Combining Data Sets | 数据合并

📚 Combining Data Sets | 数据合并

When two or more groups of data are combined, we often need to recalculate statistical measures for the merged data set. This skill is essential in IGCSE Mathematics, as it tests both your understanding of averages and your ability to manipulate data efficiently.

当两组或多组数据合并时，我们通常需要重新计算合并后数据集的统计量。这一技巧在IGCSE数学中至关重要，因为它同时考查你对平均数的理解以及高效处理数据的能力。

1. The Combined Mean | 合并平均值

The combined mean of two data sets is the average of all data points when the sets are merged into one. It is not simply the average of the two means unless the groups are the same size.

两组数据合并后的平均值，是所有数据点合并为一个集合后的平均数。除非两组数据大小相同，否则它并不是两个平均值的简单平均。

For two groups with sizes n₁ and n₂, and means x̄₁ and x̄₂, the combined mean x̄ is:

对于规模为 n₁ 和 n₂、平均值分别为 x̄₁ 和 x̄₂ 的两组数据，合并平均值 x̄ 为：

x̄ = (n₁x̄₁ + n₂x̄₂) / (n₁ + n₂)

This works because n₁x̄₁ is the sum of all values in group 1, and n₂x̄₂ is the sum of all values in group 2.

这是因为 n₁x̄₁ 是第1组所有数值的总和，n₂x̄₂ 是第2组所有数值的总和。

2. Why You Cannot Simply Average the Means | 为什么不能简单求均值

Many students incorrectly average the two means. Consider group 1 with 10 students averaging 80 marks, and group 2 with 2 students averaging 50 marks.

许多学生错误地对两个平均值再取平均。例如第1组有10名学生，平均分为80；第2组有2名学生，平均分为50。

Incorrect: (80 + 50) / 2 = 65

Correct: (10 × 80 + 2 × 50) / 12 = (800 + 100) / 12 = 75

The correct combined mean is 75, which is much closer to 80 because group 1 contributes more data points. The larger group should have a greater influence on the combined mean.

正确的合并平均值是75，它更接近80，因为第1组贡献了更多的数据点。较大的组应当对合并平均值产生更大的影响。

3. Worked Example 1 | 示例1

A class has 15 boys with mean height 170 cm and 10 girls with mean height 162 cm. Find the mean height of the whole class.

某班级有15名男生，平均身高170厘米；10名女生，平均身高162厘米。求全班平均身高。

x̄ = (15 × 170 + 10 × 162) / (15 + 10)

x̄ = (2550 + 1620) / 25 = 4170 / 25 = 166.8 cm

The combined mean height is 166.8 cm. Notice we used weighted contributions, not the simple average of 170 and 162, which would be 166.

合并平均身高为166.8厘米。注意我们使用了加权贡献，而不是170和162的简单平均，那会是166。

4. Combining Frequency Distributions | 合并频数分布表

When data is presented in frequency tables, the combined mean is found by summing the fx column (frequency × midpoint) for both tables, then dividing by the total frequency.

当数据以频数分布表呈现时，合并平均值需将两张表的 fx 列（频数 × 组中值）相加，再除以总频数。

For grouped data, use the midpoint of each class interval as the representative value.

对于分组数据，使用每个组区间的组中值作为代表值。

x̄ = (Σf₁x₁ + Σf₂x₂) / (Σf₁ + Σf₂)

5. Worked Example 2 | 示例2

Group A has the following frequency distribution:

A组具有以下频数分布：

Value	2	4	6
Frequency	3	5	2

Group B has mean 5 with total frequency 4. Find the combined mean.

B组平均值为5，总频数为4。求合并平均值。

For Group A: Σfx = 3 × 2 + 5 × 4 + 2 × 6 = 6 + 20 + 12 = 38, and Σf = 3 + 5 + 2 = 10.

对于A组：Σfx = 3 × 2 + 5 × 4 + 2 × 6 = 6 + 20 + 12 = 38，Σf = 3 + 5 + 2 = 10。

x̄ = (38 + 4 × 5) / (10 + 4) = (38 + 20) / 14 = 58 / 14 ≈ 4.14

The combined mean is approximately 4.14.

