Biology · Life, Evidence & Biological Scale

Measurements, graphs, uncertainty and clear scientific communication

Before calculating, make a prediction: if the higher-light condition increases average extension, what pattern would you expect across many comparable new cohorts? An average effect need not appear in every small cohort. A cohort here means a new group of seedlings considered together. Write your prediction before switching between the graphs.

No earlier biology course is needed. The data lesson explains its own arithmetic. Chemistry and mathematics offer useful next steps, but you can read this chapter before completing their maps.

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By the end, you can

Read the supplied data, check the averages and revise a bounded conclusion when more observations are included.

Say exactly what the numbers represent

Stem extension is final stem length minus starting stem length. The supplied values are in millimetres over seven days, not final plant height and not a daily growth rate. Imagine measuring along the same defined stem path from the same reference points on each occasion. Changing the method halfway would make the comparison harder to interpret.

The data below are constructed for this lesson. They do not report a real experiment or establish how a particular plant species responds to light. The imagined independent-pot design comes from the previous lesson. Keep that distinction when communicating the result.

Look at the observations before the averages

Constructed educational data, not measurements from a real experiment.

Circle: lower light. Diamond: higher light. Dashed mark: group mean.

The first cohort suggests a seven-millimetre difference between the group means, with overlapping individual values. Difference between group means: 7 mm.

Constructed observations: one value per independently assigned pot
Cohort	Pot within cohort	Lower light (mm)	Higher light (mm)
First cohort	1	42	49
First cohort	2	48	54
First cohort	3	51	58
First cohort	4	54	61
First cohort	5	55	63
Mean (mm)		50	57
Range (mm)		42–55	49–63

These summaries describe the supplied sample. They do not establish a population effect or statistical significance. Entries across columns are not matched pairs.

Check an average without losing the individuals

In the first cohort the lower-light values are 42, 48, 51, 54 and 55 mm. Their sum is 250 mm. Divide by the five values: the mean is 50 mm. The higher-light sum is 285 mm, so its mean is 57 mm. The difference between these sample means is 7 mm.

The ranges overlap: lower light runs from 42 to 55 mm and higher light from 49 to 63 mm. The overlap tells us that the groups do not perfectly separate at the individual level. It does not provide a significance test. On the plot, each marker is one observation; the dashed marks summarise the means.

The second cohort gives means of 55 mm for lower light and 53 mm for higher light. Its mean difference points in the opposite direction. With both cohorts included, the sums are 525 and 550 mm across ten values per condition. The combined means are 52.5 and 55 mm; their difference is 2.5 mm.

Let new evidence change the sentence

After the first cohort, an accurate statement is: 'In these constructed observations, mean extension is seven millimetres greater in the higher-light group.' After including the second cohort, revise it: 'Across the two cohorts, the mean difference is two and a half millimetres, and the cohorts differ in direction.' State what changed instead of retaining the most striking early result.

Greater stem extension is not the same measurement as greater mass, better survival or greater reproductive success. Those outcomes would need their own measurements. The word 'better' hides the choice of outcome; name the quantity you actually observed.

Variation among seedlings, limited sampling and uncertainty in measurement are different issues. A ruler's markings do not tell us the full measurement uncertainty. Repeated checks can reveal inconsistent readings; they do not necessarily reveal a systematic error in the reference point. The exercise gives no uncertainty estimates, so we should not invent them.

Make the result inspectable

A clear report includes the question, design, measured quantity, units, individual data, calculation and limitations. Keep the vertical scale consistent when switching cohorts. Do not join unrelated seedlings with a line that suggests a time sequence. Label constructed data plainly and retain the table so another reader can check the plot.

The next useful evidence might be more independently assigned pots, recorded temperature and a more precise outcome definition. Choose it because it resolves a specific uncertainty. A larger pile of readings is not automatically a better answer.

Worked example

The revision is the result

First cohort: higher-light mean minus lower-light mean = 57 − 50 = 7 mm. Second cohort: 53 − 55 = −2 mm. Combined: 55 − 52.5 = 2.5 mm. Both groups have equal numbers in each cohort, so averaging the two cohort means gives the same combined mean here. With unequal group sizes, pooling would require weighting by the number of observations.

Check your understanding

Can you keep calling the mean difference seven millimetres after combining both supplied cohorts?
Show reasoning
No. Seven millimetres describes the first cohort. The combined difference is two and a half millimetres, and the report should identify which observations it summarises.
What is wrong with 'Higher light made every seedling healthier'?
Show reasoning
The individual values do not perfectly separate; health was not measured; and a causal claim still depends on the design. The data concern stem extension in a constructed sample.
Do overlapping ranges prove there is no population effect?
Show reasoning
No. Range overlap is not a formal inferential test. This lesson makes descriptive comparisons and does not estimate a population effect.

What to carry forward

Preserve individual observations, check the arithmetic and revise conclusions at the scope the data support.

References

Resources supporting this lesson’s explanations and examples.