Measures of Central Tendency and Dispersion
This domain tests whether you can select and compute an appropriate average and pair it with a matching measure of spread. Kerala papers frequently combine a grouped frequency table with a request for mean, median, mode, standard deviation and coefficient of variation in a single question.
Averages
- Grouped mean: x̄ = Σfx/Σf, with x the class marks. Step-deviation form: x̄ = A + h(Σfu/Σf), which reduces arithmetic.
- Median = L + [(N/2 − cf)/f] × h, where L is the lower boundary of the median class and cf the cumulative frequency of the class before it.
- Mode = L + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h, the modal class having the highest frequency.
- Combined mean = (n₁x̄₁ + n₂x̄₂)/(n₁ + n₂).
Dispersion
Range = maximum − minimum. Quartile deviation = (Q₃ − Q₁)/2. Mean deviation = Σ|x − x̄|/n. Standard deviation σ = √[Σf(x − x̄)²/N] = √[Σfx²/N − x̄²]. The coefficient of variation, CV = 100σ/x̄, is unit-free and is the correct tool for comparing consistency of two data sets.
Worked example
Classes 0–10, 10–20, 20–30, 30–40, 40–50 with frequencies 4, 6, 10, 5, 5 give N = 30. N/2 = 15; cf before the median class is 10, f = 10, L = 20, h = 10, so median = 20 + (5/10) × 10 = 25. The modal class is 20–30, giving mode = 20 + (10 − 6)/(20 − 6 − 5) × 10 = 24.44.
Traps that cost marks
Using N instead of N/2 in the median formula; taking the cf of the median class itself; comparing standard deviations across different units instead of CV; forgetting that adding a constant leaves σ unchanged while multiplying by a scales σ by |a|.