daily maximum temperature ($x$) and the number of people at a Western Sydney pool ($y$) shows points rising from left to right, close to a straight line. Which statement is best?
Rising points close to a line give strong, positive and linear. The statement in context uses "tend to", because the trend doesn't hold for every single day. Temperature is explanatory, not the response.
Exam style: A student describes a scatterplot of hours of sleep ($x$) and number of errors in a test ($y$) as "strong positive linear". The points clearly fall from left to right. What is wrong?
Points that fall from left to right show $y$ decreasing as $x$ increases: more sleep tends to go with fewer errors. That is a negative direction.