Mean, Median, Mode, and Range | 平均数、中位数、众数和极差

📚 Mean, Median, Mode, and Range | 平均数、中位数、众数和极差

In everyday life and in many areas of science, we often collect numerical information – or data – and need to summarise it. Averages and the range help us describe a set of data with just a few numbers. The three main averages you need to know at KS3 are the mean, the median and the mode. The range tells you how spread out the data is. Understanding how to calculate and interpret each of these measures is a core skill in the Cambridge KS3 mathematics curriculum. This article will guide you through all the key ideas, step‑by‑step examples, common mistakes and useful tips for the checkpoint test.

在日常生活中和许多科学领域里，我们经常收集数值信息（即数据），并需要对其进行概括。平均数和极差可以帮助我们用寥寥几个数字来描述一组数据。在 KS3 阶段，你需要掌握的三种主要平均数是平均数（均值）、中位数和众数。极差则表示数据的分散程度。理解如何计算和解读这些统计量是剑桥 KS3 数学大纲中的一项核心技能。本文将带你一步步掌握所有关键概念、分步示例、常见错误，并提供对 checkpoint 考试有用的技巧。

1. What Are Averages? | 什么是平均数？

An average is a single value that represents the 'centre' or typical value of a set of data. It gives us a quick summary of a whole list of numbers. There are three commonly used averages, each calculated in a different way and useful in different situations.

平均数是一个单独的数值，用来代表一组数据的“中心”或典型值。它能让我们快速了解一整列数字的大致水平。常用的平均数有三种，每一种的计算方法不同，适用的场合也不同。

The three averages are the mean (the arithmetic average), the median (the middle value when the data is in order), and the mode (the most frequent value). Together with the range (difference between the largest and smallest), these statistics give a powerful summary of a data set's centre and spread.

这三种平均数分别是 均值（算术平均数）、中位数（按顺序排列后居中的数值）和 众数（出现次数最多的值）。再结合 极差（最大值与最小值的差），这些统计量就能有力地概括数据集的集中趋势和离散程度。

2. Calculating the Mean | 计算平均数（均值）

The mean is what most people think of as the 'average'. To find the mean, add up all the data values and then divide the total by the number of values.

均值就是大多数人所说的“平均数”。要求均值，先把所有数据值加起来，再把总和除以数据的个数。

Mean = (Sum of all data values) ÷ (Number of values)

Remember, the symbol for the mean is often x̄, but in KS3 you can simply write 'mean' , 9, 12, 3, the sum is 4 + 7 + 9 + 12 + 3 = 35, and there are 5 values. Mean = 35 ÷ 5 = 7.

请记住，均值常用 x̄ 表示，但在 KS3 阶段你只要写“mean”即可。对于数据集 4, 7, 9, 12, 3，总和为 4 + 7 + 9 + 12 + 3 = 35，一共有 5 个数值。均值 = 35 ÷ 5 = 7。

Always check your addition carefully. If the mean is not a whole number, it is fine to leave it as a decimal or fraction. The mean is sensitive to extremely high or low values (outliers), so it might not always be the best average to use.

一定要仔细检查加法。如果均值不是整数，可以保留为小数或分数。均值对极端高值或低值（异常值）很敏感，因此它不一定总是最合适的平均数。

3. Finding the Median | 求中位数

The median is the middle number when the data values are arranged in order from smallest to largest. If there is an odd number of values, the median is simply the value in the middle position. If there is an even number of values, the median is the mean of the two middle numbers.

中位数是将数据值按从小到大的顺序排列后，处于中间位置的数。如果数据个数是奇数，中位数就是正中间的那个值；如果数据个数是偶数，中位数就是中间两个数的均值。

, 10, 12 there are 6 values; the two middle ones are 6 and 8, so median = (6 + 8) ÷ 2 = 7.

