📚 Study Notes – Average (Quantitative Aptitude – Tamil Nadu Forest Guard)
Audience – Candidates preparing for the Tamil Nadu Forest Guard quantitative aptitude exam.
Depth – MIT‑level treatment from first principles, rigorous proofs, edge‑case analysis, and a challenging problem.
Format – Markdown with LaTeX equations, tables, and clear sectioning.
1️⃣ Introduction
The average (or mean) of a finite set of numbers is the most common way to describe the central tendency of the data.
In quantitative aptitude, it is used for:
- Quick comparison of groups (e.g., average speed, average cost).
- Formulating weighted summaries (e.g., average score with different weights).
- Building formulas for probability and statistics (expectation, variance).
Key Idea – The average is the value that, if all observations were equal to it, would preserve the total sum.
2️⃣ Definition (First Principles)
Let (x_1, x_2, \dots, x_n) be real numbers.
Arithmetic Mean (unweighted)
[ \operatorname{AM}(x_1,\dots,x_n) = \frac{1}{n}\sum_{i=1}^{n}x_i ]
If we wish to give each element a weight (w_i \ge 0) (not necessarily summing to 1), the weighted arithmetic mean is
Weighted Arithmetic Mean
[ \operatorname{WAM}(x_1,\dots,x_n; w_1,\dots,w_n) = \frac{\sum_{i=1}^{n} w_i x_i}{\sum_{i=1}^{n} w_i} ]
Derivation – Suppose we replace each (x_i) by a constant (c).
The total sum becomes (c,\sum_{i} w_i).
Setting this equal to the original weighted sum (\sum w_i x_i) gives
(c = \frac{\sum w_i x_i}{\sum w_i}), i.e. the weighted mean.
3️⃣ Types of Averages (Beyond Arithmetic)
| Average | Formula | When to Use | Notes |
|---|---|---|---|
| Arithmetic Mean (AM) | (\displaystyle \frac{1}{n}\sum x_i) | Equal importance | Minimizes squared error |
| Weighted AM | (\displaystyle \frac{\sum w_i x_i}{\sum w_i}) | Different importance | (w_i) may be integers, decimals, percentages |
| Geometric Mean (GM) | (\displaystyle \bigl(\prod x_i\bigr)^{1/n}) | Positive numbers, growth rates | Sensitive to zeros |
| Harmonic Mean (HM) | (\displaystyle \frac{n}{\sum 1/x_i}) | Rates, speeds | Lower than AM for positive data |
| Quadratic Mean (RMS) | (\displaystyle \sqrt{\frac{1}{n}\sum x_i^2}) | Energy, RMS voltage | Always ≥ AM |
| Median | Middle value after sorting | Skewed distributions | Not an “average” in the algebraic sense |
| Mode | Most frequent value | Categorical data | Not a numeric mean |
Why Multiple Means?
Different applications require different aggregating properties.
E.g., the harmonic mean gives the correct average speed when distances are equal but times differ.
4️⃣ Fundamental Properties of the Arithmetic Mean
-
Linearity
[ \operatorname{AM}(ax_1+by_1,\dots,ax_n+by_n)=a,\operatorname{AM}(x)+b,\operatorname{AM}(y) ] Proof: Direct expansion of the sum. -
Symmetry – Permuting the data does not change the mean.
-
Bounds – For non‑negative data, [ \min{x_i}\le \operatorname{AM}\le \max{x_i} ]
-
Minimization of Squared Deviations
[ \operatorname{AM}\ \text{minimizes}\ \sum_{i=1}^n (x_i-c)^2 ] Proof: Differentiate with respect to (