Statistics: Central Tendency and Dispersion
Domain 3, Statistics, tests summarising data and measuring spread. Expect computation of mean, median, mode, quartiles, range, variance, standard deviation and coefficient of variation, plus interpretation of what each measure says.
Core rules
- For ungrouped data, mean = Σx/n. For grouped data use class midpoints: mean = Σfx/Σf.
- Median is the middle ordered value (average of the two middle values for even n). For grouped data use the cumulative-frequency formula with the median class.
- Mode is the most frequent value; a data set may have no mode or several modes.
- Range = maximum − minimum. Population variance = Σ(x−μ)²/n; standard deviation is its positive square root. Coefficient of variation = (SD/mean)×100%, used to compare relative variability.
Worked example
For 4, 6, 8, 10, 12: mean = 40/5 = 8. Deviations are −4, −2, 0, 2, 4; squared deviations sum to 16+4+0+4+16 = 40. Treating these five values as a population, variance = 40/5 = 8 and SD = √8 ≈ 2.828. CV = (2.828/8)×100% ≈ 35.36%. The median is 8.
Traps
Sort data before finding the median. Do not use n−1 in the population variance unless the question specifies a sample. Variance is in squared units; SD returns to original units. In grouped data, use the class midpoint, not the upper limit. A zero SD means all observations are equal, not that the mean is zero.