How steady is the performance, really?
61 optional-maths marks and 10 GPA records, measured with standard deviation, the coefficient of variation and a five-number summary, to see which group holds together and which one scatters.
Where the marks actually landed
The 30–40 band holds 18 of 61 scripts (30%), the tallest pile-up. Above it the counts thin out but never vanish, so the tail on the right is long.
σ = h·√(Σfd²/N − (Σfd/N)²). Hover a bar for its count.
A coefficient of variation of 39.20% means the standard deviation is nearly 40% of the mean. That is a wide spread: an average of 44 describes the class poorly, so look at the individual bands before drawing conclusions.
A tighter cluster
Raw scores: 2.40, 3.21, 3.26, 3.31, 3.62, 3.66, 3.87, 3.88, 3.94, 3.95. Grouped into half-point bands and re-derived by the step-deviation method.
| Band | f | d | fd | fd² |
|---|---|---|---|---|
| 2.0–2.5 | 1 | −2 | −2 | 4 |
| 2.5–3.0 | 0 | −1 | 0 | 0 |
| 3.0–3.5 | 3 | 0 | 0 | 0 |
| 3.5–4.0 | 6 | 1 | 6 | 6 |
| Total | 10 | 4 | 10 |
Grouping is an approximation: the ten raw scores average 3.51, slightly above the grouped 3.45.
Shape, not just spread
Nine of the ten scores sit between 3.21 and 3.95. One score, 2.40, stretches the lower whisker to 0.86 below Q₁, while the upper whisker is only 0.07 above Q₃. The fences (dashed) show 2.40 is just inside the limit, so it is a real skew, not an outlier.
Same instrument, two very different groups
The optional-maths C.V. is about three times the GPA C.V. (39.20 vs 13.28). C.V. is unit-free, so the two can share a scale.
Two caveats: the groups differ in size (61 vs 10), and GPA is capped at 4.0, which limits how far it can spread.
Reading the instruments
Low C.V. means trust the average. High C.V. means look at the individuals.
Optional maths: check the bands, not just the mean of 43.69.
GPA: the mean is a fair guide, though one low score drags it below the median of 3.64.
Box-plot fences: the skew comes from a real score, not an error.