Weighted average

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Not to be confused with the average or mean

Definition

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

Then the weighted average, defined here as [ilmath]A[/ilmath], of the [ilmath]v_i[/ilmath] is:

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

We define the following terms:

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

Special cases

Of the average

Note that if [ilmath]\forall i\in\{1,\ldots,n\}\big[w_i\in\mathbb{N}_{\ge 1}\big] [/ilmath] 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: [ilmath]v_1[/ilmath], representing some unit [ilmath]w_1\in[0,1]\subseteq\mathbb{R} [/ilmath] units of the population
• group B: [ilmath]v_2[/ilmath], representing [ilmath]w_2\in[0,1]\subseteq\mathbb{R} [/ilmath] units of the population
• group C: [ilmath]v_3[/ilmath], representing [ilmath]w_3\in[0,1]\subseteq\mathbb{R} [/ilmath] units of the population

The average of the population surveyed, [ilmath]A[/ilmath], is:

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