Weighted Mean

In Mathematics, the weighted mean is used to calculate the average value of the data. In the weighted mean calculation, the average value can be calculated by providing different weights to some of the individual values. We need to calculate the weighted mean when data is given in a different way compared to the arithmetic mean or sample mean. Different types of means are used to calculate the average of the data values. Let’s understand what is weighted mean and how to define weighted mean along with solved examples.

Weight Definition

Weight is defined as the measure of how heavy an object is. The weights cannot be negative. Some weight can be zero, but not all of them, since division by zero is not allowed.

The data elements which have a high weight will contribute more to the weighted mean as compared to the elements with a low weight.

What is Weighted Mean?

To calculate the weighted mean of certain data, we need to multiply the weight associated with a particular event or outcome with its associated outcome and finally sum up all the products together. It is very useful in calculating a theoretically expected outcome. Apart from weighted mean and arithmetic mean, there are various types of means such as harmonic mean, geometric mean, and so on.

Define Weighted Mean

The weighted mean is defined as an average computed by giving different weights to some of the individual values. When all the weights are equal, then the weighted mean is similar to the arithmetic mean. A free online tool called the weighted mean calculator is used to calculate the weighted mean for the given range of values.

Weighted Mean Formula

To calculate the weighted mean for a given set of non-negative data x₁,x₂,x₃,...x_n with non-negative weights w₁,w₂,w₃,..., we use the formula given below.

\[WeightedMean\overline{( W )} = \frac {\sum_{i=1}^{n} w_{i}x_{i}}{\sum_{i=1}^{n} w_i}\]

Where \[\overline {W}\] is the weighted mean,

x is the repeating value,

n is the number of terms whose mean is to be calculated, and

w is the individual weights.

Uses of Weighted Means

• Weighted means are useful in a wide variety of scenarios in our daily life. For example, a student uses a weighted mean in order to calculate their percentage grade in a course. In such a case, the student has to multiply the weighing of all assessment items in the course (e.g., assignments, exams, projects, etc.) by the respective grade that was obtained in each of the categories. 

• It is used in descriptive statistical analysis, such as index numbers calculation. For example, stock market indices such as Nifty or BSE Sensex are computed using the weighted average method. It can also be applied in physics to find the center of mass and moments of inertia of an object.

• It is also useful for businessmen to evaluate the average prices of goods purchased from different vendors where the purchased quantity is considered as the weight. It gives a better understanding of his expenses.

• A customer's decision on whether to buy a product or not depends on the quality of the product, knowledge of the product, cost of the product, and service by the franchise. The customer allocates weight to each criterion and calculates the weighted average. This will help him to make a better decision on buying the product.

• To recruit a person for a job, the interviewer looks at the personality, working capabilities, educational qualification, and team working skills. Based on the profile, different levels of importance (weights) are given, and then the final selection is made.

Important Notes

• The weights can be in the form of quantities, decimals, whole numbers, fractions, or percentages.

• If the weights are given in percentage, then the sum of the percentage will be 100%.

• Weighted average for quantities (x)i having weights in percentage (P)i% is: Weighted average = ∑ (P)i% × (x)i

Solved Example of Weighted Mean

Suppose a marketing firm conducted a survey of 1,000 households to determine the average number of TVs each household owns. The data shows that there are more households with two or three TVs and a few numbers with one of four. Every household in the sample has at least one TV and not a household has more than four. Calculate the mean number of TVs per household.

Solution: Here most of the values in this data set are repeated multiple times, we can easily compute the sample mean as a weighted mean. Following are steps to calculate the weighted arithmetic mean.

Step 1: First assign a weight to each value in the dataset.

x₁=1, w₁=73

x₂=2, w₂=378

x₃=3, w₃=459

x₄=4, w₄=90

Step 2: Now compute the numerator of the weighted mean formula.

To calculate it, multiply each sample by its weight and then add the products together to get the final value

\[\displaystyle\sum\limits_{i=1}^4 i\] w_ix_i=w₁x₁+w₂x₂+w₃x₃+w₄x₄

= 1 x 73 + 2 x 378 + 3 x 459 + 4 x 90

= 73 + 756 + 1377 + 360

= 2566

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

\[\displaystyle\sum\limits_{i=1}^4 i\] w_i=w₁+w₂+w₃+w₄

= 73 + 378 + 459 + 90

= 1000

Step 4: Finally divide the numerator value by the denominator value.

\[\frac{\displaystyle\sum\limits_{i=1}^4w_{i}x_{i}}{\displaystyle\sum\limits_{i=1}^4 w_{i}}\]

=\[\frac {2566}{1000}\]

=2.566

Hence, the mean number of TVs per household in this sample is 2.566.

Note: The weighted mean can be easily influenced by an outlier in our data. If we have very high or very low values in our data set, then we cannot rely on the weighted mean.

Conclusion

Weighted Mean is a mean where some of the values contribute more than others. It represents the average of a given data. The Weighted mean is similar to the arithmetic mean or sample mean. Sometimes it is also known as the weighted average.

When the weights add to 1, we just have to multiply each weight by the matching value and sum it all up.

Otherwise, we have to multiply each weight w by its matching value x, the sum that all up, and divide it by the sum of weight.