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Statistics · Ch 3 — Measures of Central Tendency

Geometric Mean — Computation, Merits and Limitations

5

Geometric Mean — Computation, Merits and Limitations

Geometric mean (GM) is the nthn^{th} root of the product of nn positive observations. It is the correct average to use whenever data involves rates of change, ratios, growth rates or index numbers — situations where values multiply together rather than simply add.

1. Individual series:

GM=x1×x2×⋯×xnnGM = \sqrt[n]{x_1 \times x_2 \times \cdots \times x_n}

For more than two or three values, this is computed conveniently using logarithms:

log⁡GM=Σlog⁡xn⇒GM=Antilog(Σlog⁡xn)\log GM = \frac{\Sigma \log x}{n} \qquad \Rightarrow \qquad GM = \text{Antilog}\left(\frac{\Sigma \log x}{n}\right)

2. Discrete series (using logarithms):

log⁡GM=Σflog⁡xN⇒GM=Antilog(Σflog⁡xN)\log GM = \frac{\Sigma f \log x}{N} \qquad \Rightarrow \qquad GM = \text{Antilog}\left(\frac{\Sigma f \log x}{N}\right)

3. Continuous series — use the class mid-point mm in place of xx:

log⁡GM=Σflog⁡mN⇒GM=Antilog(Σflog⁡mN)\log GM = \frac{\Sigma f \log m}{N} \qquad \Rightarrow \qquad GM = \text{Antilog}\left(\frac{\Sigma f \log m}{N}\right)

A common business use: the average rate of growth over several years must always be found by geometric mean, never by arithmetic mean — e.g. if a firm's sales grow by different percentages in different years, only the GM of the year-on-year growth FACTORS gives the true average annual growth rate that, compounded over the same number of years, reproduces the actual overall growth.

Merits of geometric mean:

  • Based on all observations.
  • Rigidly defined, capable of further algebraic treatment.
  • The most suitable average for ratios, rates of change, and growth rates — gives less weight to large extreme values than the arithmetic mean does.
  • Useful in the construction of index numbers (covered in Std 12 Statistics).

Limitations of geometric mean: …

Definition 1Geometric mean

The nth root of the product of n positive observations; computed in practice via logarithms as the antilog of the average of the logarit …

Definition 2Growth factor

A value of the form (1 + rate), used so that the geometric mean of successive years' growth factors gives the correct compound …

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