Measures of Central Tendency & Dispersion
WBJEE Mathematics – Key Topic
Central tendency tells us the “typical” value in a set of data.
Dispersion tells us how spread out the data are around that typical value.
1. Simple Explanation
| Measure | What it represents | Typical usage |
|---|---|---|
| Mean | Average of all values | General “centre” when data are not heavily skewed |
| Median | Middle value when data are ordered | Robust against out‑liers |
| Mode | Most frequently occurring value | Useful for categorical data |
| Range | Difference between largest & smallest | Quick sense of spread |
| Variance | Average of squared deviations from mean | Quantifies dispersion (square units) |
| Standard Deviation (σ) | Square root of variance | Same units as data, easier interpretation |
| Coefficient of Variation (CV) | σ / mean × 100 % | Relative spread, useful when means differ |
2. Key Points
| Item | Formula | Notes |
|---|---|---|
| Mean | (\displaystyle \bar{x} = \frac{1}{n}\sum_{i=1}^{n}x_i) | For population: divide by (N); for sample: divide by (n) |
| Median | If (n) odd: (x_{(n+1)/2}); if even: (\frac{x_{n/2}+x_{n/2+1}}{2}) | (x_{(k)}) denotes kth ordered value |
| Mode | Value with max frequency | Can be none, one, or multiple |
| Range | (R = x_{\max} - x_{\min}) | Sensitive to extreme values |
| Variance | (\displaystyle s^2 = \frac{1}{n-1}\sum_{i=1}^{n}(x_i-\bar{x})^2) (sample) (\displaystyle \sigma^2 = \frac{1}{N}\sum_{i=1}^{N}(x_i-\mu)^2) (population) |
(n-1) (Bessel's correction) for unbiased estimator |
| Standard Deviation | (s = \sqrt{s^2}) (sample) (\sigma = \sqrt{\sigma^2}) (population) |
Often denoted (s) or (\sigma) |
| Coefficient of Variation | (\displaystyle CV = \frac{s}{\bar{x}}\times 100%) | Useful for comparing dispersion of different units |
Quick Tips
- Mean vs Median – use median when data have out‑liers or are skewed.
- Variance vs Standard Deviation – variance is in squared units; SD keeps the same units as the data, making it easier to interpret.
- CV – a dimensionless number; compare CVs to judge relative variability.
3. Worked Example
Dataset (in cm): 12.0, 13.5, 14.0, 15.0, 16.5, 20.0, 21.5
3.1 Central Tendency
| Measure | Calculation | Result |
|---|---|---|
| Mean | ((12.0+13.5+14.0+15.0+16.5+20.0+21.5)/7) | ( \displaystyle \bar{x}=15.93) cm |
| Median | Ordered list already sorted; middle value (4th) | (15.0) cm |
| Mode | No value repeats | None |
3.2 Dispersion
| Measure | Calculation | Result |
|---|---|---|
| Range | (21.5-12.0 = 9.5) cm | 9.5 cm |
| Variance (sample) | (\displaystyle s^2=\frac{1}{n-1}\sum (x_i-\bar{x})^2) | Compute each squared deviation: ((12-15.93)^2=15.42) ((13.5-15.93)^2=6.06) ((14-15.93)^2=3.71) ((15-15.93)^2=0.84) ((16.5-15.93)^2=0.33) ((20-15.93)^2=16.50) ((21.5-15.93)^2=31.09) Sum = 73.95 (s^2 = 73.95 / 6 = 12.33) |
| Standard Deviation | (s = \sqrt{12.33} = 3.51) cm | 3.51 cm |
| Coefficient of Variation | ( CV = \frac{3.51}{15.93}\times100 = 22.0%) | 22 % |
Interpretation
- The average height is about 15.93 cm, but the typical (median) height is 15 cm.
- The spread is moderate: a range of 9.5 cm, SD of 3.5 cm.
- A CV of 22 % indicates reasonable relative variability.
4. Quick Revision Summary
| Concept | Formula | Quick Check |
|---|---|---|
| Mean | (\bar{x} = \frac{1}{n}\sum x_i) | Add all, divide by count |
| Median | Middle ordered value (or avg of two middle) | Sort, pick centre |
| Mode | Highest frequency | Look for repeats |
| Range | (x_{\max} - x_{\min}) | Largest minus smallest |
| Variance (sample) | (s^2 = \frac{1}{n-1}\sum (x_i-\bar{x})^2) | Sum squared deviations, divide by (n-1) |
| Standard Deviation | (s = \sqrt{s^2}) | Square root of variance |
| CV | (\frac{s}{\bar{x}} \times 100 |