Statistics — Mean, Median, Mode and Range — 统计学—平均数、中位数、众数和极差

What Are Averages? | 什么是平均数？

在数学中，”平均数”这个词用来描述一组数据的中心或典型值。当你听到有人说”平均温度”或”平均分数”时，他们指的就是这个一般概念。然而，计量平均其实有三种不同的方式：平均数（Mean）、中位数（Median）和众数（Mode）。这三种方式各有优缺点，适用于不同情况。

In mathematics, the word “average” is used to describe the center or typical value of a set of data. When you hear someone talk about “the average temperature” or “the average score”, they are referring to this general idea. However, there are actually three different ways to measure the average: the Mean, the Median, and the Mode. Each has its own strengths and weaknesses, and each is useful in different situations.

What is the Mean? | 什么平均数？

平均数（Mean）是最常见的平均数类型。要计算平均数，你需要将所有数据值相加，然后除以数据的总个数。公式如下：

The Mean is the most common type of average. To calculate the mean, you add up all the data values and then divide by the total number of data points. The formula is:

Mean = (Sum of all values) / (Number of values)

平均数 = (所有值的总和) / (值的个数)

例如，如果一个学生五门考试的成绩分别为 78、85、92、88 和 76 分，那么平均数计算如下：78 + 85 + 92 + 88 + 76 = 419，然后 419 / 5 = 83.8。所以平均分是 83.8 分。

For example, if a student scores 78, 85, 92, 88, and 76 in five tests, the mean is calculated as: 78 + 85 + 92 + 88 + 76 = 419, then 419 / 5 = 83.8. So the mean score is 83.8.

平均数的优点在于它使用了所有数据，能全面反映数据集的整体水平。但它对极端值（也称为异常值）非常敏感。例如，如果班级里有一个学生每次考试都得 0 分，全班平均分会被人为拉低，无法准确反映大多数学生的真实水平。

The advantage of the mean is that it uses all the data and provides a comprehensive picture of the dataset’s overall level. However, it is very sensitive to extreme values, also called outliers. For example, if one student in a class scores 0 on every test, the class mean will be artificially lowered and won’t accurately reflect the true performance of most students.

What is the Median? | 什么是中位数？

中位数（Median）是当数据按从小到大排序时，位于中间位置的值。如果数据个数是奇数，中位数就是正中间的那个数。如果数据个数是偶数，中位数是中间两个数的平均数。

The Median is the middle value when the data is arranged in order from smallest to largest. If there is an odd number of data points, the median is the exact middle number. If there is an even number of data points, the median is the mean of the two middle numbers.

考虑以下数据集，代表 7 名学生的身高（厘米）：150, 152, 158, 160, 163, 165, 170。这里有 7 个数据点（奇数），中位数是第 4 个数：160 厘米。如果有 8 名学生：150, 152, 158, 160, 163, 165, 170, 175 – 中间两个数是 160 和 163，中位数就是 (160 + 163) / 2 = 161.5 厘米。

Consider the following dataset representing the heights (in cm) of 7 students: 150, 152, 158, 160, 163, 165, 170. There are 7 data points (odd), so the median is the 4th number: 160 cm. If there were 8 students: 150, 152, 158, 160, 163, 165, 170, 175 – the two middle numbers are 160 and 163, so the median is (160 + 163) / 2 = 161.5 cm.

中位数的主要优点是它不受异常值的影响。在收入数据等偏态分布中，中位数通常比平均数更能代表典型值。例如，如果一个国家的大多数人年收入在 2 万到 5 万英镑之间，但极少数的亿万富翁大幅拉高了平均数，中位数能更准确地反映”普通人”的收入。

The main advantage of the median is that it is not affected by outliers. In skewed distributions like income data, the median often represents the typical value better than the mean. For instance, if most people in a country earn between 20,000 and 50,000 pounds per year but a tiny number of billionaires pull the mean way up, the median gives a more accurate picture of what an “ordinary person” earns.

What is the Mode? | 什么是众数？

众数（Mode）是数据集中出现频率最高的值。一个数据集可以有一个众数（单峰分布）、多个众数（双峰或多峰分布），或者没有众数（如果所有值出现频率相同）。

The Mode is the value that appears most frequently in a dataset. A dataset can have one mode (unimodal), more than one mode (bimodal or multimodal), or no mode at all (if all values appear with equal frequency).

