Weighted Average Formula

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

This equals a weighted average cost of $1.18 per unit.

How to calculate weighted average

Weighted average differs from finding the normal average of a data set because the total reflects that some pieces of the data hold more “weight,” or more significance, than others or occur more frequently. You can calculate the weighted average of a set of numbers by multiplying each value in the set by its weight, then adding up the products.

For a more in-depth explanation of the weighted average formula above, follow these steps:

Determine the weight of each data point

You determine the weight of your data points by factoring in which numbers are most important. Teachers often weigh tests and papers more heavily than quizzes and homework, for example.

In large statistical data sets, such as consumer behavior data mining or a population census, randomized data trees are used to determine the importance of a variable in a data set. This helps ensure the distribution of importance is unbiased.

This process is typically performed with the aid of a computer program. For accounting and finance purposes, the number of units of a product is used as the weighing factor.

Example:

• You score a 76 on a test that is 20% of your final grade. The percentage of your grade is the weight it carries.

• An investor purchases 50 stocks at $100 each. The stocks purchased serve as the weight.

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.

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.

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.

For example, say an investor acquires 100 shares of a company in year one at $10, and 50 shares of the same stock in year two at $40. To get a weighted average of the price paid, the investor multiplies 100 shares by $10 for year one and 50 shares by $40 for year two, then adds the results to get a total of $3,000. Then the total amount paid for the shares, $3,000 in this case, is divided by the number of shares acquired over both years, 150, to get the weighted average price paid of $20.

This average is now weighted with respect to the number of shares acquired at each price, not just the absolute price.

How does a weighted average differ from a simple average?

How is a weighted average calculated?

What is a Weighted Average?

A weighted average is a type of mean that gives differing importance to the values in a dataset. In contrast, the regular average, or arithmetic mean, gives equal weight to all observations. The weighted average is also known as the weighted mean, and I’ll use those terms interchangeably.

Use a weighted mean when you must consider the relative significance of values in a dataset. In other words, you’re placing different weights on the values in the calculations.

For example, use a weighted average in the following situations:

• A professor weights projects, exams, and quizzes to reflect varying difficulty.
• An investor weights the share price by the number of stocks they purchase to reflect the changing prices.

In these examples, a weighted average gives differing importance to each value according to relevant criteria.

Weighted Average Formula

Calculating the mean is a simple process of summing all your values and dividing them by the number of values. That process gives each value an equal weight.

Now let’s see how that procedure contrasts with the weighted average calculation.

The weighted average formula is the following:

Where:

• w = the weight for each data point.
• x = the value of each data point.

Calculating the weighted average involves multiplying each data point by its weight and summing those products. Then sum the weights for all data points. Finally, divide the weight*value products by the sum of the weights. Voila, you’ve calculated the weighted mean!

Two broad calculation cases exist when using the weighted average formula:

• The weights sum to 1.
• They don’t sum to 1.

Notice how you divide the products by the sum of the weights in the denominator. Consequently, when the weights sum to one, the weighted average simply equals the sum of the products in the numerator. However, you’ll need to perform the division when the denominator does not equal one.

Let’s use the weighted mean formula to work through two examples.

Example Weighted Average Calculations

We’ll start with an example where the weights sum to one. This situation frequently occurs when someone intentionally builds the weighted average into a process. For example, a teacher might devise a grading system using weights and, for simplicity, has the weights sum to one.

The teacher has given weights ranging from 0.05 for quizzes to 0.4 for the group project. Because the teacher devised the weights to equal one, it’s easy to understand the importance of each observation. For instance, the group project accounts for 40% of the grade!

Let’s calculate the weighted mean for one student’s grade! In the column headers, I use notation that matches the weighted average formula above.

The student did well on the quizzes and exams but not so well on the group project. The resulting weighted average is 79.70.

The regular mean is 84.5, but the considerable importance of the group project brought their weighted average grade down to 79.7. Ouch!

Example Weighted Mean Calculations

Next, let’s work through an example where the weights don’t sum to one.

An investor is building up a particular stock in his portfolio. He purchases the same stock at different prices over time. He can calculate the weighted mean for the average share price. In this example, the prices are the values, and the numbers of stocks are the weights.

Here, the weights sum to 125. Consequently, we need to divide the sum of the products (2,985) by 125. The weighted average price per stock is $23.88.

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 x_i be the observations and w_i 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:

Solution:

From the given,

∑w_ix_i = 4 + 14 + 15 + 18 = 51

∑w_i = 1 + 2 + 3 + 2 = 8

Weighted average = (∑w_ix_i)/∑w_i

= 51/8

= 6.375

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

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.

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.

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
$w_1$= 8
$w_2$ = 5
$w_3$ = 9
$x_1$ = 10
$x_2$ = 25
$x_3$ = 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 = $w_1$ ×$x_1$ +$w_2$× $x_2$ +$w_3$× $x_3$

= 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.

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

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:

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 1^st value in array1 by the 1^st value in array2 (91*0.1 in this example), then multiply the 2^nd value in array1 by the 2^nd 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 = w₁x₁ + w₂x₂ + w₃x₃ + w₄x₄
• Weighted Avg = 10% * 5% + 20% * 10% + 30% * 15% + 40% * 20% = 0.005 + 0.02 + 0.045 + 0.08 = 15%.

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