## Weighted Mean

Weighted mean (also known as weighted average) is the average of the given data set. It is an average calculated by assigning different weights to some of the individual values. If all the values are the same, then the weighted mean is the same as the arithmetic mean.

Weighted mean is the same as the average mean or arithmetic mean. It is calculated when data is given in different ways in comparison to arithmetic mean or sample mean.

Although weighted mean generally behaves in a similar way as average mean, they do have some contradictory properties. Data values with high weights contribute to the more weighted mean than the weights with lower weighted mean. The negative weights are not possible, some may be zero but not all of them as division by 0 is not allowed.

### Weighted Average Meaning

A weighted average is the average of all the values which are arranged on a priority basis. The weighted average of values is the sum of the weight times values divided by the sum of the weights.

### Weighted Mean Formula

The weighted mean for a given non - negative set of data with non - negative weights can be derived from the weighted mean formula given below:

\[\bar{x}\] = \[\frac{w_{1}x_{1} + w_{2}x_{2} +............w_{n}x_{n}}{w_{1} + w_{2}+.......w_{n}}\]

Here,

x represents the repeating value

w represents the number of occurrence of y weights.

\[\bar{x}\] represents the weighted mean

### Weighted Arithmetic Mean Meaning

The most commonly used average i.e ordinary arithmetic mean is the same as the weighted arithmetic mean, except that rather than each data point contributes equally to the final average, some data points contribute more than others. The idea of weighted mean makes a significant contribution in descriptive statistics and also occurs in a more general form in different areas of Mathematics.

The weighted arithmetic mean will be equal to an ordinary arithmetic mean in case all the given weights are equal. While weighted means generally behave similarly to the arithmetic mean, they do have some contradiction properties.

### Define Weighted Arithmetic Mean

If each number (x) is allocated to an equivalent positive weight (w), then the weighted arithmetic mean is defined as the sum of their products divided by the sum of their weights. In such a case:

\[\bar{x}\] = \[\frac{w_{1}x_{1} + w_{2}x_{2} +............w_{n}x_{n}}{w_{1} + w_{2}+.......w_{n}}\]

### Weighted Mean Examples

1. The numbers 40, 45, 80, 75 and 10 have weights 1, 2, 3, 4, and 5 respectively. Find the weighted mean for the given data set.

Solution:

Weighted Arithmetic Mean = \[\frac{\sum w_{x}}{\sum w}\]

= \[\frac{40 \times 1 + 45 \times 2 + 80 \times 3 + 75 \times 4 + 10 \times 5}{1 + 2 + 3 + 4 + 5}\]

= \[\frac{40 + 90 + 240 + 300 + 50}{15}\]

= \[\frac{720}{15}\]

= 48

2. A firm conducted a market survey of 1000 families to determine the average number of mobile phones each family owns. The data shows that a joint family owns two or three mobile phones and a nuclear family owns one or four mobile phones. Find the mean number of mobile phones per family.

Solution:

Step 1: Assign a weight to each value in a given data set.

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: Calculate the numerator of the weighted mean formula

Multiply each grade by its counts and then add the product together

\[\sum_{i=1}^{4}\]w\[_{i}\]x\[_{i}\] = w\[_{1}\]x\[_{1}\] + w\[_{2}\]x\[_{2}\] + w\[_{3}\]x\[_{3}\]

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

= 73 + 756 + 1377 + 360

= 2566

Step 3: Now, calculate the denominator of the weighted mean formula by adding their weights.

\[\sum_{i=1}^{4}\]w\[_{i}\] = w\[_{1}\] + w\[_{2}\] + w\[_{3}\]

73 + 378 + 459 + 90

= 1000

Step 4: Divide the numerator by the denominator.

\[\frac{\sum_{i=1}^{4}w_{i}x_{i}}{\sum_{i=1}^{4}w_{i}}\] = \[\frac{2566}{1000}\]

= 2.566

The weighted arithmetic mean of mobiles per household in this sample is 2.566

Q1. Can Weights Be Negative?

Ans. No, negative weights are not possible. Some weights maybe 0 but not all as the division by 0 is not permitted.

Q2. How to Calculate the Weighted Mean?

Ans. Weighted mean is the mean where some values contribute more than others. When the sum of all the weights adds to 1, then multiply each weight by its corresponding value and sum it all up. Else, multiply each weight (w) by its corresponding value (x), sum that all up, and divide by the sum of weights.

Q3. When to Use Weighted Average?

Ans. A weighted average or weighted mean is used when some data values are considered to be more important than other values and so we want them to contribute more to the final 'average'. This basically occurs similarly as some professors or teachers assign grades for their courses.