To compare two data sets, compare their centres, compare their spreads, then write a conclusion that uses the numbers. When the means are equal, the standard deviation decides which set is more consistent.
This lesson is part of dispersion of ungrouped data. It uses the skills from finding variance and standard deviation.
What is the sentence frame?
Write three sentences in this order.
- The means: “The mean of A is … and the mean of B is …”.
- The spreads: “The standard deviation of A is … and of B is …”.
- The conclusion: “So set … is more consistent because its standard deviation is smaller.”
Worked example: two archers
The scores of two archers in five rounds are:
- Archer A: 10, 12, 14, 16, 18
- Archer B: 13, 14, 14, 14, 15
Archer A.
- Mean = 70 ÷ 5 = 14.
- Deviations: −4, −2, 0, 2, 4, squared: 16, 4, 0, 4, 16, total 40.
- Variance = 40 ÷ 5 = 8, so σ = √8 = 2.83.
Archer B.
- Mean = 70 ÷ 5 = 14.
- Deviations: −1, 0, 0, 0, 1, squared: 1, 0, 0, 0, 1, total 2.
- Variance = 2 ÷ 5 = 0.4, so σ = √0.4 = 0.63.
| Mean | Standard deviation | |
|---|---|---|
| Archer A | 14 | 2.83 |
| Archer B | 14 | 0.63 |
Conclusion. Both archers have a mean of 14, so their average scores are equal. The standard deviation of B (0.63) is much smaller than that of A (2.83), so Archer B is more consistent.
Why the mean alone is not enough
If the question had only asked for the mean, the two archers would look identical. The standard deviation shows that A swings between 10 and 18, while B stays close to 14.
A coach picking a player for a final would choose B if a steady score matters, and A if a chance of a very high score matters. The data supports either choice, and the question decides which.
The slips to avoid
The first slip is to write “Archer B is better” with no reason. The second is to compare a mean with a standard deviation, which measure different things.
A third slip is to give the two standard deviations but not say which is smaller. Use the word “consistent” and give both numbers in the same sentence. Use the graph evidence comparison lab to practise writing this sentence on invented data.
Check yourself
Shop X sells 20, 22, 24, 26, 28 cups of tea on five days. Shop Y sells 23, 24, 24, 24, 25. Compare them using the mean and the standard deviation.
Answer
Both means are 24. For X the deviations are −4, −2, 0, 2, 4, so the variance is 40 ÷ 5 = 8 and σ = 2.83. For Y the deviations are −1, 0, 0, 0, 1, so the variance is 2 ÷ 5 = 0.4 and σ = 0.63.
The means are equal, and the standard deviation of Y is smaller, so sales at Shop Y are more consistent.
What to study next
Move on to predicting the effect of changing a data set. For a harder case, see comparing two classes with equal means but different spread.
For a teacher to read your comparison sentences, see online one-to-one Mathematics tuition.