Two data sets can share a mean and still describe very different groups. The standard deviation reveals the difference, and your conclusion should say what it means for the situation.
This page is part of choosing the right summary of a data set. The basic comparison method is in comparing data sets using centre and spread.
Worked example: two classes of six
The marks of six students in each class are:
- Class X: 35, 45, 60, 60, 75, 85
- Class Y: 55, 58, 60, 60, 62, 65
Means. Class X: 360 ÷ 6 = 60. Class Y: 360 ÷ 6 = 60. The means are equal.
Class X spread. Deviations: −25, −15, 0, 0, 15, 25. Squared: 625, 225, 0, 0, 225, 625, total 1 700. Variance = 1 700 ÷ 6 = 283.3, so σ = 16.83.
Class Y spread. Deviations: −5, −2, 0, 0, 2, 5. Squared: 25, 4, 0, 0, 4, 25, total 58. Variance = 58 ÷ 6 = 9.67, so σ = 3.11.
| Mean | Median | Range | Standard deviation | |
|---|---|---|---|---|
| Class X | 60 | 60 | 50 | 16.83 |
| Class Y | 60 | 60 | 10 | 3.11 |
How to write the conclusion
The means are equal at 60, and the medians are also 60, so the centres are the same. The standard deviation of X (16.83) is much larger than that of Y (3.11), so the marks in Class X are far more spread out.
Then add the meaning: in Class X, some students are far above and some far below 60, so one teaching pace may not suit them all. In Class Y, every student is within a few marks of 60.
What the same mean hides
If a school only reported the mean of 60 for each class, both would look the same. The spread shows that Class X has students who need extra help and students who need extension.
So always report the spread alongside the mean. A centre without a spread is half a description.
The slips to avoid
The first slip is to write “Class Y is better”. The data shows it is more consistent, not that the marks are higher. The second is to quote the range alone, because the range depends on two values only.
The third slip is to forget the context sentence. A good conclusion mentions the marks or the students, not just the statistic. Compare your sentences with the graph evidence comparison lab.
Check yourself
Two sets of five test marks: P = 40, 50, 60, 70, 80 and Q = 58, 59, 60, 61, 62. Show the means are equal and find both standard deviations.
Answer
Both means are 60 (300 ÷ 5). For P the deviations are −20, −10, 0, 10, 20, squared 400, 100, 0, 100, 400, total 1 000, so the variance is 200 and σ = 14.14.
For Q the deviations are −2, −1, 0, 1, 2, squared 4, 1, 0, 1, 4, total 10, so the variance is 2 and σ = 1.41. The marks in Q are far more consistent.
What to study next
Go to explaining how an outlier changes the mean and median differently. Test the cluster with the practice set.
For a teacher to check your context sentences, see online one-to-one Mathematics tuition.