例如，考虑一个班级里学生最喜欢的颜色的数据集：红色、蓝色、蓝色、绿色、蓝色、黄色、红色。这里”蓝色”出现了三次，比其他任何颜色都多，所以众数是蓝色。另一个例子：一个鞋店出售的各种鞋码：38, 39, 39, 40, 40, 40, 41, 41, 42。众数是 40 码，因为它出现的次数最多（三次）。

For example, consider a dataset of students’ favourite colours in a class: Red, Blue, Blue, Green, Blue, Yellow, Red. Here “Blue” appears three times, more than any other colour, so the mode is Blue. Another example: shoe sizes sold in a shop: 38, 39, 39, 40, 40, 40, 41, 41, 42. The mode is size 40 because it appears most often (three times).

众数特别适用于分类数据（非数字数据），比如颜色、类型、品牌等。对于这种数据，平均数和中位数无法计算，而众数则是唯一可用的平均数指标。商店经常使用众数来决定哪种产品需要多进货，因为众数反映了”最受欢迎”的选择。

The mode is especially useful for categorical data (non-numerical data), such as colours, types, brands, etc. For such data, the mean and median cannot be calculated, and the mode is the only usable measure of average. Shops often use the mode to decide which product to stock more of, because the mode reflects the “most popular” choice.

What is the Range? | 什么是极差？

极差（Range）虽然本身不是平均数，但它通常与平均数、中位数和众数一起学习，因为它描述了数据的离散程度（Spread）。极差是数据集中最大值与最小值之间的差值：

The Range, while not an average itself, is usually taught alongside the mean, median, and mode because it describes the spread of the data. The range is the difference between the largest and smallest values in a dataset:

Range = Largest value – Smallest value

极差 = 最大值 – 最小值

例如，在考试成绩数据集 78, 85, 92, 88, 76 中，最高分是 92，最低分是 76，所以极差 = 92 – 76 = 16。这说明了分数的分散程度。极差较大意味着数据分布较广，极差较小意味着数据较为集中。

For example, in the test scores dataset 78, 85, 92, 88, 76, the highest score is 92 and the lowest is 76, so the range = 92 – 76 = 16. This tells us how spread out the scores are. A larger range means the data is more spread out; a smaller range means the data is more tightly clustered.

极差的一个缺点是它只依赖于两个值（最大值和最小值），因此对异常值高度敏感。如果数据集中有一个极端值，极差会变得非常大，不再能准确反映大部分数据的离散程度。

One disadvantage of the range is that it depends on only two values (the maximum and minimum), making it highly sensitive to outliers. If there is a single extreme value in the dataset, the range becomes very large and no longer accurately reflects the spread of most of the data.

Choosing the Right Average | 选择合适的平均数

理解何时使用每种平均数是一项重要的数学和现实生活技能。以下是一个简单的指南：

Understanding when to use each average is an important mathematical and real-life skill. Here is a simple guide:

使用平均数（Mean） 当你希望每个数据值都被考虑在内，且没有极端异常值时。适用于正态分布的数据，如考试成绩、身高、体重等。

Use the Mean when you want every data value to count, and there are no extreme outliers. It is suitable for normally distributed data such as test scores, heights, and weights.

使用中位数（Median） 当数据集包含异常值或呈偏态分布时。适用于收入数据、房价数据、反应时间数据等。

Use the Median when the dataset contains outliers or is skewed. It is suitable for income data, house price data, reaction time data, and similar.

使用众数（Mode） 当处理分类（非数字）数据时，或者当你想知道”最受欢迎”或”最常见”的选择时。

Use the Mode when dealing with categorical (non-numerical) data, or when you want to know the “most popular” or “most common” choice.

Worked Example 1 — Weather Data | 示例 1 — 天气数据

一个城镇连续 7 天的日最高气温（摄氏度）记录如下：22, 25, 19, 23, 22, 28, 21。计算平均数、中位数、众数和极差。

The daily maximum temperatures (in Celsius) in a town over 7 consecutive days were recorded as: 22, 25, 19, 23, 22, 28, 21. Calculate the mean, median, mode, and range.

解：

Solution:

平均数（Mean）：总和 = 22 + 25 + 19 + 23 + 22 + 28 + 21 = 160。数据个数 = 7。平均数 = 160 / 7 = 22.9°C（保留一位小数）。

Mean: Sum = 22 + 25 + 19 + 23 + 22 + 28 + 21 = 160. Number of data points = 7. Mean = 160 / 7 approximately equals 22.9 degrees Celsius (to 1 decimal place).

