Basic Algebra/Proportions and Proportional Reasoning/Weighted Averages

What Is a Weighted Average? Weighted average is a calculation that takes into account the varying degrees of importance of the numbers in a data set. In calculating a weighted average, each number in the data set is multiplied by a predetermined weight before the final calculation is made.

Key takeaways:


 * Weighted average is the average of a set of numbers, each with different associated “weights” or values.
 * To find a weighted average, multiply each number by its weight, then add the results.
 * If the weights don’t add up to one, find the sum of all the variables multiplied by their weight, then divide by the sum of the weights.

The weighted average method is a tool used in classrooms, statistical analysis and accounting offices, among others. A weighted average helps the user gather a more accurate look at a set of data than the normal average alone. The accuracy of the numbers you arrive at with this method is determined by the weight you give specific variables in the data set.

In this article, we explore how to calculate weighted average using two methods.

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:


 * 1) Determine the weight of each data point
 * 2) Multiply the weight by each value
 * 3) Add the results of step two together

1. Determine the weight of each data point
You determine the weight of your data points by factoring 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 weighting 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.

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2. 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

3. 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:


 * 1) Determine the weight of each number
 * 2) Find the sum of all weights
 * 3) Calculate the sum of each number multiplied by its weight
 * 4) Divide the results of step three by the sum of all weights

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1. 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 the period of 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

2. 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

3. 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

4. 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.