Weighted Mean Formula | Weighted Average | Weighted Arithmetic

What is Weighted Mean?

A weighted mean or average is same some king. Here, instead of contributing each point in the calculation of mean, some data points are contributing to calculate weight as compared to others. In case, all weights are equal to the arithmetic mean then it is named as the regular average and this concept is used commonly in statistics to study about the population in depth.

The next common term is arithmetic mean here where all numbers carry almost an equal weight. For example, if there are numbers like 1, 3, 5, 7, and 10. If you wanted to calculate their arithmetic means then you have to add these points first and divide the sum by total number of items within a set.

  1. Sum of all data points: 1 + 3 + 5 + 7 + 10 = 26.
  2. Divide by the number of items in the set: 26 / 5 = 5.2.

So, what we meant by equal weight here? Take an example of competitive exam where each question has the equal weightage. For example, if you have total 20 questions for 100-point exam then the weighted average for each question would be 5 point here. In this way, the concept is used frequently in the real-life too.

Weighted Mean Formula

The Weighted mean for given set of non-negative data x1,x2,x3,….xn with non-negative weights w1,w2,w3,….wn can be derived from the formula.

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

At the same time, there are cases when one particular number has more weightage then you need to calculated weighted mean here instead of the arithmetic mean. To calculate the weighted mean – you first need to multiply the data points by weights. Then add the results up. For that set of number above with equal weights (1/5 for each number), the math to find the weighted mean would be:

1(*1/5) + 3(*1/5) + 5(*1/5) + 7(*1/5) + 10(*1/5) = 5.2.

In most of the cases, this is easy to find the weighted mean, For complex numbers, you have to be extra careful.

Weighted Average Formula

The weighted average is defined as the average where each observation in the data set is multiplied or assigned before it summed up to the single average value. In the given process, where each quantity to be averages or assigned a weight that helps to determine the relative importance of each quantity. Weighing means many equivalent terms where same value is involved in the average.

\[\large \overline{x}=\frac{\sum_{i=1}^{n}w_{i}x_{i}}{\sum_{i=1}^{n}w_{i}}\]

Or in simple terms we can write the formula as below

\[\large Weighted\;Average=\frac{Sum\;of\;Weighted\;Terms}{Total\;Number\;of\;Terms}\]

Let us see ahead how to break down the weighted average. The weighted average is usually computed in respect to the frequency of values for a given dataset. However, a set of certain values for a dataset is given highest importance for one or other reasons where frequency of occurrence is calculated.

Investors are commonly dealing with the weighted average from years where prices are changing daily. So, this is quite tough to keep track of costs where shares are accumulated over a certain period of time. If any investor wanted to calculate the weighted average then he must multiply the total number of shares that are acquired at a particular price, you need to add those values then divide the total values with total number of shares as well.

Weighted average concept is applicable in multiple areas mainly finances where you have to find share costs, portfolio returns, inventory costs, valuation etc. If a fund holds multiple securities with 10 percent share in a year then it represents the weighed average of returns for funds with respect to the value of position in the fund.