📚 Advanced Study Notes – Average
(Quantitative Aptitude – MIT‑Level)
Audience: Graduate students in Mathematics, Statistics, or Data Science.
Prerequisites: Basic algebra, inequalities, sequences & series, and probability basics.
1. Foundations – What is an “Average?”
| Concept | Definition | Notation | Remarks |
|---|---|---|---|
| Arithmetic Mean (AM) | For a finite set (S={x_1,x_2,\dots,x_n}): (\displaystyle \overline{x} = \frac{1}{n}\sum_{i=1}^n x_i). | (\overline{x}) | The most common “average”. |
| Weighted Arithmetic Mean | (\displaystyle \overline{x}w = \frac{\sum{i=1}^n w_i x_i}{\sum_{i=1}^n w_i}), (w_i\ge 0). | (\overline{x}_w) | Weights reflect importance or frequency. |
| Geometric Mean (GM) | (\displaystyle G = \sqrt[n]{\prod_{i=1}^n x_i}), (x_i>0). | (G) | Natural for growth rates, ratios. |
| Harmonic Mean (HM) | (\displaystyle H = \frac{n}{\sum_{i=1}^n 1/x_i}), (x_i>0). | (H) | Useful for rates, speeds. |
| Quadratic Mean (Root‑Mean‑Square, RMS) | (\displaystyle Q = \sqrt{\frac{1}{n}\sum_{i=1}^n x_i^2}). | (Q) | Related to energy, standard deviation. |
| Median | Middle value after sorting; if even, average of two middle values. | (M) | Robust to outliers. |
| Mode | Most frequent value. | (Mo) | Can be multiple. |
Key Idea: An average is a summary statistic that reduces a collection of numbers to a single representative value. It is not always unique or meaningful (e.g., negative weights, zero elements).
2. From First Principles – Deriving the Arithmetic Mean
2.1 The Rationale
- Symmetry: All elements are treated equally.
- Additivity: The sum of deviations from the mean is zero: (\sum (x_i-\overline{x}) = 0).
- Linearity: Scaling or translating data scales or translates the mean linearly.
2.2 Proof of Uniqueness
Let (m) be any number satisfying (\sum (x_i-m)=0).
[
\sum (x_i-m)=\sum x_i - n m = 0 ;;\Longrightarrow;; m=\frac{1}{n}\sum x_i = \overline{x}.
]
Thus, the arithmetic mean is the unique number whose deviations sum to zero.
2.3 Alternative Characterizations
-
Optimization: (\overline{x}) minimises the sum of squared deviations: [ \overline{x} = \arg\min_{a\in\mathbb{R}} \sum_{i=1}^n (x_i-a)^2. ] Proof: Differentiate w.r.t. (a), set derivative to zero.
-
Expectation: For a discrete random variable (X) taking values (x_i) with equal probability (1/n), (\mathbb{E}[X] = \overline{x}).
3. Inequalities Involving Averages
3.1 AM–GM–HM Inequality
For positive numbers (x_i>0):
[ H \le \overline{x} \le G \le Q. ]
Proof Sketch (AM–GM):
[ \frac{1}{n}\sum_{i=1}^n x_i \ge \sqrt[n]{\prod_{i=1}^n x_i} ] by the AM–GM inequality (standard proof using convexity of (\ln)).
Equality holds iff all (x_i) are equal.
3.2 Cauchy–Schwarz for Sums
For sequences (a_i, b_i):
[ \left(\sum a_i b_i\right)^2 \le \left(\sum a_i^2\right)\left(\sum b_i^2\right). ]
When (a_i = b_i = 1), this reduces to ((\sum 1)^2 \le n \sum 1) (trivial).
More useful: choosing (a_i = x_i), (b_i = 1) gives (\left(\sum x_i\right)^2 \le n \sum x_i^2), leading to (\overline{x}^2 \le Q^2).
3.3 Jensen’s Inequality
If (f) is convex, then:
[
f!\left(\sum w_i x_i\right) \le \sum w_i f(x_i)
]
for (w_i\ge 0,;\sum w_i = 1).
This generalizes AM–GM: apply (f(x)=\ln x) to get GM ≤ AM.