Geometric Mean, Harmonic Mean, and Relationship with Arithmetic Mean

In addition to the commonly used Arithmetic Mean (AM), other types of averages like Geometric Mean (GM) and Harmonic Mean (HM) are particularly useful in specific statistical and business contexts. Understanding these means helps in analyzing growth rates, ratios, and rates of change more accurately.

Geometric Mean (GM)

Definition

Geometric Mean is the nth root of the product of n values. It is best used when dealing with multiplicative processes, such as growth rates, financial returns, and population studies.

Formula

For $n$ positive values $x_1, x_2, ..., x_n$:

$$GM=\sqrt[n]{x_1\times x_2\times \cdots \times x_n}$$

Or in logarithmic form:

$$\log(GM) = \frac{1}{n} \sum_{i=1}^{n} \log(x_i)$$
$$GM = \text{antilog} \left( \frac{1}{n} \sum \log(x_i) \right)$$

Example

Let the values be: 4, 16, and 64

$$GM=\sqrt[3]{4\times 16\times 64}=\sqrt[3]{4096}=16$$

Harmonic Mean (HM)

Definition

Harmonic Mean is the reciprocal of the arithmetic mean of the reciprocals of the values. It is especially useful for averaging rates like speed, price per unit, or cost per item.

Formula

For $n$ values $x_1, x_2, ..., x_n$:

$$HM=\frac{n}{\sum _{i=1}^n\frac{1}{x_{\mathrm{i}}}}$$

Example

Let the values be: 4, 5, and 6

$$HM=\frac{3}{\frac{1}{4}+\frac{1}{5}+\frac{1}{6}}=\frac{3}{0.25+0.20+0.1667}=\frac{3}{0.6167}\approx 4.87$$

Relationship Between AM, GM, and HM

For any two positive numbers $a$ and $b$:

• Arithmetic Mean (AM) = $\frac{a+b}{\mathrm{2}}$

• Geometric Mean (GM) = $\sqrt{ab}$

• Harmonic Mean (HM) = $\frac{2ab}{a + b}$

Inequality Relationship:

$$AM \geq GM \geq HM$$

Equality holds only when all values are equal.

Example (Illustrating the Relation)

Let $a = 4$, $b = 16$

• AM = $\frac{4+16}{2}=10$

• GM = $\sqrt{4 \times 16} = \sqrt{64} = 8$
  

• HM = $\frac{2 \times 4 \times 16}{4 + 16} = \frac{128}{20} = 6.4$
  

So,

$$AM(10)>GM(8)>HM(6.4)$$

When to Use AM, GM, and HM

Key Points 

• GM and HM are suitable only for positive values.

• GM is used for compounded growth, e.g., interest rates.

• HM is used for time-speed-distance and cost-volume problems.

• Always remember the inequality:

  $$AM\ge GM\ge HM$$

Conclusion

The Geometric Mean and Harmonic Mean offer valuable insights into specific statistical scenarios where the Arithmetic Mean may be misleading. For UGC NET Commerce, mastering the use cases, formulas, and the relationship among AM, GM, and HM is crucial. Understanding when and why to use each mean can improve data interpretation and analytical accuracy in business contexts.