This weighted average calculator combines numerical values with nonnegative weights and shows how their normalized shares contribute to the result.
Calculate a weighted average from values and weights
A weighted average gives some values more influence than others. Multiply each value by its weight, add the products, then divide by the total weight. The answer retains the unit of the values; the weights describe relative influence on a shared scale. The official R documentation for weighted.mean explains that weights are normalized to sum to one when possible. This calculator shows that normalization and each row’s contribution in Calculation details. Larger weights pull the average toward their values. With equal positive weights, the result is the ordinary arithmetic mean described by OpenStax. This page accepts nonnegative weights and requires a positive total. That restriction is the tool’s product policy, rather than a claim that every mathematical weighting scheme forbids negative coefficients. Enter actual observations and their intended relative weights. The calculation does not decide which weights are appropriate or infer missing values.
Values must measure the same quantity in the same unit. Weights must use one consistent relative scale.
At least one weight must be positive. A zero weight has no influence on the weighted result.
Check the weighted sum and its denominator
The relative-weight preset enters values 10, 20 and 30 with weights 1, 2 and 3. The engine adds the weights to 6 and the weighted products to 140. Dividing the weighted sum by the total weight gives 23.33333333.
The row weights normalize to 0.1666666667, 0.3333333333, 0.5. Those shares multiply their respective values to make the contributions shown in the working. Every accepted row is processed even when the detail display is shortened. Zero-weight rows have no weighted influence; they remain in the entered-row count and plain-mean comparison.
Giving each entered row equal influence would instead produce a plain mean of 20. The weighted result is higher by 3.333333333, because the largest value received the largest weight. This comparison describes the specified data, rather than whether that weighting choice is justified.
Normalize relative weights and percentage weights
Enter one value and weight pair per line. Decimals, scientific notation and simple fractions are accepted for either quantity. Separate a pair with spaces, commas, semicolons or a colon. Omit thousands separators, which would introduce another token.
Weights can be relative amounts, repetition counts or percentage numbers. Multiplying every weight by the same positive factor leaves the result unchanged because it scales the numerator and denominator together. The tool normalizes by the actual total; the weights need not add to 100.
The percentage-number preset enters values 80, 90 and 70 with weights 30, 50 and 20. Its total weight is 100, weighted sum 8,300 and average 83. Enter percentage numbers without a sign. Using proportional weights 3, 5 and 2 gives the same average.
A course grading policy can require a complete percentage allocation or special rules for missing assignments. Those rules are outside this generic normalization tool. It does not establish a GPA scale, investment return or tax cost basis.
Interpret zero weights, scope and numerical limits
A zero weight omits that row’s influence from the weighted numerator and denominator. A zero value with a positive weight still matters. In the zero-weight preset, the value 10 has weight 0; the other values are 20 with weight 2 and 40 with weight 1. The engine reports 3 entered pairs, 2 positive-weight pairs, total weight 3 and weighted average 26.66666667.
The plain-mean comparison gives every entered pair equal influence, including zero-weight rows. It is explicitly labeled so it cannot be mistaken for the weighted result. Here it is 23.33333333. Duplicate values can be entered on separate rows; their weights add to the same weighted influence.
For an unweighted dataset, the Mean Median Mode Calculator compares arithmetic center, ordered center and repeated values. The Standard Deviation Calculator describes spread, with separate sample and population conventions. Ratio simplifies relative terms, and Percentage computes a share of a quantity. These tools answer different mathematical questions.
Local limits are 5,000 pairs and 100,000 characters in the selected field. Each numeric token supports at most 128 characters. Nonzero values and weights must have magnitude from 1e-100 to 1e+100. Negative values are allowed; negative weights are rejected. An all-zero weight total has no defined weighted average.
The engine uses IEEE754 floating-point numbers and compensated numerator and denominator sums. Fractions may be represented approximately, and differences below machine precision cannot be recovered. Display rounding happens after calculation. Details show the first 100 pairs; the result includes every accepted row.
The weighted-minus-plain comparison uses differences from a shared origin before subtracting rounded large means. It can preserve a small change that would disappear if the displayed averages were subtracted.
The Geometric Mean Calculator averages multiplicative values or equal-period factors. It separates raw values, relative weights and percentage changes, with explicit zero conventions.
The Harmonic Mean Calculator averages positive values through their reciprocals. For speeds, it separates equal-distance averaging from equal-time arithmetic averaging and accepts relative distance weights.