Statistics: Mean, Median and Mode of Grouped Data
Statistics questions ask you to summarise grouped frequency data with a single representative value and to read a cumulative frequency curve. Marks depend on a correctly built frequency table, so copy the classes and frequencies accurately before calculating.
Mean
Direct method: x̄ = Σfᵢxᵢ/Σfᵢ, using class marks xᵢ. Assumed mean method: choose a convenient A, set dᵢ = xᵢ − A, then x̄ = A + Σfᵢdᵢ/Σfᵢ. Step deviation divides dᵢ by the class width h and multiplies the result back by h.
Median
Add a cumulative frequency column. The median class is the one whose cumulative frequency first reaches or exceeds N/2. Then median = L + [(N/2 − cf)/f] × h, where L is the lower boundary of the median class, cf the cumulative frequency of the class before it, f the frequency of the median class and h the class width.
Mode
The modal class has the highest frequency. Mode = L + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h, with f₁ the modal frequency, f₀ the preceding frequency and f₂ the following frequency.
Ogive
Plot upper class boundaries against cumulative frequency and join the points with a smooth curve; the x-value at cumulative frequency N/2 estimates the median.
Worked example
With N = 40, N/2 = 20. The class 20–30 has cumulative frequency before it 14 and frequency 12, with h = 10, so L = 20. Median = 20 + ((20 − 14)/12) × 10 = 20 + (6/12) × 10 = 25.
Traps that cost marks
Using N instead of N/2 in the median formula, inserting the class mark where L belongs, forgetting to multiply by h in step deviation, swapping f₀ and f₂ in the mode formula, and cumulating frequencies down a table that is not in ascending class order are the errors that quietly destroy the answer.