# Weighted Average Formula

## What Is An Average?

For a fact, “Average” and “Mean” means the same. They are actually synonyms for one another. What you basically do here is that you add all the observations followed by division by the number of observations. For example, in a family, there are 5 children weighing 20, 35, 80, 100, and 145 pounds respectively.  So to find their average rate, we add all their weights ( 20 + 35 + 80 + 100 + 145 = 280 ) and then divide it with the number of children whose weights we have calculated, so 280 / 5 = 145 pounds. The applications of average in our day to day life is huge. So let us unearth more about average.

### What is Weighted Average?

A weighted average is an average where assigning weights to each of the quantity or value is done and that is done according to the relative importance of each value or quantity. Let us take an example. Assume that the overall rank of a student needs to be calculated who has scored marks in different subjects. And each subject has a different percentage of the total marks. For example, the written test may have 50%, the practical test may have 30% and sports may have 20% weightage in total grade marks. Then what is the weighted average? The weight of the written test will be 0.5, the practical test will be 0.3 and sports will be 0.2. The assigned weights can be zero but can never be negative. Also, if the value will have the highest weight then it will have more effect on the weighted average and vice-versa. A weighted mean formula is usually more accurate than a simple average because, in a simple average, all the numbers in a data set are assigned an identical weight.

### Weighted Mean Formula

The weight that we assign to each variable varies from situation to situation, so does the formula. So, in general, we can write the weighted mean formula as:

$(Ax^1+Bx^2+Cx^3..............Zx^n)/n$

Suppose in a quiz, there are 10 questions and out of those 10 questions, 5 are about science, and 5 are about history. But the questions on science have been given twice the weight than that of history. If students score an average of 4 science questions and three history questions correct then what will be the simple class average?

4 science question + 3 history questions / 10 questions

4+3/10 = 7/10 = 7

But what will be the answer if we calculate the weighted average equation?

[(2)(4) + (1)(3)/10] = (8 + 1)/10 = 11

We can also simply write the weighted average equation as:

Weighted average method formula= $\frac{sum of weighted observations}{sum of weights}$

### How To Calculate Weighted Average?

There are a few steps that we can follow in order to calculate the weighted average. The steps are given below.

Step 1) Arrange all the number needed to calculate the weighted average and represent them as ‘x’

Step 2) Assign weights to the values based on their importance and represent them as ‘w’

Step 3) Make a table and arrange the values of ‘x’ and ‘w’ for convenience

Step 4) Multiply each ‘w’ with its corresponding ‘w’ to get ‘xw’

Step 5) Find the sum of all the weights $\sum w’ and \sum xw$

Step 6) Apply the formula to find the weighted average equation by dividing $\sum xw$ by  $\sum w$

### Solved Examples

Example 1) A teacher conducted a science test of 25 students out of which 10 students scored 80 and the rest scored 60. What is weighted average score of the entire class?

Solution 1) (80 marks x 10 students) + (60 marks x 15 students)

= 800 + 900

= 1700

Total number of students = 25

Weighted average formula:

Weighted average method formula= $\frac{sum of weighted terms}{total number of terms}$

= 1700/25

= 68

Example 2) Class 8 has two sections A and B. There are 25 students in section A and 30 students in section B. The average marks scored by the students in section A is 75 and the average marks scored by the students of section B is 60. Calculate the average marks obtained by the whole class.

Solution 2) (25 x 75) + (30 x 60)

= 1875 + 1800

= 3675

Weighted average formula:

Weighted average method formula= $\frac{sum of weighted observations}{sum of weights}$

= 3675 / 55

= 66.81

### Fun Facts of Weighted Average

1. A weighted average formula is usually more accurate than a simple average.

2. A weighted average formula is used by stock investors basically to track the cost basis of shares.

Question 1) What are the advantages of a Weighted Average?

Answer 1) In stock and accounting, the weighted average smoothes out the fluctuations in the market. Often in survey works such as census data, a segment of a population is either over-represented or under-represented. In such a situation, the weighted average takes the uneven population into consideration and turns them into a more balanced and equal representation of the data. This is basically done when the data deals with demographics and population size. Another advantage of the weighted average is that it considers equal values to be equivalent in proportion.

Question 2) What are the differences between Average and Weighted Average?

Answer  2) Average is a mean where observations are summed up and are divided by the total number of observations. A weighted average is where we give the weight to every observation of the data before we sum it up. In a simple average, every observation must be equally weighted. On a weighted average, every observation has its frequency.  For using a simple average, there are no specific conditions but we use a weighted average when we want to calculate the average which is based on different percentage values or when observations have its frequencies.