3. Relationship between AM, GM, and HM

This section explores the fundamental measures of central tendency: the Arithmetic Mean (AM), Geometric Mean (GM), and Harmonic Mean (HM). We will also delve into the significant mathematical relationship that connects these three important statistical concepts.

3.1 Arithmetic Mean (AM)

The Arithmetic Mean (AM), often simply called the mean or average, is calculated by summing all the values in a dataset and then dividing by the number of values. It is sensitive to outliers.

Formula:

For a dataset $x_1, x_2, \ldots, x_n$, the Arithmetic Mean (AM) is given by:

$$ AM = \frac{\sum_{i=1}^{n} x_i}{n} $$

Example:

Consider the dataset: {2, 4, 6, 8, 10} $$ AM = \frac{2 + 4 + 6 + 8 + 10}{5} = \frac{30}{5} = 6 $$

3.2 Geometric Mean (GM)

The Geometric Mean (GM) is calculated by multiplying all the values in a dataset and then taking the $n$-th root of the product, where $n$ is the number of values. It is particularly useful for data that grows exponentially or is expressed as ratios or percentages. It is only defined for non-negative numbers.

Formula:

For a dataset $x_1, x_2, \ldots, x_n$, the Geometric Mean (GM) is given by:

$$ GM = \sqrt[n]{x_1 \times x_2 \times \ldots \times x_n} = \left(\prod_{i=1}^{n} x_i\right)^{\frac{1}{n}} $$

Example:

Consider the dataset: {2, 4, 8, 16} $$ GM = \sqrt[4]{2 \times 4 \times 8 \times 16} = \sqrt[4]{1024} = \sqrt[4]{2^{10}} = 2^{\frac{10}{4}} = 2^{2.5} \approx 5.657 $$

3.3 Harmonic Mean (HM)

The Harmonic Mean (HM) is the reciprocal of the arithmetic mean of the reciprocals of the dataset. It is most useful in situations involving rates, ratios, or averages of speeds. It is only defined for non-zero numbers.

Formula:

For a dataset $x_1, x_2, \ldots, x_n$, the Harmonic Mean (HM) is given by:

$$ HM = \frac{n}{\sum_{i=1}^{n} \frac{1}{x_i}} $$

Example:

Consider the dataset: {2, 4, 8} $$ HM = \frac{3}{\frac{1}{2} + \frac{1}{4} + \frac{1}{8}} = \frac{3}{\frac{4+2+1}{8}} = \frac{3}{\frac{7}{8}} = 3 \times \frac{8}{7} = \frac{24}{7} \approx 3.429 $$

3.4 Relationship between AM, GM, and HM

There is a fundamental and consistent relationship between the Arithmetic Mean (AM), Geometric Mean (GM), and Harmonic Mean (HM) for any set of positive numbers. For any dataset of positive numbers, the following inequality holds:

$$ AM \ge GM \ge HM $$

This inequality is known as the AM-GM-HM inequality.

Key Properties of the Relationship:

• Equality Condition: The equality ($AM = GM = HM$) holds if and only if all the numbers in the dataset are equal.
• Application: This relationship is crucial in various mathematical proofs, optimization problems, and understanding the behavior of different types of averages.

Proof Outline (for two positive numbers $a$ and $b$):

Let $a$ and $b$ be two positive numbers.

1. AM vs. GM: Consider $( \sqrt{a} - \sqrt{b} )^2 \ge 0$. Expanding this, we get $a - 2\sqrt{ab} + b \ge 0$. Rearranging, $a + b \ge 2\sqrt{ab}$. Dividing by 2, $\frac{a+b}{2} \ge \sqrt{ab}$. This shows $AM \ge GM$.

2. GM vs. HM: Consider the dataset $\frac{1}{a}, \frac{1}{b}$. Using the AM-GM inequality for these reciprocals: $$ \frac{\frac{1}{a} + \frac{1}{b}}{2} \ge \sqrt{\frac{1}{a} \times \frac{1}{b}} $$ $$ \frac{\frac{b+a}{ab}}{2} \ge \frac{1}{\sqrt{ab}} $$ $$ \frac{a+b}{2ab} \ge \frac{1}{\sqrt{ab}} $$ Taking the reciprocal of both sides (and reversing the inequality sign since both sides are positive): $$ \frac{2ab}{a+b} \le \sqrt{ab} $$ The left side is the Harmonic Mean (HM) of $a$ and $b$, and the right side is the Geometric Mean (GM). Thus, $HM \le GM$, or $GM \ge HM$.

Combining these two inequalities, we get $AM \ge GM \ge HM$.

Example Illustrating the Inequality:

Consider the dataset: {2, 4, 8}

• $AM = \frac{2+4+8}{3} = \frac{14}{3} \approx 4.667$
• $GM = \sqrt[3]{2 \times 4 \times 8} = \sqrt[3]{64} = 4$
• $HM = \frac{3}{\frac{1}{2} + \frac{1}{4} + \frac{1}{8}} = \frac{3}{\frac{4+2+1}{8}} = \frac{3}{\frac{7}{8}} = \frac{24}{7} \approx 3.429$

As expected, $4.667 \ge 4 \ge 3.429$.

Example of Equality:

Consider the dataset: {5, 5, 5}

• $AM = \frac{5+5+5}{3} = 5$
• $GM = \sqrt[3]{5 \times 5 \times 5} = 5$
• $HM = \frac{3}{\frac{1}{5} + \frac{1}{5} + \frac{1}{5}} = \frac{3}{\frac{3}{5}} = 5$

Here, $AM = GM = HM = 5$.