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Class 11 Measures of Central Tendency: cheat sheet
CBSE Social Science. One value to stand for a whole set of data: the arithmetic mean (direct, assumed mean, step deviation, weighted), the median, quartiles, percentiles and the mode.
Key ideas
- The mean uses every value but is pulled by extreme values; the median is the middle value and resists them; the mode is the most frequent value.
- Use the mean for average marks, the median for a typical income where a few people are very rich, and the mode for the most popular size.
- The assumed mean and step deviation methods give the same mean with smaller numbers.
- A weighted mean multiplies each value by its importance before averaging.
- The sum of deviations from the mean is always zero.
- Quartiles split ordered data into four equal parts and percentiles into a hundred; Q2 and the 50th percentile are the median.
- Data may have one mode, two (bimodal), or none.
Remember
| Mean | ΣX ÷ N; for a series Σfx ÷ Σf (class marks for classes) |
| Assumed mean | A + Σfd ÷ Σf, where d = X − A |
| Step deviation | A + (Σfd′ ÷ Σf) × c, where d′ = (X − A) ÷ c |
| Weighted mean | ΣWX ÷ ΣW |
| Missing value | N × mean − sum of the others |
| Median position | (N + 1) ÷ 2 th item; Q1 at (N + 1) ÷ 4, Q3 at 3(N + 1) ÷ 4 |
| Median (classes) | L + ((N ÷ 2 − c.f.) ÷ f) × h |
| Mode (classes) | L + D₁ ÷ (D₁ + D₂) × h; D₁, D₂ = modal frequency minus the classes before and after |
Watch out
- Finding the median without putting the data in order first.
- For an even number of items, taking one middle value instead of the mean of the two.
- In the median formula, using the cumulative frequency of the median class itself instead of the class before it.
- Using the mean for income data with one huge value.
Worked example
Find the mode of this distribution. Class: 40–60, 60–80, 80–100, 100–120, 120–140 Frequency: 19, 27, 22, 18, 24
Working: modal class 60–80: L = 60, D1 = 8, D2 = 5, h = 20; L + D1 ÷ (D1 + D2) × h
Answer: 72.31
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