Statistics and Coordinate Geometry: grouped data and division of segments
Statistics and Coordinate Geometry (WBBSE Madhyamik)
Statistics covers mean, median, mode and the ogive for grouped frequency data; coordinate geometry covers the distance formula and internal or external division of a segment. Both reward careful tabulation more than cleverness.
Core rules
- Mean: x̄ = Σfx/Σf. With assumed mean a, x̄ = a + Σfd/Σf; with step deviation, x̄ = a + h(Σfu/Σf).
- Median of grouped data: find the class where cumulative frequency reaches N/2, then median = l + ((N/2 − cf)/f) × h, where l is the lower class boundary, cf the cumulative frequency before the median class, f its frequency and h the class width.
- Ogive: plot upper class boundaries against cumulative frequency for a 'less than' curve; the x-value at N/2 is the median, and the two ogives intersect at the median.
- Distance between (x₁, y₁) and (x₂, y₂) is √((x₂ − x₁)² + (y₂ − y₁)²).
- Internal division in the ratio m : n gives ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)); for external division replace + by −. The midpoint is the 1 : 1 case.
Worked example
The point dividing A(1, 2) and B(4, 5) internally in the ratio 2 : 1 is ((2×4 + 1×1)/3, (2×5 + 1×2)/3) = (3, 4). Check with the distance formula: AP = √(4 + 4) = 2√2 and PB = √(1 + 1) = √2, so AP : PB = 2 : 1, as required.
Traps
- Using frequency instead of cumulative frequency when locating the median class.
- Taking class marks as class limits; l and h must come from class boundaries.
- Swapping m and n in the section formula, which returns the complementary point.
- Forgetting the square root in the distance formula, or subtracting coordinates in inconsistent order.