Statistics: Mean, Median and Mode of Grouped Data
What the exam tests
Statistics questions test the mean, median and mode of grouped frequency data, cumulative frequency, and interpretation of an ogive. You must use class midpoints and class boundaries correctly.
Core rules
- Mean from a frequency table: x̄=Σfx/Σf, where x is each class midpoint.
- Cumulative frequency is the running total of frequencies.
- Median class is the class where the cumulative frequency first reaches n/2.
- Median = l + [(n/2−cf)/f]×h, where l is the lower boundary of the median class, cf is the cumulative frequency before it, f is its frequency and h is the class width.
- Mode class has the highest frequency; mode = l + [(f₁−f₀)/(2f₁−f₀−f₂)]×h.
- On an ogive, the median is the x-value at cumulative frequency n/2.
Worked example
Classes 0–10, 10–20, 20–30, 30–40, 40–50 have frequencies 4, 6, 10, 8, 2. Total n=30, so n/2=15. Cumulative frequencies are 4, 10, 20, 28, 30; the median class is 20–30. With l=20, cf=10, f=10 and h=10, median=20+[(15−10)/10]×10=25.
Traps
Use class boundaries, not class limits, when classes are continuous. The cumulative frequency before the median class excludes the median class itself. The median is not simply the middle class mark. In the mode formula, f₁ is the highest frequency, f₀ the preceding frequency and f₂ the following frequency. Keep h consistent and do not mix up n with n/2.