# Máy tính trung bình có trọng số

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

Phiên bản tương tác: https://www.calcopenly.com/vi/math/weighted-average-calculator
Chủ đề: Máy tính toán học

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.

## Dữ liệu đầu vào

- **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.

## Kết quả

- Trung bình có trọng số — kết quả chính
- Sum of weights
- Sum of value × weight
- Number of values
- Unweighted average

## Công thức

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

## Ví dụ có lời giải

### Course grade (default)

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

## Câu hỏi

### 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.

### “Máy tính trung bình có trọng số” chính xác đến mức nào?

Độ chính xác phụ thuộc vào dữ liệu nhập và giả định của phương pháp. Phép tính thập phân dùng 50 chữ số có nghĩa, nhưng ước lượng, phương pháp số và dữ liệu nguồn có thể kém chính xác hơn; làm tròn khi hiển thị không loại bỏ các giới hạn đó. Ví dụ có lời giải đã đối chiếu với nguồn độc lập: 7. Ví dụ, “Course grade (default)” được kiểm tra bằng Hand calculation: (92 × 20 + 85 × 30 + 78 × 50) / 100 = 8290 / 100 = 82.9.

### Phương pháp này lấy từ đâu?

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

## Nguồn

- [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)