Three separate datasets — exam marks, GPA records, a ten-point sample — run through the same instrument: standard deviation and the coefficient of variation, to see which group holds together and which one scatters.
Frequencies plotted against the decile bands. The 30–40 band alone accounts for nearly a third of the cohort — a visible pile-up around the middle.
A coefficient of variation of 39.24% — mathematically, dispersion this wide means the class is really two populations sharing one exam paper: a group still finding its footing, and a group already fluent in the material.
The raw scores: 2.40, 3.21, 3.26, 3.31, 3.62, 3.66, 3.87, 3.88, 3.94, 3.95. Grouped by half-point band and re-derived by the step-deviation method as a cross-check.
| Band | f | fd | fd² |
|---|---|---|---|
| 2.0–2.5 | 1 | −2 | 4 |
| 2.5–3.0 | 0 | 0 | 0 |
| 3.0–3.5 | 3 | 0 | 0 |
| 3.5–4.0 | 6 | 6 | 6 |
The five-number summary shows a longer lower whisker than upper — the sample's one weak score pulls the tail down, while the top end stays tightly packed. Fences at [2.33, 4.81] confirm it: no outliers, just a real skew.
A gap this size on the same scale isn't noise. The GPA group is roughly three times more uniform than the optional-math cohort — whatever is being taught to the latter, it is not landing evenly.
Reading the instruments
Optional math: intervene by group, not by average.
GPA sample: the mean of 3.45 is a fair description.
Five-number check: the skew is real, not an anomaly.