中位数（Median）：按升序排列：19, 21, 22, 22, 23, 25, 28。7 个数据点（奇数），中位数是第 4 个数 = 22°C。

Median: Arrange in ascending order: 19, 21, 22, 22, 23, 25, 28. 7 data points (odd), so the median is the 4th number = 22 degrees Celsius.

众数（Mode）：22°C 出现了两次，其他值只出现一次。众数 = 22°C。

Mode: 22 degrees Celsius appears twice; all other values appear once. Mode = 22 degrees Celsius.

极差（Range）：最大值 = 28，最小值 = 19。极差 = 28 – 19 = 9°C。

Range: Largest = 28, smallest = 19. Range = 28 – 19 = 9 degrees Celsius.

Worked Example 2 — Comparing Datasets | 示例 2 — 比较数据集

两个班级在数学考试中的成绩（满分 100 分）：

Two classes’ scores in a mathematics test (out of 100 marks):

A 班（Class A）：45, 52, 58, 60, 62, 65, 68, 70, 72, 85

B 班（Class B）：10, 48, 55, 60, 65, 70, 75, 80, 85, 92

计算每个班的平均数、中位数、众数和极差。比较两班的表现。

Calculate the mean, median, mode, and range for each class. Compare the performance of the two classes.

解 – A 班：

Solution – Class A:

平均数 = (45 + 52 + 58 + 60 + 62 + 65 + 68 + 70 + 72 + 85) / 10 = 637 / 10 = 63.7。

Mean = (45 + 52 + 58 + 60 + 62 + 65 + 68 + 70 + 72 + 85) / 10 = 637 / 10 = 63.7.

中位数：数据已排序。偶数个（10 个），取第 5 和第 6 个数的平均数 = (62 + 65) / 2 = 63.5。

Median: Data is already ordered. Even number (10 points), take the mean of the 5th and 6th numbers = (62 + 65) / 2 = 63.5.

众数：所有值都只出现一次，所以没有众数。

Mode: All values appear only once, so there is no mode.

极差 = 85 – 45 = 40。

Range = 85 – 45 = 40.

解 – B 班：

Solution – Class B:

平均数 = (10 + 48 + 55 + 60 + 65 + 70 + 75 + 80 + 85 + 92) / 10 = 640 / 10 = 64.0。

Mean = (10 + 48 + 55 + 60 + 65 + 70 + 75 + 80 + 85 + 92) / 10 = 640 / 10 = 64.0.

中位数 = (65 + 70) / 2 = 67.5。

Median = (65 + 70) / 2 = 67.5.

众数：无众数（所有值唯一）。

Mode: No mode (all values are unique).

极差 = 92 – 10 = 82。

Range = 92 – 10 = 82.

比较：B 班的平均数略高（64.0 vs 63.7），中位数也更高（67.5 vs 63.5），说明 B 班的整体表现更好。然而 B 班的极差也更大（82 vs 40），因为有一个异常低分 10 分 – 这表明 B 班的成绩更分散。如果不考虑 10 分这个异常值，B 班的优势会更明显。

Comparison: Class B has a slightly higher mean (64.0 vs 63.7) and a noticeably higher median (67.5 vs 63.5), indicating stronger overall performance. However, Class B also has a much larger range (82 vs 40) because of one outlier – a very low score of 10 – showing that Class B’s scores are more spread out. Without the 10-point outlier, Class B’s advantage would be even clearer.

Frequency Tables and Averages | 频率表与平均数

当数据以频率表的形式呈现时，你需要采用稍微不同的方法来计算平均数。假设你要计算以下考试分数（满分 10 分）的平均数：

When data is presented in a frequency table, you need a slightly different approach to calculate averages. Suppose we need to calculate the mean for these test scores (out of 10 marks):

分数 Score (x)	频率 Frequency (f)	分数 x 频率 f x x
4	2	8
5	3	15
6	5	30
7	6	42
8	3	24
9	1	9
Total	20	128

平均数 = (f 与 x 乘积的总和) / (总频率) = 128 / 20 = 6.4。这意味着平均每名学生得分为 6.4 分（满分 10 分）。

Mean = (Sum of f multiplied by x) / (Total frequency) = 128 / 20 = 6.4. This means the average score per student is 6.4 out of 10.

