Scatter plots: describing a relationship between two variables
What this lesson covers
- A scatter plot shows whether two numerical variables change together, and how closely.
- An association between two variables does not show that one causes the other.
Do two variables change together?
- Two numerical variables, measured on the same people.
- Screen time and sleep, for 30 Year 11 students.
- Each student gives one pair of values.
Which variable goes where
- The variable that might explain goes across.
- The response goes up the side.
- One dot per student.
| Variable | Role | |
|---|---|---|
| Across (x) | screen time, hours a day | explanatory |
| Up (y) | sleep, hours a night | response |
| Each dot | one student, both values | 30 dots |
The scatter plot
- Thirty dots, one for each student.
- Read the whole cloud, not single dots.
- It runs from top left to bottom right.
Direction
- Positive: both tend to rise together.
- Negative: as one rises, the other tends to fall.
- This one is negative.
| Direction | Example | Looks like |
|---|---|---|
| Positive | arm span and height | both rise together |
| Negative | screen time and sleep | one rises, one falls |
| None | shoe size and test score | no pattern |
Strength and form
- Strong: the dots sit close to a line.
- Weak: a loose cloud with a slight tilt.
- Here: fairly strong, and linear.
An unusual point
- Two hours of screen time, only 5.5 hours of sleep.
- Far below the students around it.
- Keep it, and suggest a reason.
Describing it in context
- Every feature, in the words of the context.
- "More screen time, less sleep" beats "negative".
- Give values from the graph.
| Feature | Here | In context |
|---|---|---|
| Direction | negative | more screen, less sleep |
| Strength | fairly strong | dots close to a line |
| Form | linear | no curve |
| Unusual | (2, 5.5) | far below the rest |
A trend line
- Drawn through the middle of the cloud, by eye.
- It passes through (0, 10) and (8, 6).
- Its equation: y = 10 − 0.5x.
Making a prediction
- Substitute 4 hours of screen time.
- y = 10 − 2 = 8 hours of sleep.
- Inside the data, so it is fairly reliable.
| Step | Working |
|---|---|
| Trend line | y = 10 − 0.5x |
| Screen time 4 h | y = 10 − 0.5 × 4 |
| Predicted sleep | 8 hours |
| Reliable? | inside the data, strong: fairly |
Stay inside the data
- The data runs from 1 to 8 hours of screen time.
- Outside that range, the pattern may not hold.
- Predicting there is extrapolation.
A relationship is not a cause
- The data shows the two go together.
- It cannot show that screens cause less sleep.
- Say "is associated with".
| What the data shows | the two go together |
| What it cannot show | that screens cause less sleep |
| Other explanations | stress, homework, late shifts |
| Wording | "is associated with", not "causes" |
Where the marks are
- Achieved: describe features with the graph.
- Merit: justify them with values and the line.
- Excellence: bring in context and limitations.
What to take away
- Direction, strength, unusual points, then a careful claim.
| Scatter plot | explanatory across, response up |
| Direction | positive, negative or none |
| Strength | how close the dots are to a line |
| Unusual points | name them, suggest a reason |
| Trend line | predict inside the data only |
| Causation | a relationship is not proof |
| Enough data | at least 30 pairs |
Your turn: four questions
- Pause here and answer each one.
- Question 2 uses the trend line from this lesson.
- Question 4 wants careful wording.
| Question | |
|---|---|
| 1 | As weekly hours of paid work rise, homework hours fall. Direction? |
| 2 | Use y = 10 − 0.5x to predict sleep for 3 hours of screen time. |
| 3 | Why is a prediction for 15 hours of screen time unreliable? |
| 4 | "Screen time causes less sleep." What can you claim instead? |
Show answer
1. Negative: as one variable increases, the other tends to decrease. 2. y = 10 − 0.5 × 3 = 8.5 hours. 3. The data only runs from 1 to 8 hours of screen time, so 15 hours is extrapolation; the line gives 2.5 hours, which the data cannot support. 4. That more screen time is associated with less sleep in the sample. The data cannot show cause, because other factors such as homework or stress could explain both.
The answers
- Negative.
- 10 − 0.5 × 3 = 8.5 hours.
- 15 hours is far outside the data: extrapolation.
- More screen time is associated with less sleep.
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