Understanding the Weighted Mean
A standard average (arithmetic mean) assumes that every data point in a set is equally important. However, in many real-world scenarios, some data points contribute more to the final outcome than others. A weighted mean solves this by multiplying each data point by a specific "weight" before averaging them out.
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The Mathematical Formula
To find the weighted average, you multiply each value ($x_i$) by its assigned weight ($w_i$), add those products together, and then divide by the sum of all the weights:
$\bar{x} = \frac{\sum_{i=1}^{n} w_i x_i}{\sum_{i=1}^{n} w_i}$
Where:
$\bar{x}$ = The Weighted Mean
$w_i$ = The weight of the $i$-th value
$x_i$ = The $i$-th data value
$\Sigma$ = The sum of
Common Real-World Examples
- School Grades: A final exam is often worth 30% of a final grade, while homework is only worth 10%. In this case, the test score carries a higher "weight" than a homework assignment.
- Finance & Portfolios: When evaluating the average return of an investment portfolio, investments with a larger amount of capital invested carry a higher weight than smaller investments.
- Statistics & Expected Value: In probability, the expected value of a random variable is essentially the weighted average of all possible outcomes, where the "weights" are the probabilities of each outcome occurring. Read more about Weighted arithmetic means on Wikipedia.