📚 Quantitative Aptitude – Average
Target Exam: Himachal Pradesh PSC
Level: MIT‑style advanced treatment (first principles, rigorous proofs, edge cases)
Format: Markdown with LaTeX for all mathematical expressions
1. Introduction
In quantitative aptitude, the term average usually refers to the arithmetic mean of a finite set of numbers.
However, the concept of average is a special case of a broader mathematical notion – the mean – which has many variants (harmonic, geometric, quadratic, etc.) and deep connections to probability, statistics, and analysis.
Why study averages rigorously?
• In exams the rules that seem “obvious” (e.g., “average of two numbers is their sum divided by 2”) hide subtle pitfalls.
• Understanding the origin of the formula and its limitations helps you spot trick questions and solve problems that involve weighted averages, averages of averages, or infinite sequences.
2. First Principles: Definition & Basic Properties
2.1 Arithmetic Mean of a Finite Set
Let (S={x_1,x_2,\dots,x_n}) with (n\ge 1).
The arithmetic mean (average) is defined by
[ \bar{x} ;=; \frac{1}{n}\sum_{i=1}^{n} x_i \qquad\qquad (1) ]
Why this formula?
The mean is the unique number (m) that minimizes the sum of squared deviations:
[ m=\arg\min_{m\in\mathbb R};\sum_{i=1}^{n}(x_i-m)^2 ]
Taking derivative w.r.t (m) and setting to zero:
[ \frac{d}{dm}\sum_{i=1}^{n}(x_i-m)^2 = -2\sum_{i=1}^{n}(x_i-m)=0 ;;\Longrightarrow;; m=\frac{1}{n}\sum_{i=1}^{n}x_i ]
Thus the arithmetic mean is the least‑squares estimate of the central tendency.
2.2 Linearity and Additivity
For any two finite sets (S={x_i}) and (T={y_j}) with sizes (n) and (m),
[ \overline{S\cup T}
\frac{n\overline{S}+m\overline{T}}{n+m} \tag{2} ]
Proof:
[
\overline{S\cup T}
= \frac{1}{n+m}\left(\sum_{i=1}^{n}x_i + \sum_{j=1}^{m}y_j\right)
= \frac{n\overline{S}+m\overline{T}}{n+m}
]
Consequence: The average of an average is not simply the average of the two numbers unless the sample sizes are equal.
2.3 Weighted Average
Suppose each element (x_i) has a weight (w_i>0).
The weighted mean is
[ \bar{x}w = \frac{\sum{i=1}^{n} w_i x_i}{\sum_{i=1}^{n} w_i} \tag{3} ]
Special cases:
- (w_i=1) for all (i): ordinary mean.
- (w_i) proportional to a frequency or importance factor.
Edge case: If (\sum w_i = 0) (e.g., all weights zero), the average is undefined.
3. Deeper Theory: Generalized Means
For real numbers (x_1,\dots,x_n >0) and a real parameter (p\neq0),
[ M_p(x_1,\dots,x_n)=\left(\