CAT 2025 Lesson : Averages - Concepts & Cheatsheet

1) Simple Average or Arithmetic Mean = $\overline{x}$ = $\dfrac{ x_1 + x_2 + x_3 + … + x_n }{n}$ = $\dfrac{ \sum\limits_{i - 1}^{n} {x_i} }{n}$

2) Where $a$ is the Assumed Mean, Simple Average = $\overline{x}$ = $a + \dfrac{ \sum\limits_{i - 1}^{n} {(x_i - a)}}{n}$

3) Average of any group of items always lie between the smallest and the largest values in the group.

4) If each of the terms are added, subtracted, multiplied or by a constant $k$, then the average also gets added subtracted, multiplied or divided by $k$.

5) Average of numbers in AP = $\dfrac{(\text{first term} + \text{last term})}{2}$

6) Weighted Average = $\overline{x}$ = $\dfrac{w_1 x_1 + w_2 x_2 + w_3 x_3 + … + w_n x_n}{w_1 + w_2 + w_3 + … + w_n}$ = $\dfrac{ \sum\limits_{i - 1}^{n} {w_i x_i}}{ \sum\limits_{i - 1}^{n} {w_i}}$

7) Where a is the Assumed Mean, Weighted Average = $\overline{x}$ = $a + \dfrac{ \sum\limits_{i - 1}^{n} {x_i - a}}{ \sum\limits_{i - 1}^{n} {w_i} }$

8) Geometric Mean (GM) = $(x_1 \times x_2 \times x_3 \times …. \times x_n)^\dfrac{1}{n}$

9) Harmonic Mean (HM) = $\dfrac{n}{ \dfrac{1}{x_1} + \dfrac{1}{x_2} + \dfrac{1}{x_3} + … + \dfrac{1}{x_n} }$

10) For 2 terms, say $a$ and $b$, GM = $\sqrt{ab}$ and HM = $\dfrac{2ab}{a + b}$

11) For any set of positive real numbers, AM $\ge$ GM $\ge$ HM

12) Median = $\left( \dfrac{n + 1}{2}\right)$^th term where $n$ is odd and mean of $\left(\dfrac{n}{2} \right)$^th  term   and $\left( \dfrac{n}{2} + 1 \right)$^th term where $n$ is even

13) For $n$ terms, mode is the term(s) with the maximum number of occurrences.

14) A set of numbers can have only $1$ mean and $1$ median. However, the set can have one or more modes.