Measures of Central Tendency
| Measure | How to find it | Example: 3, 5, 5, 7, 10 |
|---|---|---|
| Mean | Sum ÷ number of values | 30 ÷ 5 = 6 |
| Median | Middle value after sorting; for an even count, the average of the two middle values | 5 |
| Mode | Most frequent value | 5 |
| Range | Largest − smallest | 10 − 3 = 7 |
- Mean of the first n natural numbers = (n + 1)/2. The mean of 1 to 99 is 50.
- Adding k to every value adds k to the mean, median and mode but leaves the range unchanged. Multiplying every value by k multiplies all four by k.
- Combined (weighted) mean = (n₁x̄₁ + n₂x̄₂)/(n₁ + n₂). 20 staff averaging 30 years and 30 averaging 40 years: (600 + 1200)/50 = 36 years.
- A wrongly copied value: correct sum = old sum − wrong value + right value.
- Empirical relation for moderately skewed data: Mode = 3 Median − 2 Mean.
Mean of a frequency table
Mean = Σfx ÷ Σf
For class intervals, use the class mid-point as x. Classes 0-10, 10-20, 20-30 with frequencies 2, 5, 3: Σfx = 2×5 + 5×15 + 3×25 = 160 and Σf = 10, so the mean is 16.
Basic Probability
- P(event) = favourable outcomes ÷ total equally likely outcomes. It lies between 0 and 1, and P(not E) = 1 − P(E).
- One die: 6 outcomes; P(even) = 3/6 = 1/2. Two dice: 36 outcomes; P(sum 7) = 6/36 = 1/6.
- A pack of 52 cards has 4 suits of 13, 26 red and 26 black cards, 4 aces and 12 face cards (king, queen, jack).
Data Interpretation
- Read the units and the scale first (thousands, lakh, crore, per cent).
- Percentage change = (new − old)/old × 100. A rise from 40 to 50 is 25%, but a fall from 50 to 40 is 20%.
- Pie chart: a share of p% is drawn as p × 3.6 degrees. A 25% share is 90°.
- To compare ratios across years, simplify each to a decimal rather than comparing raw figures.
Common Mistakes
Finding the median without sorting; averaging the averages of groups of different sizes; and taking percentage change on the new value instead of the old one.