Weighted average

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Important to get correct, for now it's just "something"

Not to be confused with the average or mean

Definition

Let $\big(v_i\big)_{i\eq 1}^n\in V$ be a finite collection of values, for $V$ a ???; and let $\big(w_i\big)_{i\eq 1}^n\in W$ be a finite collection of weights, for $W$ a ???.

Then the weighted average, defined here as $A$, of the $v_i$ is:

• $$A:\eq\frac{\sum^n_{i\eq 1}w_i v_i} {\sum^n_{i\eq 1}w_i}$$
  • note that each $w_i$ is called the weighting of $v_i$ and each $w_iv_i$ as the weighted contribution or contribution of $v_i$

We define the following terms:

1. Weighted sum, sometimes denoted $S$ or $S_{w,v}$, as the numerator: $\sum^n_{i\eq 1}w_i v_i$, and
2. Total weight (AKA: weighting or sum of weights), sometimes denoted $w$ or $S_w$, as the denominator: $$\sum^n_{i\eq 1}w_i$$

Special cases

Of the average

Note that if $\forall i\in\{1,\ldots,n\}\big[w_i\in\mathbb{N}_{\ge 1}\big]$ then this is just a special case of the average where the weights are the frequencies of occurrences of the values. An example is given below demonstrates a weighted average that isn't an average.

Examples

Suppose we have the following measurements for a population:

• group A: $v_1$, representing some unit $w_1\in[0,1]\subseteq\mathbb{R}$ units of the population
• group B: $v_2$, representing $w_2\in[0,1]\subseteq\mathbb{R}$ units of the population
• group C: $v_3$, representing $w_3\in[0,1]\subseteq\mathbb{R}$ units of the population

The average of the population surveyed, $A$, is:

• $$A:\eq\frac{w_1v_1\ +\ w_2v_2\ +\ w_3v_3}{w_1\ +\ w_2\ +\ w_3}$$ (measured value) per unit of population