Comparing two fertiliser trials with unequal group sizes
A horticulture class grows basil under two fertiliser regimes. Group A (organic) has 12 plants and a mean leaf count of 34.2 with standard deviation 4.6. Group B (synthetic) has 8 plants and a mean leaf count of 38.1 with standard deviation 3.9. A student claims: "Group B is clearly better because its mean is higher and its spread is smaller." The teacher notes the groups differ in size and asks the class to interrogate the claim rather than accept it. One Group B plant was later found to have been measured twice and recorded as two separate plants, both with leaf count 41. The class must decide how, or whether, this affects the comparison, and must state what the standard deviations actually measure here.
1. Recompute Group B's mean after removing the duplicate record, and explain why the original mean of 38.1 was not simply wrong but was based on a contaminated sample.
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Original total for Group B: 8 × 38.1 = 304.8. The duplicated plant contributed 41 twice, so removing one copy gives total 304.8 − 41 = 263.8 over 7 plants. New mean = 263.8 ÷ 7 ≈ 37.69.
The original mean was computed correctly from the data as recorded; the error is upstream, in the sampling frame. One physical plant was counted twice, so the sample of 8 was not 8 independent observations. This inflates the apparent sample size, which matters because the standard deviation and any later inference depend on n. The corrected mean is lower, narrowing the gap with Group A from 3.9 to about 3.5 leaves.
2. The student argues the smaller standard deviation for Group B (3.9 vs 4.6) proves Group B is more consistent. Evaluate this reasoning.
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The standard deviations measure spread of leaf counts within each group, not consistency of the fertiliser's effect. Two issues weaken the claim. First, Group B has only 7 valid plants versus 12, and a smaller sample typically shows a smaller spread by chance; comparing raw standard deviations across unequal n is not a fair consistency test. Second, the duplicate record (41, 41) sat near the top of the distribution and would have pulled the Group B standard deviation downward by making the upper tail look denser than it is. Recomputing after removal would likely raise the Group B standard deviation, further eroding the claim. The honest conclusion is that the difference in spread is not established.
3. State one piece of additional information the class would need before concluding that synthetic fertiliser causes higher leaf counts.
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They need to know how plants were assigned to groups. If plants were allocated by, say, position on the bench or by prior size, the groups may differ in light, temperature, or starting vigour, so the fertiliser is confounded with those variables. Random allocation would make a causal reading more defensible; without it, only an association is supported. Other acceptable answers include whether the same person counted leaves in both groups (measurement bias) or whether the groups were grown at the same time.