合并平均值约为4.14。

6. Weighted Averages | 加权平均数

The combined mean is actually a special case of the weighted mean, where weights are the group sizes. Weighted averages are used when different data points have different levels of importance.

合并平均值实际上是加权平均数的一种特殊情况，其中权重就是各组的大小。当不同数据点具有不同重要程度时，使用加权平均数。

x̄w = Σ(wᵢxᵢ) / Σwᵢ

For example, if a course has coursework worth 30% and an exam worth 70%, a student scoring 80 on coursework and 62 on the exam has:

例如，某课程平时作业占30%，考试占70%。一名学生作业得80分，考试得62分，则加权平均为：

x̄w = (0.30 × 80 + 0.70 × 62) / 1 = 24 + 43.4 = 67.4

Weighted averages are frequently used in IGCSE statistics, particularly when combining marks from different components of an assessment.

加权平均在IGCSE统计中经常出现，特别是在合并不同考试部分的分数时。

7. Combined Median and Mode | 合并中位数与众数

The combined median cannot be calculated from the individual medians alone. You must first merge the data values (or cumulative frequency tables) and then find the middle value.

合并中位数不能仅由各组的中位数计算得出。你必须先合并数据值（或累积频数表），再找到中间值。

Similarly, the combined mode is found only by merging the data sets or identifying the highest total frequency once the frequency tables are combined.

类似地，合并众数也只能通过合并数据集合或在合并频数表后识别总频数最高的值来确定。

Example: Group 1 values {3, 5, 7}, Group 2 values {5, 5, 9}. Combined sorted data: {3, 5, 5, 5, 7, 9}. Median = (5 + 5) / 2 = 5, Mode = 5.

示例：第1组数据 {3, 5, 7}，第2组数据 {5, 5, 9}。合并后排序为 {3, 5, 5, 5, 7, 9}。中位数 = (5 + 5) / 2 = 5，众数 = 5。

8. Effect on Range and Spread | 对极差与离散程度的影响

When two data sets are combined, the range of the merged set is the difference between the overall maximum and overall minimum across both sets.

当两组数据合并时，合并后的极差是两组中整体最大值与整体最小值之差。

Range of combined set = max(all values) − min(all values)

Note that the combined range is at least as large as each individual range, and could be larger if the sets occupy different intervals.

注意，合并后的极差至少与各组极差一样大，如果各组所在区间不同，合并极差可能会更大。

9. Common Mistakes | 常见错误

Simply averaging the two means without weighting by group size.

不对组大小加权而直接对两个平均值再取平均。
Forgetting to multiply the mean by the frequency when calculating the total sum.

在计算总和时忘记将平均值乘以频数。
Using class boundaries instead of midpoints for grouped data.

对分组数据使用组边界而不是组中值。
Mixing up n₁x̄₁ with n₁ + x̄₁.

混淆 n₁x̄₁ 与 n₁ + x̄₁。

10. Exam Tips | 考试技巧

Always write down the formula for combined mean before substituting values — you earn method marks.

代入数值前先写出合并平均值公式——这样可以获得方法分。
For grouped data, always calculate Σfx for each table separately, then add.

对于分组数据，先分别计算每张表的Σfx，再相加。
Check whether your answer lies between the two individual means. If not, you have made a mistake.

检查你的答案是否落在两个单独平均值之间。如果没有，说明你算错了。
When questions give ‘total frequency’ and mean for one group, treat the product as that group’s total sum.

当题目给出某一组的’总频数’和平均值时，将它们的乘积视为该组的总和。

11. Practice Questions | 练习题

1. Set A has 8 values with mean 12. Set B has 12 values with mean 18. Find the combined mean.

1. 集合A有8个值，平均数为12；集合B有12个值，平均数为18。求合并平均值。

Answer: (8 × 12 + 12 × 18) / 20 = (96 + 216) / 20 = 312 / 20 = 15.6

2. The mean of 20 numbers is 7. Two new numbers, 5 and 9, are added. Find the new mean.

2. 20个数的平均数是7。加入两个新数5和9。求新的平均数。

Answer: (20 × 7 + 5 + 9) / 22 = (140 + 14) / 22 = 154 / 22 = 7

3. Group X has mean 25 and total frequency 40. Group Y has mean 30 and total frequency

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Combining Data Sets | 数据合并