Rainfall and Crop Yield Analysis
An agricultural scientist records monthly rainfall (in mm) and maize yield (in tonnes per hectare) across 10 farms in one season. A scatterplot shows a generally positive linear trend. The scientist calculates a correlation coefficient of r = 0.72 and fits a least-squares line y = 0.9 + 0.015x. However, one farm received 480 mm of rain (far above the others, which ranged from 120 to 260 mm) and had a yield of only 2.1 t/ha, well below the line's prediction. The scientist is unsure whether to keep this point. A colleague argues that r = 0.72 already proves rainfall causes higher yield and that the outlier should simply be deleted to improve the fit.
1. Assess the colleague's two claims: that r = 0.72 proves causation, and that the outlier should be deleted.
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First claim: correlation does not establish causation. r = 0.72 indicates a moderately strong positive linear association, not proof that rainfall causes higher yield. Confounding variables such as fertiliser use, soil quality, temperature, and irrigation could drive both rainfall patterns and yield. So the causation claim is unsupported.
Second claim: deleting a point merely to improve the fit is poor practice. The 480 mm farm is a legitimate observation and may reveal a real effect (for example, waterlogging reducing yield at very high rainfall). Removing it without justification biases the analysis.
Judgement: the outlier should be investigated, not automatically deleted. The scientist should check whether it is a recording error. If genuine, it may indicate the relationship is non-linear, which is important information rather than noise.
2. The scientist refits the line without the outlier and finds r = 0.88. Interpret what this change suggests about the outlier's influence and the shape of the relationship.
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The jump from r = 0.72 to r = 0.88 shows the outlier substantially weakens the linear correlation. This is a high-leverage point: its extreme x-value (480) gives it strong influence on the line.
The fact that removing it strengthens the linear fit, while the point sits well below the line, suggests the true relationship may curve downward at high rainfall. That is, yield rises with rainfall up to a point, then falls (possibly due to waterlogging or disease).
Judgement: the scientist should not simply report r = 0.88 as the headline result, because that hides a meaningful pattern. A better approach is to report both analyses, describe the possible non-linear effect, and recommend more data at high rainfall to confirm. The higher r is not automatically the 'better' result.