How do you compute a weighted mean of data points with weights w_i?

How do you compute a weighted mean of data points with weights w_i?

• A. Weighted mean = (Σ w_i x_i) / (Σ w_i)
• B. Weighted mean = (Σ x_i) / n
• C. Weighted mean = Σ w_i x_i
• D. Weighted mean = Σ w_i / Σ x_i

Answer: (C)

Weighted mean is the average where each value is scaled by how important it is, through its weight. To get it, multiply each data point by its weight, add all those products, and then divide by the total of the weights. That gives (Σ w_i x_i) / (Σ w_i). Dividing by the total weight is important because it normalizes the influence of all observations so the result stays on the same scale as the data. If you only sum the products (Σ w_i x_i) you haven’t normalized by the total weight, so you don’t obtain the true average unless the weights happen to sum to 1. When all weights are 1, the weighted mean becomes the ordinary average, (Σ x_i)/n.

Weighted mean is the average where each value is scaled by how important it is, through its weight. To get it, multiply each data point by its weight, add all those products, and then divide by the total of the weights. That gives (Σ w_i x_i) / (Σ w_i). Dividing by the total weight is important because it normalizes the influence of all observations so the result stays on the same scale as the data. If you only sum the products (Σ w_i x_i) you haven’t normalized by the total weight, so you don’t obtain the true average unless the weights happen to sum to 1. When all weights are 1, the weighted mean becomes the ordinary average, (Σ x_i)/n.