A distribution shows how values are spread across a group. The average of a group describes the group, not each individual in it. When two groups overlap, knowing which group an individual belongs to tells you very little about that individual’s value.
This skill uses the data ideas from continuous and discontinuous variation. The aim is a conclusion that matches the data and goes no further.
What should you check before drawing a conclusion?
- Mean, median or mode: state which average is used and what it describes.
- Range or spread: how far do values stretch on either side?
- Overlap: do the ranges share values? How much?
- Sample size: is the sample small? Small samples can mislead.
- Other causes: could the environment, age or sampling method explain the difference?
Only then write a sentence that says what the data show and no more.
Worked example
Two fictional islands each have a population of small birds. A scientist measures beak length in a sample of 10 birds from each. The data are invented.
| Island A (mm) | 10, 11, 11, 12, 12, 12, 12, 13, 13, 14 |
|---|---|
| Island B (mm) | 11, 12, 12, 13, 13, 13, 13, 14, 14, 15 |
Step 1, mean for A: 10 + 11 + 11 + 12 + 12 + 12 + 12 + 13 + 13 + 14 = 120, and 120 ÷ 10 = 12.0 mm.
Step 2, mean for B: 11 + 12 + 12 + 13 + 13 + 13 + 13 + 14 + 14 + 15 = 130, and 130 ÷ 10 = 13.0 mm.
Step 3, ranges: A is 14 − 10 = 4 mm and B is 15 − 11 = 4 mm. The spreads are equal.
Step 4, overlap: A runs from 10 to 14 and B runs from 11 to 15, so they share the values 11 to 14. Nine of the 10 Island A birds (all except the 10 mm bird) have a beak length that also appears on Island B, so there is no clean split.
Step 5, conclusion: “The mean beak length on Island B is 1.0 mm greater than on Island A in this small sample. The ranges overlap, so some Island A birds have longer beaks than some Island B birds. The difference may be due to genes, food or chance, so more data are needed.”
Notice that the conclusion never claims that every Island B bird has a longer beak. It does not name a cause as a fact either.
The mistake to watch for
Mistaken answer: “Birds on Island B have longer beaks than birds on Island A, so you can tell the island by looking at one beak.”
The difference in means is 1.0 mm, but the spread inside each island is 4 mm. A beak of 12 mm appears on both islands, so one beak cannot identify the island.
The same reasoning applies to any trait in any species, including people. Group differences in an average never justify judging an individual, and human traits are shaped by environment and by large variation inside every group. The correction is to write “on average” and “in this sample” and to mention the overlap.
Check yourself
All data are invented.
1. Sample 1 of a plant has heights 20, 22, 24 cm. Sample 2 has heights 23, 25, 27 cm. Calculate both means and say whether every plant in sample 2 is taller than every plant in sample 1.
Show answer
Sample 1 mean: (20 + 22 + 24) ÷ 3 = 66 ÷ 3 = 22 cm. Sample 2 mean: (23 + 25 + 27) ÷ 3 = 75 ÷ 3 = 25 cm. The ranges overlap at 23 and 24, so the 24 cm plant in sample 1 is taller than the 23 cm plant in sample 2. No, not every plant is taller.
2. Give two reasons why a difference in group means might not be caused by genes.
Show answer
Any two of: a difference in environment (food, light, temperature), a difference in age, a small or unrepresentative sample, or chance.
3. A student writes: “This insect is from the northern group, so its wing is longer.” Explain the problem using the idea of overlap.
Show answer
The groups overlap, so individuals in the southern group can also have long wings. A group average cannot predict one insect. The student should measure the wing itself.
Where this leads next
You have now covered the five skills, so try the module practice set to mix them. The mistake log and retest queue helps you track wording errors, and the probability tree and counting board and inheritance model board show that grids are models, not predictions about real people.
Some students find this lesson the hardest to write because the careful wording feels long. A teacher in online one-to-one Biology tuition can help you compress it into two precise sentences.