Statistics: Mean, Median and Mode of Grouped Data
Statistics asks for the mean, median or mode of grouped data and for reading an ogive, typically for 3–4 marks.
The three measures
- Mean: direct Σfx/Σf; assumed mean a + Σfd/Σf with d = x − a; step deviation a + h(Σfu/Σf) with u = (x − a)/h.
- Median: l + [(n/2 − cf)/f] × h, where l is the lower limit of the median class (the class whose cumulative frequency first reaches n/2), cf the cumulative frequency before it, f its frequency and h the class width.
- Mode: l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h, with f₁ the frequency of the modal class and f₀, f₂ the frequencies of the classes before and after.
- Empirical relation: 3 median = mode + 2 mean.
A "less than" ogive plots upper class limits against cumulative frequency; the median is the x-value at cumulative frequency n/2. The intersection of the "less than" and "more than" ogives also gives the median.
Worked example
For the class 20–30 with f₁ = 12, previous frequency f₀ = 8, next frequency f₂ = 6 and h = 10, with l = 20: Mode = 20 + [(12 − 8)/(24 − 8 − 6)] × 10 = 20 + (4/10) × 10 = 24.
Traps that cost marks
- Use the cumulative frequency before the median class, not one that includes it.
- The modal class has the highest frequency, not the highest cumulative frequency.
- Make class intervals continuous before applying any formula; a gap changes l and h.
- Pick the assumed mean from the middle of the data to keep d small, and always show the working table.