Scatter Plots & Association
Two measurements per person, one dot each. The shape of the cloud is the finding.
You collect two numbers from each child: hours practised, and test score.
A table of forty rows is hard to see anything in. So plot each row as a single dot.
Now look at the cloud rather than any one dot. It drifts upward to the right.
So children who practised more tended to score higher. Not every one of them — the cloud is loose, and one child did badly despite practising.
Which is the point: the finding lives in the shape of the cloud, never in a single dot.
And notice what the picture cannot tell you. It says practice and score went together. It cannot say which caused which, or whether something else — a quiet home, a good night's sleep — caused both.
Building the plot
Each case contributes one dot: the first variable gives the x position, the second gives the y.
Label both axes with the variable and its units. An unlabelled scatter plot says nothing.
Never join the dots. These are separate cases, not one thing changing over time.
One row, one dot
Work down the table one row at a time and put in one dot for each. Reading the table column by column is how points end up transposed.
Four questions to ask
Direction: as x goes up, does y tend to go up (positive) or down (negative)?
Form: is the pattern roughly a straight line, a curve, or nothing?
Strength: are the dots tight to the pattern (strong) or loosely scattered (weak)?
Unusual points: any outliers or separate clusters?
Answer all four
A full description names direction, form, strength and any outliers. A one-word answer like "positive" leaves most of the marks on the table.
Association is not cause
Ice cream sales and drowning deaths rise together. Ice cream does not cause drowning — hot weather drives both.
That hidden third variable is a confounder. It is the standard reason two things move together without one causing the other.
You can report an association from a scatter plot. You cannot report a cause.
The sentence to write
Say "there is a strong positive association between x and y", not "x causes y". Only a controlled experiment can support the second.
Outliers and clusters
An outlier is far from the pattern, not merely far along it. A very tall person on a height-weight plot is not an outlier if their weight fits.
Two separate clusters often mean two different groups are mixed together in one plot.
Say what you see and, if you can, suggest why. That is what a statistician would do.
Worked examples
Describing a plot
Hours of sleep against test scores: the dots rise from lower left to upper right, fairly tightly, with one student who slept 9 hours and scored very low.
- Direction: positive. More sleep goes with higher scores.
- Form: roughly linear. The dots follow a straight band, not a curve.
- Strength: fairly strong. They sit close to the band.
- One outlier: the 9-hour student with a low score. Far from the pattern, not just far along it.
A fairly strong positive linear association, with one outlier
Association or cause?
A town finds that neighbourhoods with more libraries have higher average incomes. Does building a library raise income?
- There is a positive association between libraries and income. That is what the data shows.
- But wealthier areas can afford more libraries. The cause may run the other way.
- And population size affects both. A confounding variable.
- So no — association, not cause. Only an experiment could settle it.
No. This is an association, with obvious confounders
Choosing the variables
You want to see whether studying longer helps. Which variable goes on which axis?
- Study time is the explanatory variable, so it goes on x. The thing you think does the explaining goes across.
- Test score is the response, so it goes on y. The thing being explained goes up.
- Label the axes "study time (hours)" and "score (points)". Variable and units, both.
Study time on x, score on y
Words to know
- Scatter plot
- A graph of paired data, one dot per case.
- Association
- A pattern linking the two variables.
- Cluster
- A group of dots bunched together.
- Outlier
- A dot far away from the rest of the pattern.
- Linear association
- The dots roughly follow a straight line.
Mistakes to watch for
Joining the dots. Each dot is a separate person, so a line between them would claim something exists in between — half a child scoring half a mark. Leave them as dots.
Claiming one variable causes the other. Ice cream sales and drowning deaths rise together, and neither causes the other — hot weather causes both. A third variable like that is called a confounder, and a scatter plot can never rule one out.
Calling a point an outlier because it sits at the far right. Far along the pattern is not far from it. A very tall adult is not an outlier on a height-weight plot if their weight fits the trend.
Describing only the direction. A complete answer names direction, form, strength and any unusual points — four things. "Positive" on its own leaves most of the marks on the table.
The scatter plot was popularised by Francis Galton in the 1880s while studying how tall children of tall parents turned out to be.
He noticed the children were tall, but less extreme than their parents — and called it "regression towards mediocrity."
That awkward phrase is why fitting a line to a scatter plot is still called regression today, long after anyone meant anything by the word.