Weighted Sum: Definition, Formula

What is a Weighted Sum?

A weighted sum accounts for the relative importance or influence of individual data points.

While a simple summation has each data point contribute equally, a weighted sum assigns a specific weight to each value, giving certain data points more “weight”.

A weighted sum is calculated by multiplying each data point by a weight and then summing the products.

The formula is:

weighted sum formula

Where:

S = weighted sum.
x_i = each data point in the dataset.
w_i = weight assigned to each data point.
n = total number of data points.

The weights w_i are usually non-negative real numbers and may or may not sum up to one, depending on the context of the analysis. For example:

Let’s say the weights represent the proportion of total investment in each fund. In this case they would naturally sum to one (i.e. 100% of investments).
On the other hand, let’s say you were performing an analysis on heteroscedastic data. You adjust for this by giving less weight to observations with higher variance (less reliability). The weights in this case do not sum to one and are used to minimize the weighted sum of squared residuals in the regression.

Determining Weights

Weights can be determined based on a variety of factors. For example:

Expert Judgment: Using your personal knowledge of the subject area to assign importance to particular data points.
Frequency or Probability: Using the frequency of occurrence of events/data points or probability of events as weights.
Economic Significance: You could reflect specific considerations such as cost or revenue in deciding what weights to give data.

Best Practices

Weighted sums should be used with the following points in mind to mitigate errors and bias:

Justify your Weight Selection: Clearly explain the rationale behind the chosen weights to ensure transparency in your documentation. For example, if your subject matter knowledge leads you to minimize weights for one data point, say why in your analysis.
Normalize Weights: Normalize weights so that they add up to one. This simplifies interpretation and comparison with other data.
Sensitivity Analysis: Run your results through sensitivity analysis (“What if…?”) to test how changes in weights might affect the outcome. Done correctly, this should allow you to assess the robustness of the results.