要从频率表中找中位数，你需要找到累积频率的中点位置。总频率是 20，所以中位数位于第 10 个和第 11 个数据点之间。累积频率：2（4分）, 2+3=5（5分）, 5+5=10（6分）, 10+6=16（7分）。第 10 个点在 6 分处结束，第 11 个点在 7 分处开始 – 所以中位数取第10和11个值的平均数：分数 6 和 7 之间，即中位数 = 6.5。

To find the median from a frequency table, you find the midpoint position using cumulative frequency. The total frequency is 20, so the median lies between the 10th and 11th data points. Cumulative frequencies: 2 (score 4), 2+3=5 (score 5), 5+5=10 (score 6), 10+6=16 (score 7). The 10th point ends at score 6 and the 11th point starts at score 7 – so the median is the mean of the 10th and 11th values: between scores 6 and 7, giving a median of 6.5.

众数是频率最高的分数，即 7 分（出现了 6 次）。

The mode is the score with the highest frequency, which is 7 (appearing 6 times).

Common Mistakes to Avoid | 常见错误与注意事项

1. 混淆 Mean 和 Median：当用户说”average”时，他们通常指的是 Mean。但在统计问题中，一定要明确问题是要求计算哪个指标。阅读问题时要格外仔细。

1. Confusing Mean and Median: When people say “average”, they usually mean the Mean. But in statistics questions, always be clear about which measure is being asked for. Read the question very carefully.

2. 忘记排序：计算中位数前必须先将数据从小到大排列。这是一个常见但代价高昂的错误，会直接导致答案错误。

2. Forgetting to Order: You must arrange the data from smallest to largest before finding the median. This is a common and costly mistake that leads directly to a wrong answer.

3. 混淆 Range 和 Mode：极差衡量的是数据的分散程度（减法），众数衡量的是最常见的值。两个都是统计概念，但作用完全不同。

3. Confusing Range and Mode: The range measures the spread of the data (a subtraction), while the mode measures the most common value. Both are statistical concepts but serve completely different purposes.

4. 忽略异常值：当数据包含极端值时，平均数可能具有误导性。始终考虑你的数据中是否存在异常值，以及中位数是否可能是更好的选择。

4. Ignoring Outliers: The mean can be misleading when extreme values are present. Always consider whether your data contains outliers and whether the median might be the better choice.

Real-World Applications | 实际应用

平均数和极差在现实生活中运用极为广泛。以下是一些常见的应用场景：

Averages and the range are used extensively in everyday life. Here are some common applications:

教育：学校使用平均分数来追踪学生的进步，使用中位数来看成绩分布的中心，使用极差来了解班级差异。例如，GCSE 和 A-Level 考官通过分析各学校的平均分和分数分布来评估教育质量。

Education: Schools use mean scores to track student progress, medians to see the centre of score distributions, and ranges to understand class variation. For example, GCSE and A-Level examiners analyse mean scores and score distributions across schools to evaluate educational quality.

体育：分析师使用平均数和中位数来评估运动员的表现。一名板球运动员的”平均击球率”实际上是一个平均数 – 总得分除以出局次数。在网球中，球员的发球速度通常以平均值的方式报告。

Sports: Analysts use means and medians to evaluate athlete performance. A cricketer’s “batting average” is actually a mean – total runs divided by number of dismissals. In tennis, player serve speeds are often reported as averages.

商业：企业使用平均数来预测需求、计算平均月度销售数据，用众数来识别最受欢迎的产品变体，用极差来监控供应链的变化。零售商定期分析销售数据来确定进货策略。

Business: Companies use means to forecast demand and calculate average monthly sales, modes to identify the most popular product variants, and ranges to monitor variations in supply chains. Retailers regularly analyse sales data to determine stocking strategies.

Key Vocabulary Summary | 关键词汇总结

English Term	中文术语	Definition
Mean	平均数	Sum of all values divided by the number of values
Median	中位数	The middle value when data is ordered
Mode	众数	The most frequently occurring value
Range	极差	The difference between the largest and smallest values
Spread	离散程度	How spread out the data is
Outlier	异常值	An extreme value that differs significantly from others
Frequency Table	频率表	A table showing how often each value occurs
Cumulative Frequency	累积频率	The running total of frequencies
Dataset	数据集	A collection of data values
Average	平均	A general term for the typical or central value

Practice Questions | 练习题

在继续阅读之前，尝试回答以下问题来测试你的理解：

Before moving on, try these questions to test your understanding:

1. 计算以下数据集的平均数、中位数、众数和极差：15, 18, 20, 20, 22, 25, 27, 30

1. Find the mean, median, mode, and range of this dataset: 15, 18, 20, 20, 22, 25, 27, 30

2. 一名板球运动员在 8 局比赛中的得分分别为：34, 56, 12, 78, 0, 45, 67, 23。计算他的平均击球率（mean）和中位数得分。哪个指标更能反映他的一贯表现？为什么？

2. A cricketer scores the following runs in 8 innings: 34, 56, 12, 78, 0, 45, 67, 23. Calculate his batting average (mean) and his median score. Which measure better reflects his consistent performance? Why?

