## Weighted Average Calculator

#### Weighted Average Calculator

 Number Weight

Result:

 Weighted Average: 0 Total of Weights: 0

### What is weighted average?

Weighted average is the mean of the numbers in a set of data, when some numbers of the data have more influence owing to some characteristics of the data set. In other words, the numbers have unequal importance in the set. Mathematically, weighted average is the mean of some numbers in which each number is assigned a different weight, and these weights determine how important each number is, with respect to the others.

If all the numbers to be averaged have each equal importance in the set, we simply take the arithmetic mean of the numbers. Weighted average is a special type of mean when the numbers have unequal importance (or unequal weights). For example, when the grades of a student are calculated, it may happen that some assignments or exams are assigned greater weightage than the other. For instance, a final exam score has a greater weight than a quiz conducted in class. This is one of the most common examples of weighted average.

### Properties of Weighted average

Given below are some important properties of weighted average.

Weight – This indicates the importance of each number in the set of numbers whose average is to be found.

Number – The entities in the data set which are assigned different weights according to their importance in the set.

Weighted average – The average of the numbers when each number is of unequal importance in the data set. In other words, some number contribute more (have greater importance) than the others (have lesser importance).

Arithmetic mean – When the entities to be averaged have equal weights, we simply sum up all the entities and divide that by the total number of entities.

### Weighted Average Formula:

The formula for weighted average is as follows. $$Weighted \,average \, = \; {{w_1 x_1+w_2 x_2+w_3 x_3+⋯+w_n x_n}\over{w_1+w_2+w_3+⋯+w_n}}$$

Here, x1,x2,… xn indicate the n numbers whose average is to be found and w1,w2,… wn indicate the respective weights of these numbers.

To find the weighted average, we first multiply each number with its respective weight, and then divide that sum by the total number of weights.

### Characteristics of weighted average

Given below are some key characteristics of the weighted average.

• Weighted average is used when some numbers to be averaged make unequal contributions to the data set (or have unequal importance) with respect to the others.
• The numbers to be averaged may be integers or whole numbers. The weights are usually positive integers or positive real numbers.
• The weighted average always lies between the maximum and minimum values of the set of numbers. If all the numbers being averaged are equal, the weighted average will be equal to that number itself.
• If the value of each number being averaged is increased or decreased by the same amount, the same will be reflected in the weighted average.
• If the value of each number being averaged is multiplied or divided by the same amount, the weighted average will also be multiplied or divided accordingly.
• If the numbers being averaged have the same weights, this reduces to a case of arithmetic mean. In that case, we simply sum up the entities and divide by the total number of entities.

• ### Areas of application

Some applications of the weighted average are:

• For calculating the final grade of students when each examination or assignment contributes a different amount to the final, depending on their importance.
• In manufacturing companies, the products made are sold at different costs depending on how valuable they are.
• When the average performance of a sportsperson is evaluated, a weighted average is used because each type of performance is not equally significant. For example, to calculate the batting average for a baseball player, each hit carries a different weight.

• Question: In Andrea’s class, exams, quizzes and homework assignments contribute to her final score. There were three exams, one quiz and one homework assignment. Each of the three exams contributes to 25% of the final score, the quiz and homework contribute 15% and 10% respectively. Andrea’s scores in the exams are 90, 76, 87, the quiz score is 65 and the homework grade is 91. All these grades are marked out of 100. What is her final score?

Answer: In this problem, we need to find the weighted average of a total of five quantities, namely three exams, one quiz and one homework assignment.

The values of these three exams, quiz and homework are 90, 76, 87, 65 and 91.

The weights of these are 0.25, 0.25, 0.25, 0.15 and 0.1 respectively.

We can now plug these in the formula for weighted average.

$$Weighted average = {{0.25×90+0.25×76+0.25×87+0.15×65+0.1×91}\over{0.25+0.25+0.25+0.15+0.1}}$$

$$Weighted average = {{22.5+19+21.75+9.75+9.1}\over{1}} = 82.1$$

Hence, Andrea’s final score is 82.1%

Question: The average score of a batsman in 20 innings is 48. In the next innings, he scores 51 runs. What will be his new average?

Answer: The new average can be found using the following formula.

$$New average = {{old sum+new score}\over{total number of innings}}$$

$$New average = {{(20×48)+51}\over{20+1}} = {{960+51}\over21} = {1011 \over 21} = 48.14$$

Hence, the new average of the batsman increases, it is now 48.14.