Scatter Plots & Association — Print

Two measurements per person, one dot each. The shape of the cloud is the finding.

What the practice questions cover

These questions use 10 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.

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.

one child, two measurements, one dot the sheet hours score 2 523 583.5 554 645 665.5 73 hours practised score each row of the sheet becomes exactly one 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.

Positive, negative and no association in scatter plots a scatter plot shows two measurements for each person or thing positive negative none up together one up, one down no pattern

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.

  1. Direction: positive. More sleep goes with higher scores.
  2. Form: roughly linear. The dots follow a straight band, not a curve.
  3. Strength: fairly strong. They sit close to the band.
  4. 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?

  1. There is a positive association between libraries and income. That is what the data shows.
  2. But wealthier areas can afford more libraries. The cause may run the other way.
  3. And population size affects both. A confounding variable.
  4. 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?

  1. Study time is the explanatory variable, so it goes on x. The thing you think does the explaining goes across.
  2. Test score is the response, so it goes on y. The thing being explained goes up.
  3. 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.

Problem 1

As study time increases, test scores increase. This is

  1. a positive association
  2. a negative association
  3. no association
  4. a causal proof
Worked answer

a positive association

Up together means positive

Nothing here proves cause

Problem 2

As the age of a car increases, its value decreases. This is

  1. a negative association
  2. a positive association
  3. no association
  4. a cluster
Worked answer

a negative association

Opposite directions means negative

Problem 3

You want to see whether more rainfall gives taller plants. Which variable belongs on the x-axis?

Worked answer

rainfall

Rainfall is the explanatory variable

So rainfall goes on x and height on y

Problem 4

On a plot of height against weight, one point is far above the pattern. It is

  1. an outlier
  2. a cluster
  3. the median
  4. impossible
Worked answer

an outlier

It does not follow the trend of the rest

So it is an outlier

Problem 5

The dots on a scatter plot lie close to a straight band. The form is

  1. linear
  2. curved
  3. clustered
  4. random
Worked answer

linear

Close to a straight line means linear form

Problem 6

Ice cream sales and sunburn cases rise together. The best explanation is

  1. hot weather drives both
  2. ice cream causes sunburn
  3. sunburn causes ice cream sales
  4. the data must be wrong
Worked answer

hot weather drives both

Hot weather increases both independently

That is a confounding variable

Association is not causation

Problem 7

A scatter plot has dots falling steeply from upper left to lower right, tightly packed. Describe the association in three words.

Worked answer

strong negative linear

Tightly packed: strong

Falling: negative

Straight band: linear

Problem 8

Two plots both show positive association. In plot A the dots are tight to the line; in plot B they are widely scattered. Which is stronger?

  1. A
  2. B
  3. They are equally strong
  4. Cannot be determined
Worked answer

A

Closeness to the pattern measures strength

A is tighter, so A is stronger

Problem 9

A scatter plot of a class shows 24 dots. How many students are in the class?

Worked answer

24

One dot per student

24 students

Problem 10

A point at the far right of a scatter plot lies exactly on the pattern. It is

  1. not an outlier
  2. an outlier
  3. the mean
  4. a cluster
Worked answer

not an outlier

This point follows the trend

So it is not an outlier, however extreme its x value

Problem 11

A scatter plot of shoe size against height shows a positive association. Does a large shoe size guarantee a tall person? Answer yes or no.

Worked answer

no

Positive means taller on average

Individual cases still vary

So no guarantee

Problem 12

A scatter plot of price against size for houses shows two separate clouds. The likeliest reason is

  1. two different neighbourhoods are mixed together
  2. houses have two prices
  3. the data is random
  4. there is no association
Worked answer

two different neighbourhoods are mixed together

Each group follows its own pattern

Splitting the plot would show two clean trends

Problem 13

A study finds a strong positive association between number of firefighters sent and damage done to a building. Should fewer firefighters be sent? Answer yes or no.

Worked answer

no

Bigger fires get more firefighters

Bigger fires also cause more damage

Fire size is the confounder

Sending fewer would make things worse

Problem 14

Two variables have a perfect V-shaped relationship. Is the linear association strong or weak?

Worked answer

weak

One arm rises, the other falls

The rises and falls cancel

So the linear association is weak, even though the pattern is perfect

Problem 15

Which study could establish that a new fertiliser causes taller plants?

  1. Randomly assigning plants to fertiliser or none
  2. Observing which gardeners use it
  3. Comparing two neighbourhoods
  4. Surveying gardeners about yields
Worked answer

Randomly assigning plants to fertiliser or none

Random assignment balances out other factors

Observational studies cannot rule out confounders

Problem 16

A plot of hours of television against grade shows a moderate negative linear association with two outliers. Write one sentence a statistician would accept about cause.

Worked answer

no cause can be concluded from this association

The data shows an association only

A third factor could drive both

So no causal claim is justified

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