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3.7 · Paper 2A · Statistical Techniques
Scatter graphs, PMCC, lines of best fit and using regression to predict — a data-handling powerhouse.
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A scatter diagram shows whether two variables are related. If the relationship is roughly linear, a line of best fit can be drawn through the mean point $(\bar{x}, \bar{y})$, following the trend of the data.
Correlation does not imply causation — two variables can be strongly correlated because a third variable (a confounder) affects both of them, without either one directly causing the other.
The regression line of $y$ on $x$, $y = a+bx$, is the precisely calculated line of best fit used to predict a value of $y$ from a given $x$.
Interpolation (predicting within the range of the original data) is generally reliable. Extrapolation (predicting outside that range) is unreliable, because there's no guarantee the linear relationship continues beyond the data you have.
The PMCC, $r$, gives a precise numerical measure of the strength and direction of linear correlation, with $-1 \le r \le 1$. Values close to $+1$ or $-1$ indicate strong linear correlation; values close to 0 indicate little or no linear correlation — but this doesn't rule out a strong non-linear relationship existing between the variables.
A regression line relating hours of revision, $x$ (ranging from 2 to 20 in the data), to exam score, $y$ (%), is $y = 38 + 2.6x$. The PMCC for this data is $r = 0.91$.
a) Interpret the value of $r$. b) Predict the score of a student who revises for 12 hours. c) Comment on using this line to predict the score of a student who revises for 60 hours.
a) $r=0.91$ is close to $+1$, indicating strong positive linear correlation — more revision is associated with a higher exam score.
b) $y = 38 + 2.6\times12 = 69.2\%$. This is interpolation (12 is within 2–20), so reasonably reliable.
c) $y = 38+2.6\times60 = 194\%$ — an impossible score. 60 hours is far outside the data range (extrapolation), and the model clearly breaks down here, so this prediction cannot be trusted.
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