Statistics · Ch 3 — Measures of Central Tendency

Relationship Among AM, GM and HM; Choosing the Right Average

For ANY set of two or more positive observations that are not all equal, the three means always stand in the following fixed order:

AM \geq GM \geq HM

with equality holding in all three ( $AM = GM = HM$ ) only when every observation in the data set is identical. This relationship is a useful, quick arithmetic cross-check: after computing all three means for the same data, verifying that they fall in this order (or are exactly equal for uniform data) confirms none of the three has been mis-computed.

Special case (exactly two observations): for $n = 2$ positive values $x_1$ and $x_2$ :

GM^2 = AM \times HM

This identity holds ONLY for two observations — it is not a general relationship for $n > 2$ values, a distinction the Gujarat Std 11 Commerce Statistics syllabus expects a student to state precisely rather than over-generalise.

Choosing the appropriate average — a summary table:

Situation	Best average	Why
General-purpose summary, further statistical work (dispersion, correlation) needed	Arithmetic mean	Based on all values; algebraically tractable
Data has extreme values / income-type skewed distribution	Median	Unaffected by outliers
Need the single most "typical" or most commonly occurring value	Mode	Represents the most frequent case directly
Averaging ratios, rates of change, growth rates, index numbers	Geometric mean	Correctly averages multiplicative/compounding data

Definition 1AM ≥ GM ≥ HM

For any set of positive, non-identical observations, the arithmetic mean is always the largest of the three means, the harmonic mean the smallest, and the geometric mean always lies between them; all three are equal on …

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