# Weighted Average Formula

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## What is weighted average?

A weighted average is the average of a data set that recognizes certain numbers as more important than others. Weighted averages are commonly used in statistical analysis, stock portfolios and teacher grading averages. It is an important tool in accounting for stock fluctuations, uneven or misrepresented data and ensuring similar data points are equal in the proportion represented.

## Weighted average example

Weighted average is one means by which accountants calculate the costs of items. In some industries where quantities are mixed or too numerous to count, the weighted average method is useful. This number goes into the calculation for the cost of goods sold. Other costing methods include last in, first out and first in, first out, or LIFO and FIFO respectively.

Example:

A manufacturer purchases 20,000 units of a product at $1 each, 15,000 at$1.15 each and 5,000 at $2 each. Using the units as the weight and the total number of units as the sum of all weights, we arrive at this calculation:$1(20,000) + $1.15 (15,000) +$2 (5,000) / (20,000 + 15,000 + 5,000) = ($20,000 +$17,250 + $10,000) / ($20,000 + 15,000 + 5,000) = $47,250 / 40,000 =$1.18

### Multiply the weight by each value

Once you know the weight of each value, multiply the weight by each data point.

Example:
In a data set of four test scores where the final test is more heavily weighted than the others:

• 50(.15) = 7.5

• 76(.20) = 15.2

• 80(.20) = 16

• 98(.45) = 44.1

### Add the results of step two together

Calculate the sum of all the weighted values to arrive at your weighted average.

Example:

7.5 + 15.2 + 16 + 44.1 = 82.8

The weighted average is 82.8%. Using the normal average where we calculate the sum and divide it by the number of variables, the average score would be 76%. The weighted average method stresses the importance of the final exam over the others.

 Test Score Assigned Weight Test Score Weighted Value 50 .15 7.5 76 .20 15.2 80 .20 16 98 .45 44.1 Weighted Average 82.8

## How to calculate weighted average when the weights don’t add up to one

Sometimes you may want to calculate the average of a data set that doesn’t add up perfectly to 1 or 100%. This occurs in a random collection of data from populations or occurrences in research. You can calculate the weighted average of this set of numbers by multiplying each value in the set by its weight, then adding up the products and dividing the products’ sum by the sum of all weights.

For a more in-depth explanation of the weighted average formula above when the weights don’t add up to one, follow these steps:

### Determine the weight of each number

To determine the weight of each number, consider its importance to you or the frequency of occurrence. If you are trying to calculate the average number of business leads you pursue, you may want leads that turn into sales to weigh more heavily than cold calls. To find the weighted average without added bias, calculate the frequency a number occurs as the variable’s weight. This reflects its influence over the entire data set.

Example: Calculate the average time you spend exercising four days a week over a month or four weeks. The time you spent exercising on any given day is the data set. The number of days you exercised for an average time is the weight you’ll use.

• 7 days you exercised for 20 minutes

• 3 days you exercised for 45 minutes

• 4 days you exercised for 15 minutes

• 2 days you were supposed to exercise and did not

### Find the sum of all weights

The next step to finding the weighted average of a data set that doesn’t equal 1 is to add the sum of the total weight. From our previous example, you should have a total of 16 days spent exercising:

• 7+3+4+2 = 16

### Calculate the sum of each number multiplied by its weight

Using the frequency numbers, multiply each by the time you spent exercising. The combined total gives you the sum of the variables multiplied by their respective weights.

Example:

• 20(7) = 140

• 45(3) = 135

• 15(4) = 60

• 0(2) = 0

• 140 + 135 + 60 + 0 = 335

### Divide the results of step three by the sum of all weights

The formula for finding the weighted average is the sum of all the variables multiplied by their weight, then divided by the sum of the weights.

Example:
Sum of variables (weight) / sum of all weights = weighted average

335/16 = 20.9

The weighted average of the time you spent working out for the month is 20.9 minutes.

## What Is the Purpose of a Weighted Average?

In calculating a simple average, or arithmetic mean, all numbers are treated equally and assigned equal weight. But a weighted average assigns weights that determine in advance the relative importance of each data point.

A weighted average is most often computed to equalize the frequency of the values in a data set. For example, a survey may gather enough responses from every age group to be considered statistically valid, but the 18–34 age group may have fewer respondents than all others relative to their share of the population. The survey team may weight the results of the 18–34 age group so that their views are represented proportionately.

However, values in a data set may be weighted for other reasons than the frequency of occurrence. For example, if students in a dance class are graded on skill, attendance, and manners, the grade for skill may be given greater weight than the other factors.

In any case, in a weighted average, each data point value is multiplied by the assigned weight, which is then summed and divided by the number of data points.

In a weighted average, the final average number reflects the relative importance of each observation and is thus more descriptive than a simple average. It also has the effect of smoothing out the data and enhancing its accuracy.

Weighted Average
Data PointData Point ValueAssigned WeightData Point Weighted Value
110220
1505250
1403120
TOTAL10010390
Weighted Average39

### Weighting a Stock Portfolio

Investors usually build a position in a stock over a period of several years. That makes it tough to keep track of the cost basis on those shares and their relative changes in value.