例如，对于数据集 3, 7, 9, 11, 14，有 5 个数值；中间位置是第 3 个，所以中位数 = 9。对于 2, 4, 6, 8, 10, 12，有 6 个数值；中间两个是 6 和 8，所以中位数 = (6 + 8) ÷ 2 = 7。

The median is not affected by very large or small outliers, so it is often used for house prices and incomes where a few extreme values could distort the mean.

中位数不受极大或极小异常值的影响，因此常用于房价和收入等数据，因为在这些情况下，少数极端值可能会扭曲均值。

4. Finding the Mode | 找出众数

The mode is the data value that occurs most often. A data set can have one mode, more than one mode (bimodal or multimodal), or no mode at all if all values appear equally often. The mode is the only average that can be used for non‑numerical data, such as favourite colours.

众数是出现次数最多的数据值。一个数据集可以有一个众数，也可以有多个众数（双峰或多峰），如果所有数值出现的次数相同，则没有众数。众数是唯一可用于非数值数据（如最喜欢的颜色）的平均数。

For the list 2, 5, 5, 7, 5, 9, the mode is 5 because it appears three times, more than any other number. For 3, 4, 4, 6, 6, 8, both 4 and 6 appear twice, so the data is bimodal – modes are 4 and 6.

对于数列 2, 5, 5, 7, 5, 9，众数是 5，因为它出现了三次，多于其他任何数字。对于 3, 4, 4, 6, 6, 8，4 和 6 都出现了两次，所以这组数据是双峰的，众数为 4 和 6。

In real life, the mode can be useful for shops stocking the most popular product size or for teachers seeing the most common score on a test. However, it may not represent the centre well if the most frequent value is far from the other values.

在现实生活中，众数可用于商店储备最受欢迎的尺码，或帮助教师了解考试中最常见的分数。不过，如果最常见的值与其他值差距很大，众数可能并不能很好地代表数据的中心。

5. Understanding the Range | 理解极差

The range is a measure of how spread out the data is. It is simply the difference between the largest data value and the smallest data value. A small range means the values are close together; a large range indicates they are more spread out.

极差是衡量数据分散程度的指标。它就是最大数据值与最小数据值之差。极差小意味着数值比较集中；极差大则表明数值比较分散。

Range = Largest value − Smallest value

= 10 cm. Be careful to subtract correctly and always include the unit if the data has one.

例如，五棵植物的高度（单位 cm）为 12, 19, 14, 22, 17。最大值为 22，最小值为 12，因此极差 = 22 − 12 = 10 cm。注意正确相减，如果数据带有单位，结果也要写上单位。

The range is easy to work out but it only tells you about the extremes; it does not show how the data is distributed in between. For a fuller picture, you often look at averages and the range together.

极差很容易计算，但它只反映两个极值的情况，无法展现中间数据的分布。为了获得更全面的信息，通常需要同时观察平均数和极差。

6. Comparing Data Sets Using Averages and Range | 用平均数和极差比较数据集

When you have two sets of data, you can compare them by looking at an average (to see which tends to be higher) and the range (to see which is more consistent or more varied). For instance, you might compare test scores for two classes. A class with a higher mean score performed better on average, while a class with a smaller range had more consistent results.

当你有两组数据时，可以通过观察平均数（判断哪组整体更高）和极差（判断哪组更稳定或更分散）来进行比较。例如，你可以比较两个班级的考试成绩。平均分更高的班级整体表现更好，而极差更小的班级成绩更稳定。

Always state which average you are using and explain what the range tells you. A typical answer might be: 'The mean score in Class A is 5 marks higher than in Class B, so Class A performed better on average. The range in Class A is only 6 compared to 18 in Class B, so Class A's scores are more consistent.'

回答时一定要说明你用的是哪种平均数，并解释极差说明了什么。典型答案可能是：“A 班的平均分比 B 班高 5 分，因此 A 班整体表现更好。A 班的极差仅为 6，而 B 班为 18，因此 A 班的成绩更稳定。”

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You can also talk about the mode, especially if one value occurs very frequently. Using the correct mathematical vocabulary – mean, median, mode, range – shows a good understanding.

