Scatter Plots, Correlation and Line of Regression

Positive correlation. As $x$ increases, $y$ tends to increase. Example: hours studied vs exam mark.

Negative correlation. As $x$ increases, $y$ tends to decrease. Example: car age vs car value.

No correlation. No clear trend — the dots are scattered.

Strength.

• Strong: dots lie close to a line.
• Weak: dots are spread out, but a trend is still visible.

Worked descriptions.

• "Strong positive correlation" — clear line, $y$ rises as $x$ rises.
• "Weak negative correlation" — dots loosely trend down-and-right.
• "No correlation" — no consistent trend.

Tip. Cambridge expects two-word descriptions: a STRENGTH (strong / weak) and a TYPE (positive / negative / no). Single-word answers lose marks.

Method.

1. Look at the cloud of points.
2. Draw a STRAIGHT LINE through the middle, with roughly equal numbers of dots above and below.
3. The line should NOT pass through every point (that's only possible for perfectly correlated data).
4. Cambridge accepts any reasonable line — there's some tolerance.

Tip. Pass the line through the mean point $(\bar{x}, \bar{y})$ if you can compute it — this is mathematically correct for least-squares regression and helps balance the line.

Worked. A scatter plot of $(x, y)$ data has a clearly positive trend. A line drawn through the middle has gradient $\approx 1.5$ and $y$-intercept $\approx 5$.

• Equation of best fit: $y \approx 1.5x + 5$.
• Use this to predict: for $x = 10$, $y \approx 1.5(10) + 5 = 20$.

Pitfall. Drawing a curvy line. Best fit is a STRAIGHT line. If the data has a curved pattern, mention this — but the line stays straight.

Once the line of best fit is drawn:

1. To estimate $y$ from a given $x$: go up from $x$ on the horizontal axis, hit the line, then go across to read $y$.
2. To estimate $x$ from a given $y$: same procedure in reverse.

Interpolation vs extrapolation.

• Interpolation: estimating WITHIN the range of the data. Reasonably reliable.
• Extrapolation: estimating OUTSIDE the range. UNRELIABLE — the trend may not continue.

Worked. Scatter plot of (study hours, mark) for hours $0$ to $10$. A student studied for $7$ hours; estimate their mark.

• Read off the line at $x = 7$: gives $y \approx 65\%$ (interpolation — reliable).
• A student studied for $20$ hours; estimate their mark.
• $x = 20$ is FAR outside the data range. The line of best fit predicts $y \approx 130\%$ — clearly nonsense (you can't get over $100\%$). Don't extrapolate.

Tip. When asked about reliability, mention the data range explicitly: "This estimate is reliable because $x = 7$ lies within the data range ($0$ to $10$)."

Correlation $\ne$ causation. Just because $x$ and $y$ are correlated doesn't mean changes in $x$ CAUSE changes in $y$.

Examples.

• Ice cream sales correlate with drowning rates. Does ice cream cause drowning? No — both are caused by hot weather (a lurking variable).
• Shoe size correlates with reading ability among schoolchildren. Bigger shoes don't make you read better — both relate to AGE.

Cambridge framing. Cambridge sometimes asks "explain why correlation does not imply $X$ causes $Y$". A good answer cites a possible lurking variable or alternative explanation.

Tip. Don't overclaim. A correlation tells you the variables MOVE TOGETHER. The causal story has to be argued separately.