What if the weights are zero or negative?

Weights cannot be negative in standard descriptive statistics. A weight of exactly zero means the corresponding data value is completely excluded from the calculation.

Do weights have to sum to 1 or 100?

No, weights can be any positive numbers, such as course credit hours (e.g., 3 credits, 4 credits). The step of dividing by the sum of the weights normalizes the result regardless of the scale.

What happens if I use unweighted mean by mistake?

An unweighted mean implicitly assumes all weights are equal ($w_i = 1$). This will skew the result toward values that should have had lesser importance.

How to calculate a weighted mean

The weighted mean is computed by multiplying each data value by its corresponding weight and dividing the sum of these products by the total sum of the weights. This method applies when different data points contribute unequally to the final average, such as calculating a course grade from assignments of varying importance.

The setup

Identify the data values $x_i$ and their corresponding weights $w_i$ . Ensure that every value has a defined numerical weight. The formula for the weighted mean is $\bar{x}_w = \frac{\sum x_i w_i}{\sum w_i}$ .

The steps

Multiply each value by its weight to find $x_i w_i$ . 2. Sum these products to find the total weighted value $\sum x_i w_i$ . 3. Sum all the weights to find the total weight $\sum w_i$ . 4. Divide the total weighted value by the total weight.

Checking the result

The calculated weighted mean must strictly fall between the minimum and maximum values of your dataset $x_i$ . If the weights already sum to 1 (or 100%), the denominator is 1 and the formula simplifies to the sum of the products.

Common errors

A frequent error is dividing the sum of the products by the number of data points $n$ rather than the sum of the weights $\sum w_i$ . Another error is mixing decimal and percentage representations of weights in the same calculation.

Worked example

Find the final grade for a student with the following scores: Homework is 90 with a weight of 20%, Midterm is 80 with a weight of 30%, and the Final is 95 with a weight of 50%.

First, multiply the scores by their weights: Homework is $90 imes 20 = 1800$ , Midterm is $80 imes 30 = 2400$ , and Final is $95 imes 50 = 4750$ . Second, sum the products: $1800 + 2400 + 4750 = 8950$ . Third, sum the weights: $20 + 30 + 50 = 100$ . Finally, divide the sum of the products by the sum of the weights: $\bar{x}_w = 8950 / 100 = 89.5$ . The final grade is 89.5.

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References: OpenStax Introductory Statistics, Chapter 2 · Khan Academy: Summarizing quantitative data unit