CAT 2025 Lesson : Averages - Weighted Average

3. Weighted Average

Weighted average method is applied where we need to find the average of groups of different sizes. Where $x_{1}, x_{2}, ... , x_{n}$ are the values and $w_{1}, w_{2}, ... , w_{n}$ are the weights, the weighted average is

$\overline{x} = \dfrac{w_{1} x_{1} + w_{2} x_{2} + w_{3} x_{3} + ... + w_{n} x_{n}}{w_{1} + w_{2} + w_{3} + ... + w_{n}} = \dfrac{\displaystyle\sum_{i=1}^n w_{i} x_{i}}{\displaystyle\sum_{i=1}^n w_{i}}$

Average is to be found for the values, i.e., the $\bm{x}$ -terms. Weights, i.e., the $\bm{w}$ -terms are the number of occurrences of these $x$-terms.

(Note: To find the weighted average of two $x$-values, we can use the alligation method detailed in the Mixtures & Alligations lesson.)

3.1 Weighted Average using Assumed Mean

This is similar to the assumed mean in the previous section. The difference here is the application of weights.

Where $a$ is the Assumed Mean, $x_{i}$ is a value and $w_{i}$ is a weight,

Weighted Average $= a + \dfrac{\displaystyle\sum_{i=1}^n (x_{i} - a) \times w_{i}}{\displaystyle\sum_{i=1}^n w_{i}} = a + \dfrac{\displaystyle\sum_{i=1}^n d_{i} \times w_{i}}{\displaystyle\sum_{i=1}^n w_{i}}$