Mr Kidman's Classroom
Revise the key ideas
Relationships and prediction
- Paired plots — Plot each pair once with explanatory x and response y, axes, units and appropriate scales. A point (4, 30) records response 30 for explanatory value 4, not two unrelated observations. Plot the raw pairs before summarising the relationship.
- Direction and strength — Positive correlation means y tends to increase as x increases; negative means it tends to decrease. Stronger correlation has points closer to the trend, not necessarily a steeper slope. Zero correlation describes no identified association of the specified kind, not identical values.
A fitted trend with scatter. Original illustrative diagram; numerical datasets are fictional worked examples. Enlarge diagram - Causation and confounding — An association does not establish that changing x causes y. A third variable may affect both, the direction may be reversed, or the association may be accidental. Compare study design and plausible mechanisms; multiple interacting influences can explain a pattern.
- Double mean point — Calculate the mean of all x values and separately of all y values. A by-eye line of best fit passes through (x̄, ȳ) and follows the overall linear trend with balanced scatter. It need not pass through every point or through the origin.
A bar over a letter means its mean: x̄ is the mean of the x values and ȳ is the mean of the y values.
- Interpolation — Estimate y for an x within the observed range by reading the fitted line. This is interpolation, not an exact result for an individual. A weak trend gives a less dependable estimate; retain units and acknowledge the scatter.
- Extrapolation — Estimating beyond observed x values is extrapolation. The relationship may change outside the range, and a line can predict impossible negative counts. Check context and distance from the sample; a convincing fit inside the range does not guarantee distant predictions.
- Gradient and intercept — Gradient is response change per one unit increase in x; intercept is fitted y at x=0. Their units differ. An intercept may have no practical meaning if x=0 lies outside the data or is impossible in context; do not force a causal interpretation of slope.
- Given rank coefficient — Spearman's rank coefficient lies between −1 and +1. Values near +1 indicate strong agreement in ordering; near −1 indicate opposite ordering. A value near zero indicates little rank association, without proving independence or absence of every possible pattern.
Independence means knowing one variable’s value does not change the probabilities for the other.
Higher — coefficients and regression
- Regression equation — Use a given y = a + bx to predict response for an explanatory value. If y=18+2.5x and x=4, predicted y=28. Interpret a and b in context and observe interpolation limits; this y-on-x model should not simply be reversed to predict x without justification.
Here a is the predicted y-value when x = 0, and b is the predicted change in y for each one-unit increase in x.
- Rank and differences — Rank both variables in the same direction, pair the ranks for each item and calculate d as their difference. Square every difference and sum d². Edexcel does not test tied ranks in the Spearman calculation; ordinary distinct ranks run from 1 to n in each column.
- Spearman calculation — Use the given formula rₛ = 1 − 6Σd²/[n(n²−1)]. For n=5 and Σd²=4, rₛ=1−24/120=0.8. Check the result lies in [−1,1], explain its sign and avoid rigid invented boundaries for strong and weak.
n is the number of paired observations. Σd² means add all the squared rank differences.
- PMCC — The Pearson product moment correlation coefficient (PMCC), r, measures linear association. A value −0.9 suggests strong negative linear correlation; r=0 means no linear correlation, but a curved relationship may still exist. PMCC calculation is not required in this course.
- Compare coefficients — Spearman uses ranks and can reflect a monotonic curved trend (one that keeps the same direction, increasing or decreasing); PMCC uses the original numerical values and measures closeness to a straight line. A perfect increasing curved relationship can have Spearman +1 while PMCC is below +1. Neither coefficient alone establishes causation.
Higher — coefficients and regression
- Build the rank table — Five people have X values 10, 20, 30, 40, 50 and corresponding Y values 12, 18, 15, 24, 30. Ascending X ranks are 1, 2, 3, 4, 5; the paired Y ranks are 1, 3, 2, 4, 5. The differences are 0, −1, 1, 0, 0, giving Σd² = 2. Spearman's coefficient is 1 − (6 × 2)/(5 × 24) = 0.9. Keep each person's measurements paired when ranking; sorting the columns independently would invent perfect agreement.
