Finding the weighted average is a mathematical calculation used to determine the average of a set of numbers that have different levels of significance or “weight”. It’s a bit more complex than calculating a simple arithmetic mean but with some basic math skills, you can easily figure out a weighted average.
In this comprehensive guide we’ll walk through the steps and examples to help you understand the weighted average formula and how to apply it in real-world situations, whether you’re calculating a grade average or analyzing financial data.
What is a Weighted Average?
A weighted average is an average calculated by giving some values in a data set more influence or weight in determining the final average. It takes into account the importance, or weight, of each data value when calculating the average.
For example, say you took 3 tests in math class. On the first test you scored 80%, on the second test you scored 90%, and on the final exam you scored 75%. However, the first two tests were worth 15% of your grade each, while the final exam was worth 70% of your total grade.
To find your overall average, you can’t just take the arithmetic mean of your 3 scores. You have to account for the different weights assigned to each test. This is where calculating a weighted average comes in handy.
Why Use Weighted Average?
Weighted averages are useful any time you need to average data values that don’t contribute equally to the overall average. The weighted average formula lets you assign varying degrees of importance to each value before averaging them.
Some common examples where weighted averages are used include:
- Calculating overall grades when assignments are worth different percentages
- Computing a stock portfolio’s performance when positions have different values
- Determining average salary when employee pay varies significantly
- Finding a city’s average home price when neighborhoods have diverse values
- Averaging product ratings when some reviews are more credible than others
By applying weights, you can ensure that values with greater importance or reliability have more influence over the final average.
How to Calculate Weighted Average Step-by-Step
Follow these steps to easily calculate a weighted average:
1. Identify the Numbers to Average
First, gather the full set of numbers you want to average. For example, let’s say you want to calculate your overall grade based on 3 test scores:
- Test 1 score: 82%
- Test 2 score: 90%
- Final exam score: 76%
Write down the numbers you’ll be averaging.
2. Determine the Weight of Each Number
Next, identify the relative weight of each number in the average. Usually, the weights are expressed as percentages or decimals that add up to 1 or 100%.
For our example grade calculation, let’s assign weights as follows:
- Test 1 weight: 15% or 0.15
- Test 2 weight: 15% or 0.15
- Final exam weight: 70% or 0.70
The weights add up to 100% (0.15 + 0.15 + 0.70 = 1).
3. Multiply Each Number by Its Weight
To apply the weights, multiply each number by its corresponding weight:
- Test 1: 82 x 0.15 = 12.3
- Test 2: 90 x 0.15 = 13.5
- Final exam: 76 x 0.70 = 53.2
This scales each number proportional to its importance in the overall average.
4. Sum the Weighted Values
Add up the weighted values to find the weighted average:
- 12.3 + 13.5 + 53.2 = 78.5
Therefore, your weighted grade average is 78.5%.
And that’s it! By following these 4 steps, you can easily calculate a weighted average for any set of numbers.
Weighted Average Formula
The weighted average formula expresses the mathematical steps above concisely:
Weighted Average = (x1 * w1) + (x2 * w2) + … + (xn * wn)
Where:
- x1, x2, …, xn are the data values
- w1, w2, …, wn are the corresponding weights of each value
- n is the total number of values
This formula lets you efficiently calculate the weighted average in one step once you have the values and weights.
Weighting Factors That Don’t Add to 1
Sometimes, the weights don’t necessarily add up to 1 or 100%. In this case, you adjust the formula slightly:
Weighted Average = (x1 * w1) + (x2 * w2) + … + (xn * wn) / Total of all weights
The only difference is dividing the sum of the weighted values by the total of all weights.
For example, say you took 4 tests with the following scores and weights:
- Test 1: 82%, weight = 2
- Test 2: 90%, weight = 2
- Test 3: 76%, weight = 1
- Test 4: 68%, weight = 3
The weights add up to 2 + 2 + 1 + 3 = 8.
To find the weighted average:
- (82 x 2) + (90 x 2) + (76 x 1) + (68 x 3) = 328
- 328 / 8 = 82
The weighted average test score is 82%.
Weighted Average Examples
Let’s look at a few more examples of calculating weighted averages for class grades, stock portfolios, and statistical samples:
Weighting Class Test Scores
Maggie takes 4 tests in history class with the following scores and weights:
- Test 1: 93%, weight 15%
- Test 2: 81%, weight 15%
- Test 3: 77%, weight 10%
- Final Exam: 84%, weight 60%
What is her weighted average test score?
(93 x 0.15) + (81 x 0.15) + (77 x 0.1) + (84 x 0.6) = 83.3
Her weighted test average is 83.3%
Weighting Stock Positions
Bob invests money in 3 stocks with the following portfolio weights:
- Stock A: $5,000, weight 30%
- Stock B: $8,000, weight 50%
- Stock C: $2,000, weight 20%
What is the weighted value of Bob’s portfolio?
($5,000 x 0.3) + ($8,000 x 0.5) + ($2,000 x 0.2) = $7,400
The weighted value of the portfolio is $7,400. Stock B has the highest weight, so it contributes most to the portfolio value.