你也可以提到众数，尤其当某个值出现得非常频繁时。使用正确的数学词汇——均值、中位数、众数、极差——可以体现你的良好理解。

7. Choosing the Best Average | 选择最佳平均数

Different situations call for different averages. The mean is often the best choice when the data is fairly symmetrical and has no extreme outliers, because it uses all the data values. The median is better when there are outliers or the data is skewed, as it better reflects the typical value. The mode is helpful for finding the most popular item or category.

不同的情况适用不同的平均数。当数据大致对称且没有极端异常值时，均值往往是最佳选择，因为它用到了所有数据值。当存在异常值或数据偏斜时，中位数更合适，因为它能更好地反映典型值。众数则适用于找出最流行的物品或类别。

For example, a shoe shop wants to know which shoe size to order the most; the mode is the right measure. A school wants to compare the overall performance of two year groups; the mean might be used. A property website reports house prices; they often quote the median because a few very expensive houses could inflate the mean.

例如，鞋店想知道应该最多订购哪种鞋码，众数就是合适的指标。学校想比较两个年级的整体表现，可能会用均值。房产网站报告房价时，常常引用中位数，因为少数几处非常昂贵的房产可能会拉高均值。

When answering a question, think about what you want to find out and whether any extreme values are present. That will help you decide whether the mean, median or mode is most appropriate.

在回答问题时，想一想你想了解什么，以及是否存在极端值。这将有助于你判断均值、中位数还是众数最合适。

8. Outliers and Their Effect | 异常值及其影响

An outlier is a data value that is much larger or much smaller than the rest of the data. Outliers can have a big effect on the mean, pulling it towards the extreme value. The median and mode are usually not affected, which is why they are called resistant statistics.

异常值是指远大于或远小于其他数据的数据值。异常值会对均值产生很大影响，将其拉向极端值。中位数和众数通常不会受到影响，所以它们被称为耐抗性统计量。

For instance, consider the weekly pocket money of five friends: £2, £3, £3, £4, £25. The mean is (2+3+3+4+25) ÷ 5 = £ which does not represent a typical amount. The median (arranging gives 2, 3, 3, 4, 25) is £3, which better reflects the group. The mode is £3 as well.

例如，考虑五个朋友每周的零花钱：2 英镑、3 英镑、3 英镑、4 英镑、25 英镑。均值是 (2+3+3+4+25) ÷ 5 = 7.40 英镑，这并不能代表典型金额。中位数（排列为 2, 3, 3, 4, 25）是 3 英镑，更好地反映了这组人。众数也是 3 英镑。

In your analysis, you should identify if an outlier exists and explain how it would distort the mean. This is a common exam question at KS3 level.

在分析时，你应该指出是否存在异常值，并解释它会如何扭曲均值。这是 KS3 阶段常见的考题。

9. Working with Frequency Tables | 使用频数表

Data is often given in a frequency table rather than as a simple list. A frequency table shows each data value (or group) and how many times it occurs (the frequency). You can still find the mode, median and mean from a frequency table; the process just needs a few extra steps.

数据常常以频数表的形式给出，而不是简单的列表。频数表显示每个数据值（或分组）及其出现的次数（频数）。你依然可以从频数表中找到众数、中位数和均值，只是过程需要一些额外的步骤。

For example, a table might show the number of pets owned by students: 0 pets (frequency 8), 1 pet (frequency 12), 2 pets (frequency 5), 3 pets (frequency 2). You can see this as a convenient way of presenting data without writing out every single observation.

例如，一个表格可能显示学生拥有的宠物数量：0 只宠物（频数为 8），1 只（频数 12），2 只（频数 5），3 只（频数 2）。你可以将这种表格视为一种无需写出每个观测值的便捷数据呈现方式。

10. Mean from a Frequency Table | 从频数表求平均数

To calculate the mean from a frequency table, you need to find the total of all data values and the total number of data values. Multiply each data value by its frequency to get the sub‑totals, add these together for the overall total, and then divide by the sum of the frequencies.

