Study Notes – Average (Quantitative Aptitude)
Audience: IIT‑JEE, JEE‑Main, and other competitive exams
Key Idea: The average (or mean) is a concise way to describe the central tendency of a set of numbers. Mastery of averages is essential because almost every “average‑based” problem can be reduced to a simple algebraic equation once you know the right formula and the right way to manipulate it.
1. Quick Reference
| Type | Definition | Formula | Notes |
|---|---|---|---|
| Arithmetic Mean (AM) | Sum of numbers ÷ count | (\displaystyle \bar{x}=\frac{\sum x_i}{n}) | Most common |
| Weighted Mean (WM) | Sum of (value × weight) ÷ sum of weights | (\displaystyle \bar{x}_w=\frac{\sum w_i x_i}{\sum w_i}) | Use when items have different importances |
| Harmonic Mean (HM) | Reciprocal of average of reciprocals | (\displaystyle H=\frac{n}{\sum \frac{1}{x_i}}) | Useful for rates, speeds |
| Geometric Mean (GM) | (n^{th}) root of product of numbers | (\displaystyle G=\sqrt[n]{\prod x_i}) | For proportional growth |
| Median | Middle value after sorting | — | Not an “average” in the strict sense; useful for skewed data |
| Mode | Most frequent value | — | — |
In exam questions “average” always means arithmetic mean unless stated otherwise.
2. Derivations & Intuition
2.1 Arithmetic Mean
Let the data set be ({x_1,x_2,\dots,x_n}).
The average (\bar{x}) satisfies the property that the total deviation from the mean is zero:
[ \sum_{i=1}^n (x_i-\bar{x}) = 0 ]
Expanding,
[ \sum_{i=1}^n x_i - n\bar{x} = 0 ;;\Rightarrow;; \bar{x} = \frac{\sum x_i}{n} ]
Key takeaway: The arithmetic mean “balances” the dataset.
2.2 Weighted Mean
Suppose each value (x_i) has weight (w_i) (importance, frequency, duration, etc.).
The weighted mean (\bar{x}_w) is defined by the same balancing equation:
[ \sum_{i=1}^n w_i(x_i-\bar{x}_w)=0 ]
Expanding,
[ \sum w_i x_i - \bar{x}_w \sum w_i = 0 ;;\Rightarrow;; \bar{x}_w = \frac{\sum w_i x_i}{\sum w_i} ]
Why? Think of each pair ((x_i, w_i)) as a “mass” (w_i) at location (x_i); the weighted mean is the centre of mass.
2.3 Harmonic Mean
The harmonic mean is the reciprocal of the arithmetic mean of reciprocals:
[ H = \frac{1}{\displaystyle \frac{1}{n}\sum_{i=1}^n \frac{1}{x_i}} = \frac{n}{\displaystyle \sum_{i=1}^n \frac{1}{x_i}} ]
When to use? Average of rates (e.g., average speed over equal distances, average price per unit in multiple purchases).
2.4 Geometric Mean
For positive numbers, the geometric mean reflects proportional change:
[ G = \sqrt[n]{\prod_{i=1}^n x_i} \quad\text{or}\quad \ln G = \frac{1}{n}\sum_{i=1}^n \ln x_i ]
Useful in: Growth problems, compound interest, average rates of increase.
3. Properties that Help in Exams
| Property | Formula | When to Use |
|---|---|---|
| Linear Combination | (\displaystyle \frac{aS_1 + bS_2}{a+b}) | Two groups with known averages |
| Average of Two Averages | (\displaystyle \frac{n_1\bar{x}_1+n_2\bar{x}_2}{n_1+n_2}) | Merge two datasets |
| Inverse Relationship | If (x) and (y) are inversely proportional, (\displaystyle \frac{1}{\bar{x}}=\frac{1}{\bar{y}}) | Average speed/distance problems |
| Adding a New Element | (\displaystyle \bar{x}{new} = \frac{n\bar{x}{old} + x_{new}}{n+1}) | Incremental data |
| Removing an Element | (\displaystyle \bar{x}{new} = \frac{n\bar{x}{old} - x_{removed}}{n-1}) | Deleting data |