Statistics: Measures of Central Tendency and Dispersion
What the exam tests
This domain tests mean, median and mode for raw and grouped data, quartiles, range, mean deviation, standard deviation, variance and coefficient of variation. The exam often asks which average is appropriate and how dispersion compares two series.
Core rules
- Mean (=\frac{\sum x}{n}) for raw data; for grouped data use class midpoints. The assumed-mean and step-deviation methods give the same result.
- Median of raw data requires ordered values. For grouped data, use the median class and the cumulative-frequency formula.
- Mode is the most frequent value; for grouped data, use the modal class formula.
- Standard deviation (\sigma=\sqrt{\frac{\sum(x-\bar x)^2}{n}}); variance is (\sigma^2).
- Coefficient of variation (=\frac{\sigma}{\bar x}\times100%); a larger CV means greater relative variability.
Worked example
For (2,4,6,8,10), the mean is (\frac{30}{5}=6). Deviations are (-4,-2,0,2,4), whose squares sum to (16+4+0+4+16=40). Thus variance (=\frac{40}{5}=8) and standard deviation (=\sqrt8\approx2.83). The coefficient of variation is (\frac{2.83}{6}\times100\approx47.1%).
Traps
Order the data before finding the median. Use class boundaries, not just class marks, in grouped-data formulas. The mode is not the modal-class mark. Standard deviation uses squared deviations, so it cannot be negative. Compare series by CV only when their units or means differ substantially; otherwise compare SD directly.