Scatter Graphs and Correlation

Scatter Graphs and Correlation

A scatter graph plots pairs of values \( (x, y) \) for bivariate data — two variables measured on the same subject. The pattern of the plotted points reveals whether and how the variables are related (their correlation).

The Five Types of Correlation

Strong Positive Correlation

x y strong positive

Strong Negative Correlation

x y strong negative

Weak Positive Correlation

x y weak positive

No Correlation

x y no correlation

Line of Best Fit and the Mean Point

When correlation exists, draw a line of best fit by eye through the cloud of points, ensuring roughly equal numbers of points above and below the line. It must pass through or near the mean point \( (\bar{x},\ \bar{y}) \).

x y 2 4 6 8 10 12 14 \( (\bar{x},\bar{y}) \) x=9 y? interpolation extrapolation (less reliable) data range

Describing Correlation

Always describe correlation using two words: direction (positive or negative) and strength (strong or weak). If there is no pattern, state "no correlation".

Examples of context language: "There is a strong positive correlation between revision time and exam score — as revision time increases, exam score tends to increase."

Causation Warning

Correlation does not prove causation. Two variables can be correlated because both are driven by a third variable, by coincidence, or through a complex chain of effects. Always consider whether a causal link is genuinely supported before drawing conclusions.

Worked Example

A scatter graph plots the number of hours slept the night before an exam against the exam score. Points slope upward with moderate scatter. Describe the correlation and suggest a reason why it might not be causal.

Correlation: moderate positive — as hours of sleep increase, exam score tends to increase.

Causation caveat: students who sleep well may also revise more, eat better, or have lower anxiety — the sleep may not directly cause the higher score; other factors are likely at work simultaneously.

A line of best fit passes through \( (4,\ 20) \) and \( (12,\ 52) \). Predict the value of \( y \) when \( x=8 \) (interpolation) and comment on a prediction at \( x=20 \) (extrapolation).
\[ m = \frac{52-20}{12-4} = \frac{32}{8} = 4 \implies y = 4x + 4 \] \[ x=8:\ y = 4(8)+4 = 36 \quad \text{(interpolation — reliable, within data range)} \]

\( x=20 \): this lies well beyond the data range — the linear relationship may not continue; treat with caution.

 Key Takeaways

  • Scatter graphs show the relationship between two quantitative variables — each point is one data pair.
  • Describe correlation with two words: direction (positive/negative/none) and strength (strong/weak).
  • Line of best fit: drawn by eye through the mean point \( (\bar{x},\bar{y}) \) with roughly equal scatter either side.
  • Interpolation (within data range) is more reliable than extrapolation (outside).
  • Correlation does not imply causation — always consider alternative explanations.