# Weighted average calculator

> Weighted average calculator: enter values with weights, one pair per line, for the weighted mean, sum of weights and each value's contribution.

Interactive version: https://www.calcopenly.com/math/weighted-average-calculator
Subject: Math calculators

A weighted average multiplies each value by its weight, adds the products and divides by the sum of the weights: Σwx ÷ Σw. Values with larger weights pull the result toward themselves, and with equal weights it is the ordinary mean.

Course grades are the usual case. With homework at 92 counting 20%, the midterm at 85 counting 30% and the final at 78 counting 50%, the products add up to 8,290 and the weights to 100, so the weighted average is 82.9, while the plain average of the three scores is 85. Weights can be credits for a GPA, share counts for an average purchase price, or any relative importance.

Weights only need to be in proportion: 20%, 30% and 50% give the same answer as 2, 3 and 5. They must not be negative, and at least one must be above zero. A label before the numbers on a line, such as “Final exam”, is carried into the table.

## Inputs

- **Values and weights**: One pair per line: the value, then its weight. Separate them with a space, comma or tab, and don't use thousands separators. A label in front is optional.

## Results

- Weighted average — main result
- Sum of weights
- Sum of value × weight
- Number of values
- Unweighted average

## Formula

$$
\bar x_w = \frac{\sum_{i=1}^{n} w_i x_i}{\sum_{i=1}^{n} w_i}
$$

## Worked examples

### Course grade (default)

- Values and weights: Homework 92 20% / Midterm 85 30% / Final exam 78 50%
- **Weighted average: 82.9**
- **Sum of weights: 100 %**
- **Sum of value × weight: 8,290**
- **Number of values: 3**
- **Unweighted average: 85**
- Checked against: Hand calculation: (92 × 20 + 85 × 30 + 78 × 50) / 100 = 8290 / 100 = 82.9

### GPA by credit hours (Inch Calculator example)

- Values and weights: 3.0 4 / 4.0 3 / 3.7 4 / 3.3 3
- **Weighted average: 3.48**
- **Sum of weights: 14**
- **Sum of value × weight: 48.7**
- Checked against: Inch Calculator weighted average page: (3.0×4 + 4.0×3 + 3.7×4 + 3.3×3) / 14 = 48.7 / 14, GPA 3.48

### GPA with four classes (Omni Calculator example)

- Values and weights: A 4 4 / B 3 4 / A 4 3 / C+ 2.3 2
- **Weighted average: 3.430769**
- **Sum of weights: 13**
- **Sum of value × weight: 44.6**
- **Unweighted average: 3.325**
- Checked against: Omni Calculator weighted average page: (4·4 + 3·4 + 4·3 + 2.3·2) / 13 = 3.43; Python Fraction(446, 130) = 3.4307692…

### Average share price

- Values and weights: 100 10 / 110 20
- **Weighted average: 106.666667**
- **Sum of value × weight: 3,200**
- **Sum of weights: 30**
- **Unweighted average: 105**
- Checked against: Hand calculation: (100 × 10 + 110 × 20) / 30 = 3200 / 30 = 106.67

### Equal weights give the plain mean

- Values and weights: 10 1 / 20 1 / 30 1
- **Weighted average: 20**
- **Unweighted average: 20**
- Checked against: Definition: with wᵢ = 1 the weighted mean is Σx / n = 60 / 3

### A zero weight drops the value (edge case)

- Values and weights: 50 0 / 80 1
- **Weighted average: 80**
- **Sum of weights: 1**
- **Number of values: 2**
- **Unweighted average: 65**
- Checked against: Definition: a value with weight 0 adds nothing to Σwx or Σw, so x̄ = 80 / 1

## Questions

### How do you calculate a weighted average?

Multiply each value by its weight, add the products, and divide by the total weight. For scores of 92, 85 and 78 weighted 20%, 30% and 50%: 92 × 20 + 85 × 30 + 78 × 50 = 8,290, and 8,290 ÷ 100 = 82.9. In a spreadsheet the same formula is =SUMPRODUCT(values, weights)/SUM(weights).

### Do the weights have to add up to 100%?

No. Dividing by the sum of the weights rescales them automatically, so weights of 2, 3 and 5 give the same result as 20%, 30% and 50%. If percentage weights add up to 90% because a grade item is still missing, the result is the average of the work done so far, with each item's weight divided by 0.9.

### How do you calculate a weighted GPA?

Use the grade points as values and the credit hours as weights. For a B (3.0) in a 4-credit class, an A (4.0) in 3 credits, an A− (3.7) in 4 credits and a B+ (3.3) in 3 credits, the quality points are 12 + 12 + 14.8 + 9.9 = 48.7 over 14 credits, a GPA of 3.48. The GPA calculator converts letter grades for you.

### What is the difference between a weighted average and a simple average?

A simple average gives every value the same weight, 1/n. A weighted average lets some values count more. For the default grades the simple average is (92 + 85 + 78) ÷ 3 = 85, but the final exam, worth half the grade, pulls the weighted average down to 82.9. The two agree only when all weights are equal.

### How do you find the average price of shares bought at different prices?

Weight each price by the number of shares bought at it. Buying 10 shares at $100 and 20 at $110 costs $1,000 + $2,200 = $3,200 for 30 shares, an average of $106.67 per share, not the $105 you get by averaging the two prices. This weighted average is the cost basis per share used to work out a gain or loss.

### How accurate is the weighted average calculator?

Accuracy depends on your inputs and the method's assumptions. Decimal arithmetic uses 50 significant digits, but estimates, numerical methods and source data can be less precise; the displayed rounding does not remove those limits. It is checked against 7 worked examples whose answers come from independent sources; for example, “Course grade (default)” is checked against Hand calculation: (92 × 20 + 85 × 30 + 78 × 50) / 100 = 8290 / 100 = 82.9.

### Where does the method come from?

Wolfram MathWorld: Weighted mean; NIST Dataplot reference manual: WEIGHTED SUM (Σ wᵢxᵢ).

## Sources

- [Wolfram MathWorld: Weighted mean](https://mathworld.wolfram.com/WeightedMean.html)
- [NIST Dataplot reference manual: WEIGHTED SUM (Σ wᵢxᵢ)](https://itl.nist.gov/div898/software/dataplot/refman2/auxillar/weigsum.htm)