Statistics
This domain tests the mean, median and mode of grouped data, cumulative frequency, and reading ogives.
Core rules
- Direct mean: x̄ = Σfᵢxᵢ / Σfᵢ, where xᵢ is the class mark.
- Assumed-mean method: x̄ = A + (Σfᵢdᵢ / Σfᵢ), with dᵢ = xᵢ − A.
- Step-deviation method: x̄ = A + h(Σfᵢuᵢ / Σfᵢ), with uᵢ = (xᵢ − A)/h.
- Median = l + [(n/2 − cf)/f] × h, where l is the lower boundary of the median class, cf the cumulative frequency before it, f its frequency and h the class width.
- Mode = l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h.
- The median is the x-coordinate of the intersection of the two ogives.
Worked example
For a distribution with n = 40, median class 20–30, cumulative frequency before the class = 12, frequency of the class = 10 and h = 10: Median = 20 + [(40/2 − 12)/10] × 10 = 20 + (8/10) × 10 = 28. The lower boundary is used, not the upper boundary, because the median lies inside the class interval.
Traps
- Using the upper boundary instead of the lower boundary of the median class.
- Treating class marks as data values.
- Forgetting to make the classes continuous before using the formulas.
- Reading the ogive intersection as the mean rather than the median.