A weighted mean is an average that gives different amounts of influence to different values. It matters whenever some data points count more than others, such as course grades, prices based on quantities, or survey results from groups of different sizes. A simple mean treats every value equally, but a weighted mean reflects importance, frequency, or size.
This makes it a more accurate summary when the data do not all have the same weight.
The weighted mean is found by multiplying each value by its weight, adding those products, and then dividing by the total weight. The weights can be points, credits, percentages, quantities, or frequencies, as long as they match the meaning of the problem. For example, a final grade should count more than a short quiz if it has a larger weight in the course.
In pricing, buying more of one item pulls the average price closer to that item's price.
Key Facts
- Weighted mean = (sum of value times weight) / (sum of weights)
- Formula: x̄w = (w1x1 + w2x2 + ... + wnxn) / (w1 + w2 + ... + wn)
- A simple mean is used when all values have equal weight.
- If weights are percentages, their total should usually be 100% or 1.
- A value with a larger weight pulls the weighted mean closer to itself.
- For grouped data with frequencies, weighted mean = (sum of value times frequency) / total frequency.
Vocabulary
- Weighted mean
- An average in which each data value is multiplied by a weight before the average is calculated.
- Weight
- A number that shows how much importance, frequency, or influence a data value has.
- Simple mean
- An average found by adding all values and dividing by the number of values, with every value counted equally.
- Frequency
- The number of times a value or category appears in a data set.
- Weighted sum
- The total found by adding the products of each value and its weight.
Common Mistakes to Avoid
- Averaging the values without using the weights is wrong because it treats all data points as equally important.
- Dividing by the number of data values instead of the sum of the weights is wrong because the weights determine the total amount being averaged.
- Using percentages as whole numbers inconsistently is wrong because 25% must be treated as either 25 with all weights summing to 100 or 0.25 with all weights summing to 1.
- Forgetting to multiply each value by its own weight is wrong because the weighted mean depends on matching every value with its correct influence.
Practice Questions
- 1 A student's grade is based on homework 20%, quizzes 30%, and a final exam 50%. The scores are 90, 80, and 86. What is the weighted mean grade?
- 2 A store sells 3 notebooks at 5 each. What is the weighted mean price per notebook?
- 3 A class has one test worth 10% and another test worth 90%. Explain why a simple mean of the two test scores may not represent the student's actual course performance.