The investor can calculate a weighted average of the share price paid for the shares. To do so, multiply the number of shares acquired at each price by that price, add those values, then divide the total value by the total number of shares.

## What is the weighted average?

Weighted average is an average in which each quantity to be averaged is assigned a weight. These weightings determine the relative importance of each quantity on average. Weightings are the equivalent of having that many like items with the same value involved in the average.

## Formula for Weighted average

Let xi be the observations and wi be the weights of the observations; the formula of the weighted average is given below.

To find the weighted term, multiply each term by its weighting factor, which is the number of times each term occurs.

### Solved Example

Example 1: A class of 25 students took a science test. 10 students had an average score of 80. The other students had an average score of 60. What is the average score of the whole class?

Solution:

Step 1: To get the sum of weighted terms, multiply each average by the number of students that had that average and then add them up.

80 × 10 + 60 × 15 = 800 + 900 = 1700

i.e. Sum of weighted terms = 1700

Step 2: Total number of terms = Total number of students = 25

Step 3: Using the formula,

Answer: The average score of the whole class is 68.

Example 2:

Calculate the weighted average for the following data:

 Data values 4 7 5 9 Weights 1 2 3 2

Solution:

From the given,

 Data values (xi) 4 7 5 9 Weights (wi) 1 2 3 2 wixi 4 14 15 18

∑wixi = 4 + 14 + 15 + 18 = 51

∑wi = 1 + 2 + 3 + 2 = 8

Weighted average = (∑wixi)/∑wi

= 51/8

= 6.375

Therefore, the weighted average of the given data is 6.375.

## Real-Life Examples on Weighted Average

A few real-life examples would help us better understand this concept of weighted average.

A teacher evaluates a student based on the test marks, project work, attendance, and class behavior. Further, the teacher assigns weights to each criterion, to make a final assessment of the performance of the student. The image below shows the weight of all the criteria that help the teacher in her assessment. The average of the weights helps in showing a clear picture.

A customer’s decision to buy or not to buy a product depends on the quality of the product, knowledge of the product, cost of the product, and service by the franchise. Further, the customer assigns weight to each of these criteria and calculates the weighted average. This will help him in making the best decision while buying the product.

For appointing a person for a job, the interviewer looks at his personality, working capabilities, educational qualification, and team working skills. Based on the job profile, these criteria are given different levels of importance(weights) and then the final selection is done.

## Weighted Average Formula

The weighted average formula is more descriptive and expressive in comparison to the simple average as here in the weighted average, the final average number obtained reflects the importance of each observation involved. In the weighted average, some data points in the data set contribute more importance to the average value, unlike in the arithmetic mean. It can be expressed as:

Weighted Average = Sum of weighted terms/Total number of terms

Let us look at an example to understand this better.

Example: The below table presents the weights of different decision features of an automobile. With the help of this information, we need to calculate the weighted average.

QuantityWeight
Safety – 8/1040%
Comfort – 6/1020%
Fuel mileage – 5/1030%
Exterior looks – 8/1010%

Solution: Let us now calculate the final rating of the automobile using the concept of weighted average.

Important Notes

1. The weights given to the quantities can be decimals, whole numbers, fractions, or percentages.
2. If the weights are given in percentage, then the sum of the percentage should be 100%.
3. Weighted average for quantities (x)i having weights in percentage (P)i% is:
Weighted Average = ∑ (P)i% × (x)i

## Weighted Average Examples

Example 2: In a 50 over cricket match, the average runs scored by a team for different sessions of the innings are given below. Find the average runs scored by the team in that innings.

First ten overs – 8 runs per over
10 to 35 overs – 5 runs per over
Last 15 overs – 9 runs per over

Solution:

To find: Average runs scored.
Given: Total overs = 50
= 8
= 5
= 9
= 10
= 25
= 15
Now, to find the sum of weighted terms, multiply the average runs scored in the respective session and then add them up.
Sum of weighted terms = × +× +×

= 8(10) + 5(25) + 9(15) = 80 + 125 + 135 = 340
Now, using the weighted average formula,
Weighted Average = Sum of weighted terms/Total number of terms
= 340/50
= 6.8

Therefore, the average runs scored in that innings by the team = 6.8.

Example 3: Ron has a supermarket and he earns a profit of $5000 from his groceries,$2000 from vegetables and $1000 from dairy products. He wants to predict his profit for the next month. He assigns weights of 6 to groceries, 5 to vegetables, and 8 to dairy products. Can you help Ron on how to calculate weighted average of his profits? Solution: Let us first present the profits and the weightage in a table. ProfitsWeights Groceries –$50006
Vegetables – $20005 Diary Products –$10008

Further applying the formula of weighted average to the above data, we have:

### What is Weighted Average Cost of Capital?

The weighted average cost of capital helps to find the capital value of the company. The capital includes fixed assets, cash in hand, goods, brand value. All of these are assigned certain weights and the weighted average formula is used to calculate the weighted average cost of capital.

