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Weighted Mean Formula

Weighted Mean is an average computed by giving different weights to some of the individual values. If all the weights are equal, then the weighted mean is the same as the arithmetic mean.

It represents the average of a given data. The Weighted mean is similar to arithmetic mean or sample mean. The Weighted mean is calculated when data is given in a different way compared to an arithmetic mean or sample mean.

Whereas weighted means generally behave in a similar approach to arithmetic means, they do have a few counter-instinctive properties. Data elements with a high weight contribute more to the weighted mean than do elements with a low weight.

The weights cannot be negative. Some may be zero, but not all of them; since division by zero is not allowed. Weighted means play an important role in the systems of data analysis, weighted differential and integral calculus.

Formula of weighted Mean

The Weighted mean for given set of non-negative data ${x_{1},\:x_{2},\:x_{3},….x_{n}}$ with non-negative weights ${w_{1},\:w_{2},\:w_{3},….w_{n}}$ can be derived from the formula.

\[\large \overline{x}=\frac{w_{1}x_{1}+w_{2}x_{2}+…..+w_{n}x_{n}}{w_{1}+w_{2}+…..w_{n}}\]

Where,

$x$ is the repeating value
$w$ is the number of occurrences of $x$ weight
$\overline{x}$ is the weighted mean

Solved examples of weighted mean

Question: Suppose that a marketing firm conducts a survey of 1,000 households to determine the average number of TVs each household owns. The data show a large number of households with two or three TVs and a smaller number with one or four. Every household in the sample has at least one TV and no household has more than four.

Solution:

Here’s the sample data for the survey:

Number of TVs per Household Number of Households
1 73
2 378
3 459
4 90

As many of the values in this data set are repeated multiple times, you can easily compute the sample mean as a weighted mean. Follow these steps to calculate the weighted arithmetic mean:

Step 1: Assign a weight to each value in the dataset:

$x_{1}=1,\:w_{1}=73$
$x_{2}=2,\:w_{2}=378$
$x_{3}=3,\:w_{3}=459$
$x_{4}=4,\:w_{4}=90$

Step 2: Compute the numerator of the weighted mean formula.

Multiply each sample by its weight and then add the products together:

$\sum_{i=1}^{4}w_{i}\,x_{i}=w_{1}\,x_{1}+w_{2}\,x_{2}+w_{3}\,x_{3}+w_{4}\,x_{4}$

= (1)(73)+(2)(378)+(3)(459)+(4)(90)

=2566

Step 3: Now, compute the denominator of the weighted mean formula by adding the weights together.

$\sum_{i=1}^{4}w_{i}=w_{1}+w_{2}+w_{3}+w_{4}$

= 73 + 378 + 459 + 90

=1000

Step 4: Divide the numerator by the denominator

$\large \frac{\sum_{i=1}^{4}w_{i}\,x_{i}}{\sum_{i=1}^{4}w_{i}}$

$=\frac{2566}{1000}$

$=2.566$

The mean number of TVs per household in this sample is 2.566.

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