Weighting Survey Responses
A survey asks 100 people to rate a product from 1 to 5 stars. The results are:
- 40 people give 5 stars
- 30 give 4 stars
- 20 give 3 stars
- 7 give 2 stars
- 3 give 1 star
What is the weighted average rating?
*(40 x 5) + (30 x 4) + (20 x 3) + (7 x 2) + (3 x 1) = 432
Weighted average = 432 / 100 = 4.32 stars
The weighted average accounts for the different frequencies of each rating.
Weighted Averages in Spreadsheets
Many spreadsheet programs like Excel have built-in functions to easily calculate weighted averages.
For example, Excel’s SUMPRODUCT function performs the weighting multiplication and summation in one step:
=SUMPRODUCT(values, weights) / SUM(weights)Where values and weights are the ranges containing the numbers and weights to average.
Spreadsheet tools make finding weighted averages a breeze.
When to Use Weighted Averages
- Grade calculations – Weight test and assignment scores
- Stock portfolios – Weight asset values by percentage
- Indexes – Weight components proportional to market cap
- Customer surveys – Weight responses by sample size
- Economic indicators – Weight sub-metrics by relevance
- Research studies – Weight data points by reliability
Anytime values being averaged contribute disproportionately, apply weights to calculate a more representative average.
Limitations of Weighted Averages
While useful in many cases, weighted averages also have some limitations to be aware of:
- Subjective weighting – Weights can be arbitrary and lack statistical meaning
- Weighting distortions – Strong weights on outliers can skew results
- Complex calculations – Weighting requires more work than simple averages
- Lack of transparency – May not be clear how weights were determined
- Difficult comparisons – Weighted averages can’t easily be compared
When these drawbacks apply, an unweighted mean may be preferred for simplicity.
By following the 4-step weighted average calculation process, you can easily find the weighted mean of any data set. This average accounts for varying levels of importance assigned to each data value through weighting factors. Just be sure the weights accurately reflect the relative significance of each number for your situation.
Weighted averages have many useful applications in statistics, business, economics, and more. But they also have some limitations to consider before use. With a solid grasp of the weighted average formula though, you’ll be prepared to apply it
Advantages and Disadvantages of Weighted Average
Weighted average provides a more accurate representation of data when different values within a dataset hold varying degrees of importance. By assigning weights to each value based on their significance, weighted averages ensure that more weight is given to data points that have a greater impact on the overall result. This allows for a more nuanced analysis and decision-making process.
Next, weighted averages are particularly useful for handling skewed distributions or outliers within a dataset. Instead of being overly influenced by extreme values, weighted averages take into account the relative importance of each data point. This means you can “manipulate” your data set so its more relevant, especially when you dont want to consider extreme values.
Thirdly, weighted averages offer flexibility in their application across various fields and disciplines. Whether in finance, statistics, engineering, or manufacturing, weighted averages can be customized to suit specific needs and objectives. For instance, like we discussed above, weighted averages are commonly used to calculate portfolio returns where the weights represent the allocation of assets. Weighted averages can also be used in the manufacturing process to determine the right combination of goods to use.
How Does a Weighted Average Differ From a Simple Average?
A weighted average accounts for the relative contribution, or weight, of the things being averaged, while a simple average does not. Therefore, it gives more value to those items in the average that occur relatively more.
How to Calculate Weighted Average
How do you find a weighted average?
Once you’ve multiplied each number by its weighting factor and added the results, divide the resulting number by the sum of all the weights. This will tell you the weighted average. For example: 98/15 = 6.53. This means you slept an average of 6.53 hours each night over the course of 15 weeks. How do I find a weighted average?
How can one know if they have an ideal weight?
The BMI is the tool most commonly used to estimate and screen for overweight and obesity in adults. BMI is a measure based on your weight in relation to your height. You can easily calculate your BMI(your weight in kgs divided by the square of height in metres) Normal or healthy weight: A person with a BMI of 18. 5 to 24. 9 is in the normal or healthy range. Overweight: A person with a BMI of 25 to 29. 9 is considered overweight. Obesity: A person with a BMI of 30 to 39. 9 is considered to have obesity. Extreme obesity:A person with a BMI of 40 or greater is considered to have extreme obesity. Because BMI doesn’t measure actual body fat, a person who is very muscular, like a bodybuilder, may have a high BMI without having a lot of body fat. Also, some groups who tend to have a lower BMI, such as Asian men and women or older adults, may still have high amounts of body fat even if they are not overweight.
How does a weighted average calculator work?
To understand how a weighted average calculator works, you must first understand what a weighted average is. Weighted average has nothing to do with weight conversion, but people sometimes confuse these two concepts. The typical average, or mean, is when all values are added and divided by the total number of values.
What is a weighted average?
As the name suggests, a weighted average is one where the different numbers you’re working with have different values, or weights, relative to each other. For example, you may need to find a weighted average if you’re trying to calculate your grade in a class where different assignments are worth different percentages of your total grade.