### How to Calculate Weighted Average Using Weighted Average Formula?

To calculate the weighted average we need to follow the following steps given below:

• Observe the weight of individual items given in the problem
• Determine the individual weight of data or items given.
• Multiply the weight individually by each value and add the results together
• Now apply the weighted average formula that is (Sum of weighted terms/Total number of terms).

### How To Calculate the Sum of Weighted Terms Using the Weighted Average Formula?

If the weighted average of items is known along with a total number of terms then we can easily calculate the weighted average by:

• Determining the individual weight of items given.
• Multiplying the weight individually by each value and sum up the results together

### What Is the Use of Weighted Average Formula?

The weighted average formula is used to calculate the mean weighted value of the data with n terms. It is described as (Sum of weighted terms/Total number of terms).

## How to calculate weighted average in Excel

he tutorial demonstrates two easy ways to calculate weighted average in Excel – by using the SUM or SUMPRODUCT function.

In one of the previous articles, we discussed three essential functions for calculating average in Excel, which are very straightforward and easy-to-use. But what if some of the values have more “weight” than others and consequently contribute more to the final average? In such situations, you’ll need to calculate the weighted average.

Although Microsoft Excel doesn’t provide a special weighted average function, it does have a couple of other functions that will prove useful in your calculations, as demonstrated in the formula examples that follow.

## What is weighted average?

Weighted average is a kind of arithmetic mean in which some elements of the data set carry more importance than others. In other words, each value to be averaged is assigned a certain weight.

Students’ grades are often calculated using a weighted average, as shown in the following screenshot. A usual average is easily calculated with the Excel AVERAGE function. However, we want the average formula to consider the weight of each activity listed in column C.

In mathematics and statistics, you calculate weighted average by multiplying each value in the set by its weight, then you add up the products and divide the products’ sum by the sum of all weights.

In this example, in order to calculate the weighted average (overall grade), you multiply each grade by the corresponding percentage (converted to a decimal), add up the 5 products together, and divide that number by the sum of 5 weights:

((91*0.1)+(65*0.15)+(80*0.2)+(73*0.25)+(68*0.3)) / (0.1+0.15+0.2+0.25+0.3)=73.5

As you see, a normal average grade (75.4) and weighted average (73.5) are different values.

## Calculating weighted average in Excel

In Microsoft Excel, weighted average is calculated using the same approach but with far less effort because Excel functions will do most of the work for you.

### Calculating weighted average using SUM function

If you have basic knowledge of the Excel SUM function, the below formula will hardly require any explanation:

=SUM(B2*C2, B3*C3, B4*C4, B5*C5, B6*C6,)/SUM(C2:C6)

In essence, it performs the same calculation as described above, except that you supply cell references instead of numbers.

As you can see in the screenshot, the formula returns exactly the same result as the calculation we did a moment ago. Notice the difference between the normal average returned by the AVERAGE function (C8) and weighted average (C9).

Although the SUM formula is very straightforward and easy to understand, it is not a viable option if you have a large number of elements to average. In this case, you’d better utilize the SUMPRODUCT function as demonstrated in the next example.

### Finding weighted average with SUMPRODUCT

Excel’s SUMPRODUCT function fits perfectly for this task since it is designed to sum products, which is exactly what we need. So, instead of multiplying each value by its weight individually, you supply two arrays in the SUMPRODUCT formula (in this context, an array is a continuous range of cells), and then divide the result by the sum of weights:

=SUMPRODUCT(values_range, weights_range) / SUM(weights_range)

Supposing that the values to average are in cells B2:B6 and weights in cells C2:C6, our Sumproduct Weighted Average formula takes the following shape:

=SUMPRODUCT(B2:B6, C2:C6) / SUM(C2:C6)

To see the actual values behind an array, select it in the formula bar and press the F9 key. The result will be similar to this:

So, what the SUMPRODUCT function does is multiply the 1st value in array1 by the 1st value in array2 (91*0.1 in this example), then multiply the 2nd value in array1 by the 2nd value in array2 (65*0.15 in this example), and so on. When all of the multiplications are done, the function adds up the products and returns that sum.

To make sure that the SUMPRODUCT function yields a correct result, compare it to the SUM formula from the previous example and you will see that the numbers are identical.

When using either the SUM or SUMPRODUCT function to find weight average in Excel, weights do not necessarily have to add up to 100%. Nor do they need to be expressed as percentages. For example, you can make up a priority / importance scale and assign a certain number of points to each item, as demonstrated in the following screenshot:

### Example

Let’s take a simple weighted average formula example to illustrate how we calculate a weighted average.

Ramen has invested an amount into four types of investments: 10% in Investment A, 20% in Investment B, 30% in Investment C, and 40% in Investment D. The rates of return for these investments are 5%, 10%, 15%, and 20%. But, first, calculate the weighted average of the rates of return Ramen would receive.

In this weighted average example, we have both w and x.

Using the weighted average formula, we get the following:

• Weighted Avg = w1x+ w2x+ w3x+ w4x4
• Weighted Avg = 10% * 5% + 20% * 10% + 30% * 15% + 40% * 20% = 0.005 + 0.02 + 0.045 + 0.08 = 15%.

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