Biased Quadrat Sampling on the School Field
Miss Owusu takes her 30 SHS 2 students to the school field for a quadrat survey of a small herb. The field is 20 m by 10 m (200 m²). Each pair has a 0.5 m × 0.5 m quadrat. Because the afternoon is hot, the students place all 15 quadrats in the shaded strip under two mango trees, where the herb looks abundant. They count a mean of 12 plants per quadrat. A student calculates the population estimate as 12 × (200 ÷ 0.25) = 9,600 plants and the class celebrates. Miss Owusu notices that the shaded strip covers only about 40 m² of the field and that students trampled some plants while placing quadrats. The school has no random number table, but the students have phones with a random number app and a 30 m tape.
1. Identify two sources of bias in the sampling and explain the direction of their effect on the 9,600 estimate.
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The first source is placement bias: all quadrats were placed in the shaded strip because it was cooler and the herb looked abundant. If the herb is denser in shade than in the open field, the mean of 12 plants per quadrat overestimates the field-wide density, and 9,600 is too high. The shaded strip is only about 40 m², one-fifth of the field, so the sample represents a small and atypical habitat.
The second source is trampling. Students walked to and around the quadrats, damaging plants before or during counting. This would tend to reduce the counts in sampled areas, pulling the estimate downward. However, this bias is probably smaller than the placement bias because the quadrats were deliberately placed where the herb was visibly abundant. The two biases act in opposite directions, but that does not cancel them reliably; the estimate remains unreliable and cannot be given a valid confidence interval. The sampling is also not random or independent, so the arithmetic 12 × 800 is correct only for the shaded strip, not for the whole field.
2. Design a more defensible sampling procedure using the available equipment, and state how the estimate would be calculated.
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A defensible procedure would use the tape to mark out the field boundaries and a coordinate system: 0–20 m along the length and 0–10 m across the width. Students should use the random number app to generate 20 pairs of coordinates, rejecting any point within 0.25 m of the boundary to avoid edge effects. At each coordinate, they place the 0.25 m² quadrat without choosing the spot, and count the herb before moving around the area. If the class wants to compare shaded and open areas, it should stratify: sample randomly within the 40 m² shade and randomly within the remaining 160 m², then weight each mean by its area.
The estimate is calculated as: mean number per quadrat × (total field area ÷ quadrat area). With 0.25 m² quadrats and a 200 m² field, the multiplier is 800. For example, if 20 random quadrats give a mean of 7 plants, the estimate is 7 × 800 = 5,600 plants. Students should report the mean, the number of quadrats and the variation, not just the final figure.
3. How should Miss Owusu respond to the student's celebrated calculation without dismissing the correct arithmetic?
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She should acknowledge that the arithmetic is correct: 0.5 m × 0.5 m = 0.25 m², 200 ÷ 0.25 = 800, and 12 × 800 = 9,600. The problem is not the multiplication; it is the assumption behind it. She can ask: 'What would have to be true about the field for 9,600 to be a valid estimate?' Students should identify that every part of the field would need the same density as the shaded strip. That makes the hidden assumption visible.
She can then ask two pairs to place quadrats in the open area and compare means. The resulting difference is a teachable moment about bias, not a failure. She should introduce the distinction between sampling error, which random sampling addresses, and bias, which more quadrats in the same place cannot fix. Finally, she can ask the class to rewrite the statement 'We estimate 9,600 plants' as 'We estimate 9,600 plants in the shaded strip; a field-wide estimate requires random sampling across the whole field.' This preserves the student's calculation while correcting the inference.