要根据频数表计算均值，你需要求出所有数据值的总和与数据值的总个数。先将每个数据值乘以其频数得到各组小计，将这些小计加在一起得到总和，然后除以频数之和。

The sum of frequencies is 8 + 12 + 5 + 2 = 27. So mean = 28 ÷ 27 ≈ 1.04 pets per student.

以宠物例子说明：(0×8) + (1×12) + (2×5) + (3×2) = 0 + 12 + 10 + 6 = 28。频数之和为 8 + 12 + 5 + 2 = 27。因此均值 = 28 ÷ 27 ≈ 1.04 只/人。

In your working, you can add a column to the frequency table labelled 'value × frequency' to help keep track. This method is often tested, so practise setting up the table neatly.

在解题时，你可以在频数表中增加一列“数值 × 频数”以便记录。这个方法常被考到，所以要多练习工整地构建表格。

11. Median and Mode from Frequency Tables | 从频数表求中位数和众数

Finding the mode from a frequency table is straightforward: it is the data value with the highest frequency. In the pets table, the highest frequency is 12, which corresponds to 1 pet, so the mode is 1 pet.

从频数表找众数很简单：频数最高的数据值就是众数。在宠物表中，最高频数是 12，对应 1 只宠物，所以众数是 1 只。

he frequencies: the first 8 values are 0s, the next 12 values are 1s (positions 9 to 20). So the 14th value falls within the 1s. Therefore the median number of pets is 1.

求中位数时，你需要找出当所有数据按顺序排列时中间那个值的位置。总频数为 27，因此中位数是第 (27 + 1) ÷ 2 = 14 个数值。逐个数频数：前 8 个值是 0，接下来的 12 个值是 1（位置为 9 到 20）。因此第 14 个值落在 1 之中。所以宠物数的中位数是 1。

If the total frequency is even, the median will be the mean of two middle values, just as in a simple list. This method tests your understanding of cumulative frequency without the formal graph.

如果总频数为偶数，中位数将是中间两个值的均值，就像处理简单列表一样。这种方法考查的是你对累计频数的理解，而不需要绘制正式的累计频数图。

12. Real-Life Applications and Key Tips | 实际应用与关键技巧

Averages and range appear in many everyday contexts: comparing weather data, analysing sports statistics, reviewing school test results, or making business decisions. When you solve problems, always read the question carefully: what exactly is being asked? Should you find the mean, the median, the mode, or the range? Do you need to compare data sets?

平均数和极差出现在许多日常场景中：比较天气数据、分析体育统计、审视学校考试成绩或做出商业决策。解答问题时，一定要仔细审题：究竟在问什么？是求均值、中位数、众数还是极差？是否需要比较数据集？

Here are some key tips for success in KS3 exams:
1. Always order the data before finding the median.
2. Show your working – write down the sum and the division for the mean.
3. For the range, remember: largest minus smallest.
4. If the mean seems unrealistic, check for an outlier.
5. When comparing, comment on both an average and the spread.

以下是一些在 KS3 考试中取得成功的关键技巧：
1. 在找中位数之前，一定要先把数据排序。
2. 展示解题过程——写出求均值的加总和除法步骤。
3. 计算极差时，记住是最大值减最小值。
4. 如果均值看起来不切实际，检查是否存在异常值。
5. 进行比较时，既要评论平均数，也要评论分散程度。

Use clear statements and include units where applicable. With plenty of practice, you will be able to calculate and interpret averages and range confidently and accurately.

用清晰的陈述作答，并适时标明单位。只要多加练习，你就能自信而准确地计算和解读平均数与极差。

Published by TutorHao | Maths Revision Series | aleveler.com

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