Statistics Beyond the Average: Range, CV, Harmonic and Weighted Means
Statistical range, coefficient of variation, harmonic mean, and weighted average — four ways to describe a data set beyond a simple average.
The arithmetic mean is only one way to summarize a data set — these four calculations capture spread, relative variability, rate-based averaging, and unequal contribution.
Range: the simplest measure of spread
Range is simply maximum minus minimum. A data set from 4 to 42 has a range of 38 — the crudest possible measure of spread, since it ignores everything about the distribution between those two extremes.
Coefficient of variation: comparing variability across different scales
CV expresses standard deviation as a percentage of the mean: (std dev / mean) × 100. A standard deviation of 5 with a mean of 50 gives a CV of 10% — this normalization is what lets you compare variability between data sets with completely different units or scales, which raw standard deviation can't do.
Harmonic mean: the right average for rates
For averaging rates like speed over equal distances, harmonic mean gives a more accurate result than the arithmetic mean. For values 2, 4, and 8, the harmonic mean is 3 / (1/2 + 1/4 + 1/8) ≈ 3.43 — noticeably lower than the arithmetic mean of 4.67, since harmonic mean weights smaller values more heavily.
Weighted average: when contributions aren't equal
A weighted average multiplies each value by its weight before averaging. Scores of 90, 80, and 70 weighted 0.5, 0.3, and 0.2 respectively give a weighted average of 83 — different from the simple average of 80, reflecting that the first score counted for more.
Choosing the right summary statistic
Range, CV, harmonic mean, and weighted average each answer a specific question a plain average can't. Try the range calculator, coefficient of variation calculator, harmonic mean calculator, and weighted average calculator.