Weighted Quadratic Mean
The weighted quadratic mean is a variation of the quadratic mean in which each term is multiplied by a specific weight. It is calculated using the following formula: $$ \mu = \sqrt{ \frac{ \sum x^2_i \cdot w_i}{ \sum w_i} } $$
The choice of weights wi assigned to each term is entirely up to you.
In general, higher weights are assigned to more significant terms, and lower weights to less important ones.
These weights can also represent the absolute frequencies of the terms.
A Practical Example
Suppose a math exam is divided into three sections: A, B, and C.
A student scores 18 on section A, 22 on section B, and 26 on section C.
The simple quadratic mean of the three scores is 21.24:
$$ \mu = \sqrt{ \frac{18^2 + 22^2 + 26^2}{3} } $$
$$ \mu = \sqrt{ \frac{324 + 484 + 676}{3} } $$
$$ \mu = \sqrt{ \frac{1484}{3} } $$
$$ \mu = 22.24 $$
However, the teacher assigns different weights to each section:
$$ w_A = 4, \quad w_B = 3, \quad w_C = 2 $$
In this case, to find the mean, we need to use the weighted quadratic mean formula:
$$ \mu = \sqrt{ \frac{ \sum x^2_i \cdot w_i}{ \sum w_i} } $$
$$ \mu = \sqrt{ \frac{18^2 \cdot 4 + 22^2 \cdot 3 + 26^2 \cdot 2}{4 + 3 + 2} } $$
$$ \mu = \sqrt{ \frac{324 \cdot 4 + 484 \cdot 3 + 676 \cdot 2}{9} } $$
$$ \mu = \sqrt{ \frac{1296 + 1452 + 1352}{9} } $$
$$ \mu = \sqrt{ \frac{4100}{9} } $$
$$ \mu = 21.34 $$
Thus, the weighted quadratic mean of the scores is μ = 21.34.
In this scenario, the weighted mean is lower than the simple quadratic mean (μ = 22.24) because section A's score has a higher weight than the others.
And so forth.
