Lesson 1.3 · Unit 1 · NGSS practice 4
Sample size, variation, and why one organism proves nothing
Two plants of the same variety, grown side by side in the same soil with the same water, will not reach the same height. That is not experimental error. Living things vary, and the variation is as real as the average. Any conclusion drawn from a single organism is a conclusion about that organism.
- Biological variation is signal, not noise. Individuals of a species differ genetically and developmentally, and that difference is the raw material evolution acts on. It cannot be eliminated, only accounted for.
- A sample of one tells you about one. With no measure of how much individuals normally differ, you cannot tell whether a gap between two organisms is caused by your treatment or is ordinary.
- Replication means repeating the measurement on different individuals, not measuring the same plant three times. Measuring one plant repeatedly tells you about your ruler.
- The mean summarizes the center and is the sum of the values divided by how many there are.
- The spread matters as much as the mean. The range is the largest value minus the smallest. Two groups with identical means and very different spreads are not the same result.
- Overlapping spreads weaken a claim. If the tallest untreated plant is taller than the shortest treated plant, the groups overlap, and a difference in means needs more individuals before it can be trusted.
Where students lose marks: reporting a mean with no indication of spread, then treating a small difference as established. "The treatment group was taller" is not a finding until you say by how much and how much the individuals varied.
The data. Five bean seedlings per group, height in centimeters after fourteen days. The figures are constructed so the arithmetic is checkable.
| Group | Heights (cm) | ||||
|---|---|---|---|---|---|
| No fertilizer | 12 | 14 | 11 | 13 | 15 |
| Fertilizer | 16 | 21 | 14 | 19 | 20 |
Step one: find each mean. Untreated: 12 plus 14 plus 11 plus 13 plus 15 is 65, and 65 divided by 5 is 13.0 cm. Treated: 16 plus 21 plus 14 plus 19 plus 20 is 90, and 90 divided by 5 is 18.0 cm.
Step two: state the difference. The treated mean is 5.0 cm greater, which is an increase of about 38 percent over 13.0 cm.
Step three: find each range. Untreated runs from 11 to 15, a range of 4 cm. Treated runs from 14 to 21, a range of 7 cm.
Step four: check whether the groups overlap. The tallest untreated plant is 15 cm and the shortest treated plant is 14 cm, so the groups overlap between 14 and 15 cm. One treated plant is shorter than three untreated ones.
Step five: say what the overlap does and does not mean. It does not mean the fertilizer had no effect. It means that if you had grown one plant per group and happened to pick the 15 cm untreated plant and the 14 cm treated plant, you would have concluded the opposite. The mean difference survives the overlap; a single comparison would not have.
Step six: notice that the treated group varies more. Its range is nearly twice the untreated range. That is itself a result worth reporting: the fertilizer may help some individuals far more than others, which a mean alone hides.
Step seven: state the conclusion honestly. In this trial, the fertilized group had a mean height 5.0 cm greater, with more variation between individuals and some overlap between the groups. Five plants per group is a small sample, and repeating with more plants would be the obvious next step.
Questions 1 to 5 are recall. Questions 6 to 10 ask for reasoning.
- Calculate the mean of 4, 7, 7, 10 and 12.
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The sum is 40, and 40 divided by 5 is 8.0
- What is the range of that set?
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12 minus 4, which is 8
- What does replication mean in an experiment?
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Repeating the measurement on different individuals, not remeasuring the same one
- In the worked example, which group varied more, and by how much?
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The fertilized group, with a range of 7 cm against 4 cm
- Why is measuring one plant three times not replication?
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It measures the precision of your ruler, not the variation between individuals
- Two groups both have a mean height of 15 cm. Group A ranges from 14 to 16 cm
and group B from 4 to 26 cm. Explain why these are not the same result.
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The means are identical but the groups behave completely differently. In group A every individual is close to 15 cm, so the mean describes each plant well and a prediction about a new plant would be reliable. In group B individuals range from very small to very large, so the mean describes almost nobody and predicting a new plant from it would be nearly worthless. The spread tells you how much the mean can be trusted as a summary, which is why reporting it is not optional. The mean summarizes group A well and group B badly
- A student concludes that a plant food works because their treated plant grew 3
cm taller than their untreated plant. Explain the flaw.
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With one plant in each group there is no way to know how much two untreated plants would have differed from each other. Ordinary variation between individuals of the same variety can easily produce a 3 cm gap, as the worked example shows, where untreated plants ranged across 4 cm with no treatment at all. The comparison cannot separate an effect of the plant food from normal variation, so the conclusion is unsupported regardless of how carefully the two plants were measured. One plant per group gives no measure of ordinary variation
- Explain why increasing the number of individuals per group strengthens a
conclusion.
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Each individual carries its own random departure from the group's typical value, and those departures partly cancel when averaged. With more individuals the mean of the sample sits closer to the value that would be obtained from the whole population, so the same measured difference between groups becomes less likely to have arisen from an unlucky selection of individuals. Larger samples also give a better picture of the spread itself, which is what a claim about a difference has to be judged against. Random individual differences cancel out, so the sample mean is a better estimate
- The treated group in the worked example contained one plant shorter than three
untreated plants. Does this refute the conclusion? Explain.
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No. The claim is about the average effect on a group, not a promise that every treated individual will exceed every untreated one. Individuals vary for many reasons at once, including seed quality and position, and the treatment shifts the distribution rather than setting each plant's height. A single overlapping individual is exactly what an overlapping range describes. What the overlap does mean is that a conclusion drawn from one plant per group would have been unreliable, which is an argument for larger samples rather than against the finding. The claim concerns group means, and overlap between individuals is expected
- Explain why biological variation should not be described as experimental error.
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Experimental error is a failure of measurement or procedure that better technique could reduce, such as a misread ruler or an unevenly watered tray. Variation between individuals is not a failure at all: organisms of the same species genuinely differ in their genes and their development, and they would still differ under perfect technique. Treating that variation as error suggests it could be removed, when in fact it is the material natural selection acts on and one of the central facts of the subject. It is a real property of living things, not a flaw in the method