The Weighted Mean Formula -

Formula

\bar{x}_w = \frac{\sum_{i=1}^{n} w_i x_i}{\sum_{i=1}^{n} w_i}

Used to calculate: The average of a set of values when individual values contribute different amounts, or weights, to the result.

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What Is the Weighted Mean Formula?

The weighted mean formula calculates an average in which some values contribute more to the result than others. Each value is multiplied by its corresponding weight, the weighted values are added together, and their sum is divided by the sum of the weights.

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Formula and Variables

\bar{x}_w = \frac{\sum_{i=1}^{n} w_i x_i}{\sum_{i=1}^{n} w_i}

Symbol	Meaning	Unit
$\bar{x}_w$	Weighted mean	Same unit as the values
$x_i$	Individual value	Depends on the data
$w_i$	Weight assigned to an individual value	Depends on the weighting system
$n$	Number of values	None
$\sum$	Summation of the specified terms	Depends on the terms being summed

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How to Use the Formula

Assign or identify a weight for each value. Multiply each value by its corresponding weight and add the resulting products. Add all of the weights separately, then divide the sum of the weighted values by the sum of the weights.

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When Is This Formula Used?

The weighted mean formula is used when the values in a data set do not contribute equally to the overall average.

Common uses:

– Calculating grades when assignments have different weights
– Finding averages from values with different frequencies or importance
– Calculating weighted financial and statistical averages

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Important Notes

– Values with larger weights have a greater influence on the weighted mean.
– The sum of the weights cannot equal zero because it appears in the denominator.
– The weighted mean has the same unit as the values when the weights are dimensionless.
– A common point of confusion is dividing by the number of values instead of the sum of the weights.

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Alternate Forms

Alternate form:

\bar{x}_w = \frac{w_1x_1 + w_2x_2 + \cdots + w_nx_n}{w_1 + w_2 + \cdots + w_n}

When normalized weights sum to one:

\bar{x}_w = \sum_{i=1}^{n} w_i x_i

with:

\sum_{i=1}^{n} w_i = 1

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