What is the formula for the weighted mean?

• A. sum(w_i * x_i) / sum(w_i)
• B. sum(x_i) / n
• C. sum(w_i) * x_i
• D. (sum w_i) / sum(x_i)

Answer: (A)

When you want a mean that reflects different levels of importance, you weight each value by its weight and then normalize by the total weight. Multiply each data point by its weight, sum those products, and divide by the sum of all weights. This gives you a measure that gives more influence to values with higher significance.

This is why the formula sum(w_i * x_i) / sum(w_i) is the correct one. The numerator accounts for each value’s influence, and the denominator scales the total so the result stays in the same units as the x values. If every weight is the same, this reduces to the ordinary mean, since you're effectively counting each observation equally.

For example, with values 2, 5, and 4 and weights 3, 1, and 2, the weighted mean would be (32 + 15 + 2*4) / (3+1+2) = 19/6 ≈ 3.17. This can differ from the simple average, which would be (2+5+4)/3 = 11/3 ≈ 3.67, showing how weights shift the result toward more influential values.

The other forms don’t fit the concept: dividing by the number of observations assumes equal weighting; summing the weighted values without dividing by total weight ignores normalization; and taking the ratio of total weight to total value isn’t an average at all.