Weighted Mean Calculator

Enter the values and weights into the calculator to determine the weighted mean. This calculator can also evaluate any of the variables given the others are known.

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Weighted Mean Formula

A weighted mean is an average that gives different levels of importance to different values. Instead of treating every observation equally, each value is multiplied by its assigned weight before the average is computed.

WM = \frac{\sum_{i=1}^{n} w_i x_i}{\sum_{i=1}^{n} w_i}

• WM = weighted mean
• w_i = weight of each value
• x_i = value being averaged
• n = number of values

If your weights are already normalized so their total equals 1, the denominator simplifies and the calculation becomes even shorter:

WM = \sum_{i=1}^{n} w_i x_i \quad \text{when } \sum_{i=1}^{n} w_i = 1

The weighted mean has the same units as the original values. The weights themselves only control influence; they do not change the unit of the result.

How to Calculate a Weighted Mean

1. List each value and its matching weight.
2. Multiply every value by its corresponding weight.
3. Add all weighted products together.
4. Add all weights together.
5. Divide the total weighted products by the total weight.

This process is useful when some observations matter more than others, such as grade categories, portfolio allocations, survey adjustments, or prices tied to different quantities.

Example

Suppose the values are 10, 15, and 20, with weights 0.2, 0.3, and 0.5. The weighted mean is:

WM = \frac{(0.2)(10) + (0.3)(15) + (0.5)(20)}{0.2 + 0.3 + 0.5}

WM = \frac{2 + 4.5 + 10}{1.0} = 16.5

Because the largest weight is attached to the value 20, the final average is pulled upward toward 20 more than it is toward 10 or 15.

Solving for a Missing Input

This calculator is especially useful when the weighted mean is known and one input is missing. Rearranging the formula lets you solve for an unknown value or an unknown weight.

Unknown Value

If one value is missing and its weight is known, you can isolate that value:

x_k = \frac{WM \sum_{i=1}^{n} w_i - \sum_{i \ne k} w_i x_i}{w_k}

Unknown Weight

If one weight is missing and the related value is known, you can isolate that weight:

w_k = \frac{\sum_{i \ne k} w_i (x_i - WM)}{WM - x_k}

For either rearrangement, the denominator must not be zero. Also, the total of all weights must not be zero, or the weighted mean is undefined.

When to Use a Weighted Mean

Weighted Mean vs. Regular Mean

A regular mean assumes every value contributes equally. A weighted mean changes that assumption by assigning more influence to some observations than others. If all weights are equal, the weighted mean reduces to the ordinary arithmetic mean.

WM = \frac{x_1 + x_2 + \cdots + x_n}{n} \quad \text{when all } w_i \text{ are equal}

Key Properties

• Higher weight means greater influence. A value with a large weight pulls the result toward itself.
• Zero weight means no influence. That value does not affect the final weighted mean.
• Positive weights keep the result bounded. With all positive weights, the weighted mean stays between the smallest and largest value.
• Equal weights reproduce the standard average. Weighting only changes the result when the weights differ.
• Scaling all weights by the same factor does not change the answer. Doubling every weight leaves the weighted mean unchanged.

Common Mistakes

• Mismatching values and weights. Each weight must correspond to the correct value.
• Forgetting to divide by total weight. This is a common error when weights do not add to 1 or 100%.
• Mixing decimals and percentages incorrectly. Use a consistent format for all weights.
• Assuming weights must add to 1. They do not have to, as long as you divide by the total of the weights.
• Using a total weight of zero. A denominator of zero makes the result undefined.

Weighted Mean FAQ