Weighted Average

A weighted average is a type of average in which each data point is assigned a weight based on its importance or frequency. Before calculating the final average, every value is multiplied by its weight.
In a simple average, all values are equal. But in a weighted average, the values with higher importance or frequency influence the result more. This makes the weighted average a more accurate way to represent in real-life situations where not all numbers matter equally.

Formula for Weighted Average:

Weighted Average =  \(\frac{\sum w_i\ \times\ x_i}{\sum w_i}\)

Where:
w_i  = weight of each value
x_i = the value
∑ = sum of all terms

If the weights add up to 1 (or 100%), the formula becomes:

Weighted Average =  \((w_i \times\ x_i)\)

For example: A buyer wants to rate a laptop based on different essential features. Each feature has a score out of 10 and a weight reflecting its importance to the buyer.
  

Solution:

Weighted Average:

\(= 50\% \times \frac{9}{10}\)
 

\(25\% \times \frac{6}{10}\)
 

\(15\% \times \frac{7}{10}\)
 

\(10\% \times \frac{8}{10}\)
 

Convert percentages to decimals:
 

\(= 0.5 × 0.9\)
 

\(0.25 × 0.6\)
 

\(0.15 × 0.7\)
 

\(0.1 × 0.8\)
 

Now multiply:
 

\(= 0.45\)
 

\(0.15\)
 

\(0.105\)
 

\(0.08\)
 

Add them up:
 

= 0.785
 

Convert to a rating out of 10:

\(= \frac{7.85}{10}\)
 

Now, let’s see how to calculate the weighted average. It is found by dividing the sum of the weighted values by the total weight. The weighted terms are the product of the value with the assigned weight.

So, weighted average = Sum of weighted terms ÷ Total weight.

In other words, if we consider the terms as\(x_1, x_2, x_3, x_4, \ldots, x_n\) and the assigned weight as \(w_1, w_2, w_3, w_4, \ldots, w_n\). Then the weighted average = \(\frac{x_1 w_1 + x_2 w_2 + x_3 w_3 + x_4 w_4 + \ldots + x_n w_n} {w_1 + w_2 + w_3 + w_4 + \ldots + w_n}\). It can be simplified into, 

Weighted Average = \(\frac{\sum (w_i \times x_i)}{\sum w_i}\)

Now, let's see the step-by-step process of finding the weighted average. 

Step 1: Arrange the data 

Step 2: Finding the weighted term, that is, the product of the value with the weight. Then, sum up the weighted values 

Step 3: Find the total number of terms

Step 4: Divide the sum of weighted values by the total weight 

For example, the grades of the students in the assessments and the weightage are given below, find the weighted average.

Step 1: Arrange the data for easier calculation

Step 2: Finding the weighted term, that is, the product of the value with the weight. Then, sum up the weighted values. 

The weighted term of homework = \(85 × 0.2 = 17\)

The weighted term of midterm = \(78 × 0.3 = 23.4\)

The weighted term of final exam = \(92 × 0.5 = 46\)

The sum of the weights values = \(17 + 23.4 + 46 = 86.4\)
 

Step 3: Identify the overall weight

Here, the total weight = \(0.2 + 0.3 + 0.5 = 1\)

Step 4: Divide the sum of the weighted values with the total number of values. 

Weighted average = the sum of the weighted values / total number of values 
= \(\frac{86.4}{1} = 86.4\).

There are different methods to find the average based on the purpose. For instance, we can calculate the weighted average, arithmetic mean, and geometric mean based on the objective. Now let’s discuss the difference between them:

• Use Proportions for Simplicity: If weights are given as percentages, make sure they add up to 100. This saves calculation errors.
   
• Break Down Large Problems: For multiple items, calculate weighted sums in smaller groups first, then combine them.
   
• Remember: Higher Weight = Higher Impact: The number with the highest weight influences the average more strongly. Keep this in mind when estimating answers.
   
• Check with Simple Cases: Always test your result against a smaller scenario (like two numbers) to see if the weighted average falls between the lowest and highest values.
   
• Use Intuition with Extreme Values: When calculating a weighted average, the result will always lie between the smallest and largest values. If your answer falls outside this range, it’s definitely incorrect. This quick check saves time and prevents calculation mistakes.
   

• Use everyday situations, such as calculating the average marks for subjects with different weightings or finding the average grocery cost, to illustrate why weighted averages matter.
   

• Help children understand that some numbers may count more than others. For example, in exams, final tests may have more weight than assignments.
   

• Parents and teachers can use an online weighted average calculator to double-check answers or demonstrate how the formula works in real time. This builds confidence and accuracy.

Now, let’s explore what are the applications of weighted averages. 

• The weighted average is used to find the average when one value has more importance than the other. That is, it helps to compare the value when each has different importance. 

• To handle the skewed distribution and outliers, we use weighted averages.

• The weighted average is widely used as it gives flexibility in the application in various fields. 

• To track the basic cost of the investment, we use a weighted average

• To make decisions and analyze data, we use weighted averages.