Advanced Study Notes on Average (Arithmetic Mean)
Quantitative Aptitude – MIT‑Level Treatment
Author: [Your Name]
Date: 2026‑11‑06
Table of Contents
- First Principles & Definition
- Elementary Properties
- Generalized Means (Power Mean)
- Weighted Averages & Convex Combinations
- Relation to Variance & Central Limit Theorem
- Edge Cases & Pathologies
- Advanced Theory & Inequalities
- Challenging Problem & Solution
- References & Further Reading
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1. First Principles & Definition
Let (x_1, x_2, \dots, x_n) be real numbers.
The arithmetic mean (or average) of this finite set is
[ \mu = \frac{1}{n}\sum_{i=1}^{n} x_i . ]
1.1 Interpretation as Expected Value
If we consider the discrete random variable (X) that takes the value (x_i) with probability (1/n), then
[ \mathbb{E}[X] = \sum_{i=1}^{n} x_i ,\mathbb{P}(X=x_i) = \frac{1}{n}\sum_{i=1}^{n} x_i = \mu . ]
Thus the arithmetic mean is the expected value of a uniform distribution over the given points.
1.2 First‑Principle Proof of Uniqueness
Assume a function (f:\mathbb{R}^n \to \mathbb{R}) satisfies:
- Symmetry: (f(x_{\sigma(1)},\dots,x_{\sigma(n)}) = f(x_1,\dots,x_n)) for every permutation (\sigma).
- Translation Invariance: (f(x_1+t,\dots,x_n+t) = f(x_1,\dots,x_n)+t).
- Homogeneity of Degree 1: (f(\lambda x_1,\dots,\lambda x_n) = \lambda f(x_1,\dots,x_n)).
- Additivity for Singleton: (f(x) = x).
Under these axioms, the only possible (f) is the arithmetic mean, as can be shown by evaluating (f) on a basis of (\mathbb{R}^n). This gives a first‑principles characterization.
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2. Elementary Properties
| Property | Statement | Proof Sketch |
|---|---|---|
| Linearity | (\displaystyle \frac{1}{n}\sum (a x_i + b) = a\mu + b) | Use distributive law of sums. |
| Boundedness | (\min_i x_i \le \mu \le \max_i x_i) | Since each (x_i) contributes positively to the sum. |
| Monotonicity | If (x_i \le y_i) for all (i) then (\mu_x \le \mu_y). | Immediate from linearity. |
| Additive Invariance | (\mu_{x+y} = \mu_x + \mu_y) | Follows from definition. |
| Variance Decomposition | (\displaystyle \sum (x_i-\mu)^2 = \sum x_i^2 - n\mu^2). | Expand ((x_i-\mu)^2) and sum. |
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3. Generalized Means (Power Mean)
For (r \in \mathbb{R}) and (x_i > 0), the power mean of order (r) is
[ M_r(x_1,\dots,x_n) = \left(\frac{1}{n}\sum_{i=1}^{n} x_i^r\right)^{1/r} \quad (r\neq 0), ] and (M_0) is defined as the geometric mean.
3.1 Monotonicity in (r)
[ r < s \implies M_r \le M_s . ]
Proof uses Hölder's inequality; equality holds iff all (x_i) are equal.
3.2 Special Cases
| (r) | Mean | Notable Inequality |
|---|---|---|
| (-1) | Harmonic | (H \le G \le A) |
| (0) | Geometric | (G \le A) |
| (1) | Arithmetic | (A = \mu) |
| (2) | Quadratic | (A \le Q) |
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4. Weighted Averages & Convex Combinations
Given weights (w_i \ge 0) with (\sum w_i = 1), the weighted mean is
[ \mu_w = \sum_{i=1}^n w_i x_i . ]