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Geometric mean and harmonic mean

Business Statistics · Unit II · Topic 6 of 15 · 4 min read

Formulas, uses, merits and demerits of the GM and HM, the relation AM ≥ GM ≥ HM, and the January 2026 and June 2025 problems solved with log tables.

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On this page
  1. 1. Geometric mean
  2. 2. Harmonic mean
  3. 3. Worked: harmonic mean, discrete (January 2026)
  4. 4. Worked: geometric mean, grouped (January 2026)
  5. 5. Worked: both GM and HM (June 2025)
  6. How this is examined
Cases checked to exist. Read the judgment before quoting a holding.

Every case cited here has been checked to exist — the citation resolves and names the right parties. The propositions attributed to each case have not been verified line by line against the judgment. Case names link through to the full judgment: read it before quoting a holding in an examination answer, and never carry a proposition from notes into court on trust.

The examinable core. GM uses logarithms: GM = antilog (Σf log x / N). HM uses reciprocals: HM = N / Σ(f/x). Both questions are pure table work. Set out four columns (x, f, log x, f log x or 1/x, f/x), carry four decimal places, and check at the end that AM > GM > HM.

1. Geometric mean

The geometric mean of n positive values is the nth root of their product: GM = (x1 × x2 × … × xn)1/n. Taking logarithms:

Uses: averaging rates of change, ratios and percentages (growth of population, rates of interest, depreciation); index numbers (Fisher’s index is the GM of Laspeyres and Paasche). Example: sales rise by 10% in year 1 and 40% in year 2. The average growth factor is √(1.10 × 1.40) = √1.54 = 1.241, i.e. 24.1% a year, not the arithmetic 25%.

Merits: rigidly defined; based on all items; gives less weight to large items than the AM; suitable for rates and ratios; capable of algebraic treatment. Demerits: hard to compute without logs; cannot be found if any value is zero or negative; may be a value not in the series.

2. Harmonic mean

The harmonic mean is the reciprocal of the arithmetic mean of the reciprocals:

Uses: averaging speeds, rates and prices where the numerator is fixed (distance fixed, time varies; money fixed, quantity varies). Example: a car goes 60 km at 60 km/h and returns at 40 km/h. Average speed = 2 / (1/60 + 1/40) = 2 / (5/120) = 48 km/h, not 50. (Check: 1 hour + 1.5 hours = 2.5 hours for 120 km = 48 km/h.)

Merits: rigidly defined; based on all items; gives the right answer for rates; gives most weight to small items. Demerits: not easily understood; cannot be found if any value is zero; gives too much weight to small values for most other purposes.

Relation: for any set of positive, unequal values, AM > GM > HM (equal only when all values are equal). For two values, GM = √(AM × HM).

3. Worked: harmonic mean, discrete (January 2026)

xf1/xf/x
12450.0080650.040323
129180.0077520.139535
134200.0074630.149254
13970.0071940.050360
14430.0069440.020833
Total530.400305

HM = 53 / 0.400305 = 132.40. (For comparison, AM = 7,027/53 = 132.58; the HM is a little lower, as it must be.)

4. Worked: geometric mean, grouped (January 2026)

Classfmlog mf log m
0–10550.69903.4950
10–207151.17618.2327
20–3015251.397920.9685
30–4025351.544138.6025
40–508451.653213.2256
Total6084.5243

Σf log m / N = 84.5243 / 60 = 1.4087. GM = antilog 1.4087 = 25.63. (AM = 29.0, so GM < AM, as expected.)

5. Worked: both GM and HM (June 2025)

Classfmlog mf log mf/m
100–20041502.17618.70440.026667
200–30062502.397914.38740.024000
300–400103502.544125.44100.028571
400–500124502.653231.83840.026667
500–600115502.740430.14440.020000
600–700136502.812936.56770.020000
Total56147.08330.145905

GM = antilog (147.0833 / 56) = antilog 2.6265 = 423.1.

HM = 56 / 0.145905 = 383.8.

Check: AM = 25,500 / 56 = 455.4 > GM 423.1 > HM 383.8.

August 2024 set the same pair on marks 0–10 to 50–60 (f = 7, 12, 4, 25, 30, 22). Follow the same layout with mid-points 5, 15 … 55.

Trap. When reading the antilog of 2.6265, the characteristic 2 means the answer has three digits before the decimal point (423.1, not 42.31 or 4,231). Write the characteristic and mantissa separately to avoid a slip.

How this is examined

HM from a discrete table and GM from a grouped table: January 2026 (six marks each, as alternatives). GM and HM from one table: June 2025 and August 2024 (ten marks). A theory answer on “measures of central tendency” (January 2025, August 2024) should include a paragraph each on GM and HM, their uses and the relation AM ≥ GM ≥ HM.

Now try the real questions: Business Statistics past question papers.