Posing the question and reading the scatter plot

Posing a relationship question

• Your question must name both variables, name the population, and ask about a relationship.
• A good question:
  • "What is the relationship between a car's engine size and its fuel consumption, for the vehicles in this data set?"
• Weak questions and why they fail:
  • "Do bigger engines use more fuel?" — a yes/no question, not a relationship question.
  • "What affects fuel consumption?" — no specific explanatory variable.
  • "Is engine size related to things?" — no response variable, no population.

Choosing which variable is which

• The response variable ($y$) must be continuous — the standard requires it. Height, mass, time, price, temperature.
• The explanatory variable ($x$) may be discrete or continuous — number of bedrooms, engine size, year.
• Decide from the context, not from the data. Ask which variable could plausibly influence the other:
  • Engine size → fuel consumption ✓ (fuel consumption cannot change engine size)
  • House size → house price ✓
  • Altitude → temperature ✓
• If both directions are plausible, say so. Hours of study and exam mark could run either way, and noting that is a contextual insight worth making.

The scatter plot

• Explanatory on $x$, response on $y$. Always.
• Label both axes with the variable name and units. An unlabelled scatter plot cannot earn credit as a display.
• Look at it before you fit anything. Everything you say about the relationship comes from here.

The four things to describe

• Work through these in order, every time. They are what "identifying features" means for bivariate data.

Feature	What to say
Trend / direction	Positive (rising), negative (falling), or none
Nature / form	Linear (straight) or non-linear (curved)
Strength	How tightly the points cluster around the pattern
Unusual features	Outliers, groupings, changing scatter

• Then explain each in context. That pairing — feature, then explanation — is the habit that produces Merit evidence throughout the report.

Describing strength without $R^2$

• The standard says $R^2$ is not expected, so strength is described in words from the plot:
  • Strong — points lie close to the pattern, with little scatter
  • Moderate — a clear pattern but noticeable scatter
  • Weak — a pattern is discernible but the scatter is large
• Support your description with what you can see: "the relationship is moderately strong — the upward pattern is clear, but at any given engine size fuel consumption varies by around 2 L/100 km."
• That sentence is better than a number, because it says what the scatter means for a prediction.

Unusual features to look for

• Outliers — points far from the general pattern. Ask whether the point is a data error, a special case, or a genuine extreme. Never delete one without saying so and giving a reason.
• Groupings or clusters — two separate clouds usually mean two different sub-populations are mixed in the data. This is a major finding, not a nuisance: it often means the analysis should be done separately for each group.
• Changing scatter (fanning) — the spread growing as $x$ increases means predictions are more reliable at one end than the other, which must be said when you make a prediction.
• Curvature — if the pattern bends, a straight line is the wrong model.