Arithmetic
Means
Pythagoras's Three Strings: How Mathematics Became Music
Legend has it: **Pythagoras** was walking past a blacksmith's shop and heard hammers ringing in harmony. He weighed the hammers and discovered - their weights were in the ratio 6:4:3. These are the arithmetic, geometric, and harmonic means! Pythagoras understood: **music is mathematics**.
All is number. - Pythagoras
The inequality **AM ≥ GM** is one of the most important in mathematics. From it follow the Cauchy, Jensen, and isoperimetric inequalities. A simple observation - (a−b)² ≥ 0 - spawns an entire branch of analysis. Pythagoras found it by listening to strings. Mathematics in its purest form.
You drive 60 km at 30 km/h, then return 60 km at 60 km/h. What's the average speed? Your intuition says 45 km/h. But the correct answer is 40! The arithmetic mean doesn't work here. You need the harmonic mean.
- **Finance:** average return on investments (geometric)
- **Physics:** average speed, parallel resistances
- **Statistics:** robustness against outliers
Arithmetic Mean
The **arithmetic mean** is the most familiar average. It's the sum of all values divided by their count. But it's not the only mean - and not always the best one.
**Formula:** A = (a₁ + a₂ + ... + aₙ) / n **For two numbers:** A(a, b) = (a + b) / 2 **Property:** A lies exactly halfway between a and b.
The arithmetic mean is linear. It treats every value equally, but is sensitive to extreme data.
What is the arithmetic mean of 4 and 16?
Geometric Mean
The **geometric mean** is the nth root of the product of all values. It's ideal for growth rates, percentages, and ratios.
**Formula:** G = ⁿ√(a₁ × a₂ × ... × aₙ) **For two numbers:** G(a, b) = √(a × b) **Property:** a/G = G/b (equal ratios)
The geometric mean is always ≤ the arithmetic mean. Equality holds only when all numbers are identical.
An investment grew 100% in the first year and fell 50% in the second. What is the average annual growth?
Harmonic Mean
The **harmonic mean** is the reciprocal of the mean of reciprocals. It's used for average speeds, frequencies, and parallel resistances.
**Formula:** H = n / (1/a₁ + 1/a₂ + ... + 1/aₙ) **For two numbers:** H(a, b) = 2ab / (a + b) **Property:** H ≤ G ≤ A (always!)
The harmonic mean is the most "conservative". It always leans closer to the smaller value.
What is the harmonic mean of 4 and 16?
The Inequality of Means
The **AM-GM inequality** is one of the most important in mathematics. The arithmetic mean is always greater than or equal to the geometric mean.
**Generalized inequality:** For any positive a₁, a₂, ..., aₙ: (a₁+...+aₙ)/n ≥ ⁿ√(a₁×...×aₙ) This is used in optimization, information theory, and statistics.
The AM-GM inequality is a powerful tool for proofs and finding extrema without calculus.
The mean always refers to the arithmetic mean
Different types of means suit different situations
The arithmetic mean works well for additive data. The geometric mean is for growth rates and ratios. The harmonic mean is for speeds and frequencies. Choosing the wrong mean gives a wrong answer. Average speed (equal distances) requires the harmonic mean, not the arithmetic one!
For which values of a and b does A(a,b) = G(a,b)?
Key Ideas
- A = (a+b)/2 - arithmetic (additive)
- G = √(ab) - geometric (multiplicative)
- H = 2ab/(a+b) - harmonic (for reciprocals)
- H ≤ G ≤ A (inequality of means)
Related Topics
Means connect to other areas:
- Square Roots — Geometric mean
- Proportions — Geometric mean
- Progressions — AP - arithmetic, GP - geometric
Вопросы для размышления
- Why does average speed require the harmonic mean?
- How does the AM-GM inequality help solve optimization problems?
- Which mean should you use for a school grade average?