How to use the simulator
The main control is the correlation r slider. It runs from to in steps of 0.05 and starts at .
- Scatterplot. Sixty points plotted on axes labelled Variable X and Variable Y, with a dashed line of best fit. The points are generated so that their correlation matches the slider exactly.
- r readout. The large number (for example r = +0.70) is recalculated from the plotted points. Underneath, a sentence describes the relationship in words, such as "strong positive relationship — as X rises, Y tends to rise".
- New sample. Draws a fresh set of 60 random points with the same . The cloud changes but does not.
- Another example. Steps through four third-variable examples. A small diagram shows a hidden variable Z with arrows to both X and Y, and a dashed line between X and Y labelled "observed correlation". The four pairs are ice-cream sales and drowning deaths, children's shoe size and reading ability, hours of sleep and school grades, and coffee intake and stress.
The verbal labels follow the simulator's own cut-offs. An absolute value of below 0.10 is described as no linear relationship, then weak up to 0.35, moderate up to 0.65, strong up to 0.90, and very strong above that. Textbooks draw these lines slightly differently.
The red Correlation ≠ causation banner never changes. Even at , with every point on the line, it still applies.
The key ideas
The correlation coefficient always lies between and and describes a linear relationship between two measured variables.
- Sign gives direction. In a positive correlation both variables tend to rise together. In a negative correlation, as one rises the other tends to fall.
- Size gives strength. The closer is to 1, the more tightly the points cluster around a straight line. An of is a stronger relationship than .
- Zero means no linear pattern. An near 0 can still hide a curved relationship, so look at the scatterplot as well as the number.
A correlation, however strong, leaves at least three explanations open:
- X causes Y.
- Y causes X (the directionality problem).
- A third variable Z causes both, which is the case the simulator's diagram shows.
Only an experiment, where the researcher manipulates one variable and randomly assigns participants, can separate these. Correlations are still useful because they allow prediction: knowing X lets you estimate Y even when you cannot explain the link. Watch out too for an illusory correlation, where people believe two things are related because a few memorable cases stick in the mind.
Worked example
Scenario: A study of 200 teenagers finds that daily hours on social media and nightly hours of sleep correlate at .
Step 1: Read the number. The negative sign means teens who spend more time on social media tend to sleep less. A magnitude of 0.45 is a moderate relationship, so plenty of teens do not fit the trend.
Step 2: Model it. Drag the slider to . The readout shows r = -0.45 and "moderate negative relationship — as X rises, Y tends to fall". The dashed line slopes down, and the points form a loose band around it rather than a tight one.
Step 3: List the explanations.
X causes Y: late-night scrolling cuts into sleep.
Y causes X: teens who cannot sleep pick up their phones.
Third variable: anxiety, or a naturally late body clock, could raise screen time and reduce sleep at the same time.
X causes Y: late-night scrolling cuts into sleep.
Y causes X: teens who cannot sleep pick up their phones.
Third variable: anxiety, or a naturally late body clock, could raise screen time and reduce sleep at the same time.
Step 4: Say what would test causation. Randomly assign teens either to stop using social media after 9 p.m. for two weeks or to carry on as usual, then compare their measured sleep. A difference between the groups would support a causal claim. The correlation alone cannot.
Common mistakes on the AP exam
- Reading a negative r as weak. The sign shows direction only. An of is a strong relationship.
- Using causal language for correlational data. Words like "causes", "leads to" and "increases" claim more than a correlation shows. Write "is associated with" or "predicts" instead.
- Naming a third variable that only affects one side. A genuine third variable must plausibly affect both X and Y. Age works for shoe size and reading because it drives both.
- Saying r = 0 means the variables are unrelated. It means there is no linear relationship. A curved pattern can still exist.
- Reading strength from the slope. A steep best-fit line does not mean a strong correlation. Strength depends on how closely the points cluster around the line.
- Calling a study an experiment because it reports r. Correlation coefficients come from measuring variables, not manipulating them.
When the AP exam uses this
Correlation draws on two AP Psychology science practices: evaluating non-experimental designs (Practice 2) and interpreting data and statistics (Practice 3). Multiple-choice questions often show a scatterplot or quote a value of and ask for its direction and strength, or ask what conclusion the data support. In the Article Analysis Question you may be asked to say whether a study's findings support a causal claim, and the answer depends on whether it used random assignment.