Advanced Study Notes – Average (Arithmetic Mean)
Quantitative Aptitude – MIT‑Level Treatment
These notes aim to give a rigorous, first‑principles view of the concept of an average, its generalizations, edge‑case behavior, and advanced applications. They are written for students preparing for high‑level aptitude exams (e.g., Nagaland Police Constable) but are also suitable for anyone wishing to deepen their mathematical understanding.
1. Introduction
The average is a measure of central tendency. In its simplest form it is the arithmetic mean of a finite collection of real numbers. The concept extends naturally to weighted collections, infinite sequences, continuous distributions, and more. Understanding its properties and edge cases is essential for solving a wide range of quantitative problems.
2. Arithmetic Mean – First Principles
2.1 Definition
Given a finite multiset (S = {x_1, x_2, \dots, x_n}) with (n \ge 1),
[ \bar{x} ;=; \frac{1}{n}\sum_{i=1}^{n} x_i ]
Interpretation: (\bar{x}) is the unique number that satisfies the balance property
[ \sum_{i=1}^{n} (x_i - \bar{x}) = 0 ] i.e., the sum of deviations from (\bar{x}) is zero.
2.2 Derivation from Balance
Starting from the balance equation:
[ \sum_{i=1}^{n} (x_i - \bar{x}) = 0 ;;\Longrightarrow;; \sum_{i=1}^{n} x_i - n\bar{x} = 0 ;;\Longrightarrow;; \bar{x} = \frac{1}{n}\sum_{i=1}^{n}x_i ]
Thus the arithmetic mean is the unique value that balances the deviations of all observations.
3. Fundamental Properties
| Property | Statement |
|---|---|
| Linearity | (\overline{a x_i + b} = a \bar{x} + b) for constants (a,b). |
| Monotonicity | If (x_i \le y_i) for all (i), then (\bar{x} \le \bar{y}). |
| Translation | Adding the same constant to all terms increases the mean by that constant. |
| Scaling | Multiplying all terms by a positive constant scales the mean by the same factor. |
| Symmetry | Ordering of terms does not affect the mean. |
Proof Sketch: The linearity follows immediately from distributivity of sums. Monotonicity follows by subtracting the two sums and observing non‑negativity.
4. Weighted Average
4.1 Definition
Given weights (w_i \ge 0) with (\sum_{i=1}^{n} w_i = 1),
[ \bar{x}w ;=; \sum{i=1}^{n} w_i x_i ]
If the weights sum to (W \neq 1), the average is
[ \bar{x}w ;=; \frac{1}{W}\sum{i=1}^{n} w_i x_i ]
4.2 Properties
| Property | Statement |
|---|---|
| Convexity | If all (w_i \ge 0) and (\sum w_i=1), then (\min x_i \le \bar{x}_w \le \max x_i). |
| Extremal Principle | (\bar{x}_w) is a convex combination of the (x_i). |
| Additivity | For two independent data sets, the overall weighted mean is the weighted sum of the subgroup means. |
Proof Sketch: Use the convexity inequality (0 \le \sum w_i (x_i - \bar{x}_w)^2).
5. Generalized Means
5.1 Power Means
For real (p \neq 0),
[ M_p(x_1,\dots,x_n) = \left(\frac{1}{n}\sum_{i=1}^{n} x_i^p\right)^{1/p} ]
Special cases:
- (p=1): Arithmetic mean.
- (p=0): Geometric mean (limit as (p\to 0)).
- (p=-1): Harmonic mean.
5.2 Inequalities
AM–GM–HM Inequality
For non‑negative (x_i),
[ \text{HM} ;\le; \text{GM} ;\le; \text{AM} ]
Proof (sketch): Apply convexity