3. 以下是足球比赛中进球数的频率表。计算平均数、中位数和众数：

3. Here is a frequency table of goals scored in football matches. Calculate the mean, median, and mode:

进球数 Goals	比赛场数 Matches
0	4
1	7
2	8
3	3
4	2
5	1

Choosing Between Mean and Median — A Deeper Look | 如何在平均数和中位数之间选择 — 深入分析

有时候，决定使用平均数还是中位数并不总是一目了然。以下是一个决策框架，帮助你做出正确选择：

Sometimes, deciding whether to use the mean or the median is not always obvious. Here is a decision-making framework to help you choose correctly:

步骤 1 – 检查数据的分布形状：如果数据大致对称（像一座钟形山），平均数和中位数会很接近，使用平均数通常是安全的。如果数据呈偏态分布（像一座一侧山坡陡峭的山），中位数更可靠。你可以通过画一个简单的点图或茎叶图来直观判断。

Step 1 – Check the shape of the distribution: If the data is roughly symmetric (like a bell-shaped hill), the mean and median will be close together and the mean is usually safe to use. If the data is skewed (like a hill with one steep side), the median is more reliable. You can check this visually by drawing a simple dot plot or stem-and-leaf diagram.

步骤 2 – 寻找异常值：寻找与其他数据相比看起来异常高或异常低的值。即使只有一个异常值，也可能显著改变平均数。中位数几乎不受影响。这就是为什么房价总是报中位数而不是平均数 – 少数豪宅就能大幅拉高平均数。

Step 2 – Look for outliers: Look for values that seem unusually high or low compared to the rest of the data. Even a single outlier can shift the mean significantly. The median is barely affected. This is why house prices are always reported as medians rather than means – a handful of luxury mansions can pull the mean way up.

步骤 3 – 考虑数据的目的：问自己：”我为什么会用到这个平均数？” 如果你需要计算总值（例如预算），平均数更好。如果你需要描述”典型”情况，中位数更合适。例如，老师计算全班预算时使用平均分数，但向家长报告”典型”分数时使用中位数。

Step 3 – Consider the purpose of the data: Ask yourself: “Why do I need this average?” If you need to calculate totals (e.g., a budget), the mean is better. If you need to describe the “typical” case, the median is more appropriate. For example, a teacher uses the mean to calculate a class budget but reports the median to parents as the “typical” score.

在 KS3 考试中，你可能会被问到一个问题，要求你解释为什么在一个特定的场景中，中位数比平均数更合适（反之亦然）。这类问题考察你判断选择什么统计指标的能力，而不仅仅是计算技巧。记住：如果数据有异常值或呈偏态分布，中位数几乎总是更好的选择。

In KS3 examinations, you may be asked to explain why the median is more appropriate than the mean in a particular scenario (or vice versa). These questions test your ability to justify your choice of statistical measure, not just your calculation skills. Remember: if the data has outliers or is skewed, the median is almost always the better choice.

Summary | 总结

平均数（Mean）、中位数（Median）和众数（Mode）是描述一组数据中心趋势的三种基本统计指标。平均数使用所有数据但容易被异常值影响；中位数不受异常值影响，适用于偏态分布数据；众数适用于分类数据，反映最常见值。极差（Range）衡量数据的离散程度，但不是一种平均数。在 KS3剑桥数学课程中，这些概念构成了数据分析和统计推理的基础。掌握这些工具，你将能够更好地理解图表、趋势以及日常生活中遇到的各种数据展示。

The Mean, Median, and Mode are three fundamental statistical measures that describe the central tendency of a dataset. The mean uses all data but is easily influenced by outliers; the median is robust against outliers and suitable for skewed distributions; the mode works for categorical data and reflects the most common value. The Range measures the spread of data but is not an average. In the KS3 Cambridge Mathematics curriculum, these concepts form the foundation of data analysis and statistical reasoning. By mastering these tools, you will be better equipped to understand charts, trends, and all kinds of data presentations you encounter in everyday life.

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