# 加权平均数计算器

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

交互版本：https://www.calcopenly.com/zh/math/weighted-average-calculator
主题：数学计算器

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.

## 输入

- **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.

## 结果

- 加权平均数 — 主要结果
- Sum of weights
- Sum of value × weight
- Number of values
- Unweighted average

## 公式

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

## 计算示例

### Course grade (default)

- Values and weights: Homework 92 20% / Midterm 85 30% / Final exam 78 50%
- **加权平均数: 82.9**
- **Sum of weights: 100 %**
- **Sum of value × weight: 8,290**
- **Number of values: 3**
- **Unweighted average: 85**
- 核验来源：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
- **加权平均数: 3.48**
- **Sum of weights: 14**
- **Sum of value × weight: 48.7**
- 核验来源：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
- **加权平均数: 3.430769**
- **Sum of weights: 13**
- **Sum of value × weight: 44.6**
- **Unweighted average: 3.325**
- 核验来源：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
- **加权平均数: 106.666667**
- **Sum of value × weight: 3,200**
- **Sum of weights: 30**
- **Unweighted average: 105**
- 核验来源：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
- **加权平均数: 20**
- **Unweighted average: 20**
- 核验来源：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
- **加权平均数: 80**
- **Sum of weights: 1**
- **Number of values: 2**
- **Unweighted average: 65**
- 核验来源：Definition: a value with weight 0 adds nothing to Σwx or Σw, so x̄ = 80 / 1

## 常见问题

### 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.

### “加权平均数计算器”有多准确？

准确性取决于输入值和方法的假设。十进制运算使用50位有效数字，但估算、数值方法和源数据的精度可能较低；显示时的舍入并不能消除这些限制。 已按独立来源核验的计算示例：7。 例如，“Course grade (default)”根据Hand calculation: (92 × 20 + 85 × 30 + 78 × 50) / 100 = 8290 / 100 = 82.9进行核验。

### 这种方法出自哪里？

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

## 来源

- [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)