Measures of Central Tendency and Dispersion
Measures of central tendency and dispersion summarise a business data set by a typical value and by how spread out it is. The exam tests mean, median and mode for raw and grouped data, quartiles, range, mean deviation, variance, standard deviation and the coefficient of variation.
Core rules
- Mean = Σx/n for raw data; for grouped data use Σfx/Σf with class midpoints.
- Median is the middle value after ordering; for grouped data use the median class and interpolation. Mode is the most frequent value; for grouped data it lies in the modal class.
- Range = largest − smallest. Quartile deviation = (Q3 − Q1)/2.
- Population variance σ² = Σ(x − μ)²/N; sample variance uses n − 1. Standard deviation is the positive square root.
- Coefficient of variation CV = (SD/mean) × 100%, a unit-free measure useful for comparing consistency between series with different units or means.
Worked example
For the data 2, 4, 6, 8, 10, the mean is 30/5 = 6. The median is 6 and there is no mode. Taking these as a population, deviations from 6 are −4, −2, 0, 2, 4; squared deviations are 16, 4, 0, 4, 16, with sum 40. Population variance = 40/5 = 8, so SD = √8 = 2.828. The CV is (2.828/6) × 100 = 47.14%. If the same figures were a sample, the variance would be 40/4 = 10 and the SD would be 3.162 - state which formula the question expects.
Traps
- Using class limits instead of midpoints for grouped means.
- Forgetting to order the data before finding the median.
- Dividing by n instead of n − 1 for a sample variance.
- Reporting CV without a percentage, or comparing SDs across series measured in different units.