We need to calculate averages for many purposes such as for determining the trends, taking some cost decisions, profit decisions, etc. The measures of determining averages are the measures of central tendency. The arithmetic mean, Geometric mean, Median and mode are some of the common measures. While calculating the average using the Arithmetic mean, we can either use simple mean formula or the weighted average formula. The calculation of the weighted average is a little bit more complex than that of the simple average. However, here we shall discuss the weighted average formula. Let’s begin learning!
Weighted Average Formula
What is Weighted Average?
Weighted average or weighted mean is an average or mean in which we assign weights to each of the quantity or value. We assign the weights according to the relative importance of each value or quantity. For example, we need to calculate the overall rank of a student who has scored marks in different subjects. Now, each subject has a different percentage of the total marks. Say, the written test has 50%, the practical test has 30% and sports have 20% weightage in total grade marks. So, the weight of the written test will be 0.5, the practical test will be 0.3 and sports will be 0.2.
Where all the observations or values have the same weight, then the weighted mean and arithmetic mean are the same. The weights that we assign to the values cannot be negative but can be zero. However, not all weights can be zero. Also, the value having the highest weight will have more effect on the weighted average and vice-versa.
Formula for Weighted Average:
\(\bar{x}\) = \(\frac{\sum_{i=1}^{n}{w_i x_i}}{\sum_{i=1}^{n}{w_i}}\)
Where,
| x | Observations or values |
| w | Weights of the observations or values |
| \(\bar{x}\) | Weighted mean or average |
In simple words, the formula can be:
Weighted Average = \(\frac{Sum of weighted observations}{Sum of weights}\)
Steps for Calculating the Weighted Average
- Collect the values or numbers for which we need to calculate the weighted average. We know these values as ‘x’.
- Determine and assign the weights to the values according to their importance. We know these weights as ‘w’.
- Arrange the values ‘x’ and the weights ‘w’ in a table.
- Multiply each weight ‘w’ with its corresponding value ‘x’. After multiplying, we get ‘xw’.
- Find the sum of all weights \(\sum w’ and ‘\sum xw\)
- Lastly, apply the formula and find the weighted average by dividing \(\sum xw by \sum w\)
Solved Examples
Q.1. Calculate the weighted average from the following data:
| Number of vehicles per household | Number of households |
| 1 | 500 |
| 2 | 450 |
| 3 | 300 |
| 4 | 200 |
| 5 | 100 |
Solution:
| No. of vehicles per household (x) | No. of households (w) | xw |
| 1 | 500 | 500 |
| 2 | 450 | 900 |
| 3 | 300 | 900 |
| 4 | 200 | 800 |
| 5 | 100 | 500 |
| \sum w = 1550 | \sum xw = 3600 |
\(\bar{x} = \frac{\sum_{i=1}^{n}{w_i x_i}}{\sum_{i=1}^{n}{w_i}}\)
\(= \frac{3600}{1550}\)
= 2.32
This implies that on an average each household has 2.32 vehicles.
Q.2. The commerce batch of a school has two sections, A and B. The number of students in section A is 25 and their average marks are 75 whereas section B has 30 students and their average marks are 60. Find the average marks of the whole batch.
Solution: In this question, we need to first find x1w1 and x2w2.
So, x1w1 = \(25 \times 75 = 1875\)
\(x2w2 = 30 \times 60 = 1800\)
We know that, w1 = 25 and w2 = 30.
\(Weighted Average = \frac{Sum of weighted observations}{Sum of weights}\)
= \(\frac{x1w1 +x2w2}{w1 + w2}\)
= \(\frac{3675}{55}\)
= 66.81
I get a different answer for first example.
I got Q1 as 20.5
median 23 and
Q3 26