Graphical Representations Between Two Quantitative Variables
Can a graph help us see how two measurements are related? Learn to build a scatterplot, describe its pattern and support a claim with evidence from the points.
By the end of this lesson, you should be able to:
- Match measurements from the same individual and construct a clearly labelled scatterplot.
- Describe form, direction, strength and unusual features in context.
- Use a scatterplot to justify a claim while respecting what the study design can show.
Before you start: Review quantitative and categorical variables. Be comfortable reading axes and locating an ordered pair. This lesson starts Unit 5; no regression equation is needed yet.
First time learning this? Follow the practice-time example, then compare the different patterns in the visual gallery.
Here to revise? Use the scatterplot checklist, then try the practice questions before revealing the solutions.
The concept in 60 seconds
A scatterplot puts two measurements from the same individual into one point. One measurement gives the horizontal position; the other gives the vertical position. Looking at all the points helps us see a relationship that a list of numbers may hide.
For example, a student’s weekly practice time and quiz score form an ordered pair. A point at can mean that the student practised for and scored points.
Read the whole pattern: ask what shape the points follow, which way the pattern moves, how closely the points follow it, and whether any groups or unusual points stand out.
A pattern describes what tends to happen. It does not mean every point follows the trend, and an observational association by itself does not show that changing one variable causes the other to change.
Why must the measurements belong to the same individual?
The point represents one individual. Pairing one student’s practice time with a different student’s score would create a relationship that was never observed. Keep each row together.
A real situation: practice time and quiz scores
A teacher asks volunteers from one class to record how long they practised during the previous week. The teacher also records each volunteer’s score on the same quiz. These are invented teaching data, with scores on a -point scale.
The question is: Do students who report more practice time tend to have higher quiz scores in this group? Practice time is the explanatory variable, and quiz score is the response.
| Student ID | Practice time (hours) | Quiz score (points) |
|---|---|---|
| A | ||
| B | ||
| C | ||
| D | ||
| E | ||
| F | ||
| G | ||
| H | ||
| I | ||
| J | ||
| K |
Student E:
Student K:
Invented observations from the same volunteers. Student K is marked by a diamond and a label; the distinction does not rely on colour alone.
Most of the points follow a fairly tight, increasing, approximately straight pattern. Student K’s pair sits well below that pattern, even though is not the lowest score in the table.
A useful first description: For these sampled students, practice time and quiz score show a positive, approximately linear association. Most points follow the pattern closely, with one unusual pair at . Students reporting more practice time generally tend to have higher scores, although that tendency does not hold for every student.
These volunteers were not randomly selected from the whole school, and students were not randomly assigned practice times. Describe the observed group; do not claim that this graph proves a school-wide effect or a causal benefit of practice.
Key ideas and notation
Bivariate quantitative data
Two quantitative measurements collected from each individual. “Quantitative” means a numerical measurement or count whose arithmetic has meaning.
Practice time and quiz score work. Student ID is a label, even when written using digits.
Ordered pair:
Both measurements belong to individual . The first value is horizontal; the second is vertical.
For student E, .
Explanatory variable:
The variable used to explain or predict the response in the question. Put it on the horizontal axis.
“Explanatory” is a role in the analysis; the name alone does not establish cause.
Response variable:
The outcome being explained or predicted. Put it on the vertical axis.
When predicting quiz score from practice time, quiz score is the response.
Association
Certain values of one variable tend to occur with certain values of the other. A scatterplot lets us investigate that pattern.
Use “tends to” or “generally” when a relationship has variability.
Unusual features
Look for clusters, gaps, and points far from the overall pattern. A point can be unusual relative to the relationship without being an extreme value on either axis.
What if neither variable is clearly explanatory?
Some questions simply ask whether two measurements are related, such as hand span and foot length. State which variable you place on each axis and describe the association. Do not invent a causal role.
Build a scatterplot from paired data
- Identify the individual. Is each row a student, car, plant or city? Check that the two measurements come from that same individual.
- Check both variable types. A scatterplot needs two quantitative variables. For a categorical group and a quantitative response, a comparison graph such as side-by-side boxplots is usually more suitable.
- Choose the axes from the question. The explanatory variable goes horizontally; the response goes vertically. Include variable names and units.
- Choose clear scales. Use equally spaced numerical increments on each axis and include the observed range. The horizontal and vertical increments may differ. A scatterplot axis does not have to start at , but the scale must be clearly labelled and must not hide relevant points.
- Plot every observed pair. For , move to on the practice-time axis and on the score axis. Put one point where they meet.
- Inspect the whole graph. Check the point count, variable order, missing entries, scale and unusual features. Do not connect successive students’ points: they are separate individuals rather than a chronological path.
Identical pairs overlap at one location. If some observations share the same pair, the visible number of dots can be smaller than the number of individuals. A note, frequency-sized marker or another transparent display can make that repetition visible.
Protect the pairing: Sorting a table by practice time is fine if each complete row moves with it. Sorting the practice-time and score columns separately destroys the observations.
Check your axes: predicting mass from height
Height is explanatory, so put it horizontally. Mass is the response, so put it vertically. For an individual with height and mass , the pair is with those axis units.
Describe form, direction and strength
Use FDSU: form, direction, strength, unusual features. The order of the letters is a memory aid; a complete response also names the variables and the group.
| Feature | Question to ask | Useful language |
|---|---|---|
| Form | Does the overall pattern look straight, curved or unclear? | Approximately linear, nonlinear, or no clear pattern |
| Direction | As the explanatory variable increases, what tends to happen to the response? | Positive, negative, or no single overall direction |
| Strength | How closely do points follow the overall pattern? | Strong, moderate or weak; support the word with what you see |
| Unusual features | Are there clusters, gaps or points away from the pattern? | Locate and describe the feature in context |