📚 Averages and Range: Mean, Median, Mode | 平均数与极差：均值、中位数、众数
In KS3 Cambridge Mathematics, you will often meet questions that ask you to summarise a list of numbers or a frequency table. Three key measures of central tendency are the mean, the median and the mode, while the range measures how spread out the data are. This revision guide explains each measure, shows how to calculate it step by step, and helps you avoid common mistakes.
在 KS3 剑桥数学中，你经常会遇到要求概括一组数字或频数表的问题。三个重要的集中趋势度量是均值、中位数和众数，而极差则衡量数据分散的程度。本复习指南逐一解释每种度量，展示逐步计算的方法，并帮助你避免常见错误。
1. What is an average? | 什么是平均数
An average is a single value used to represent the centre of a set of data. It gives a quick summary of a whole list of numbers without having to quote every value. There are three common averages: the mean, the median and the mode. They are useful in different situations, so you need to know how to calculate each one and when to use it.
平均数是一个用来代表一组数据中心位置的单一数值。它能在不必列出每一个值的情况下快速概括整组数字。常见的平均数有三种：均值、中位数和众数。它们在不同情境下各有用途，因此你需要知道如何计算每一种，以及什么时候该用哪一种。
Averages answer the question 'what is a typical value here?'. The range answers a different question: 'how spread out are these values?'. Together, one average plus the range gives a compact description of a data set.
平均数回答的是"这里的一个典型值是多少"。极差回答的是另一个问题："这些数值有多分散？"把某个平均数与极差放在一起，就能简洁地描述一组数据。
- Mean | 均值: add all values, divide by how many there are. 把所有数值相加，再除以数值的个数。
- Median | 中位数: the middle value when the data are put in order. 把数据从小到大排列后处于中间位置的值。
- Mode | 众数: the value that appears most often. 出现次数最多的那个数值。
- Range | 极差: largest value minus smallest value. 最大值减去最小值。
Note that the range is not an average. It is a measure of spread, and it tells you nothing about the centre of the data.
注意，极差不是平均数。它是一种离散程度的度量，完全不能说明数据的中心在哪里。
2. The Mean: Formula and Method | 均值：公式与方法
The mean is the most commonly used average. It uses every value in the data set, which is both its strength and its weakness: it reflects all the data, but a single extreme value can pull it a long way.
均值是最常用的平均数。它会用到数据集中的每一个数值，这既是它的优点也是它的缺点：它能反映全部数据，但一个极端值就可能把它拉得很远。
Mean = (sum of all values) ÷ (number of values)
x̄ = Σx ÷ n
Here x̄ means the mean (read as 'x bar'), Σ means 'add up all of', x stands for each individual value, and n is the number of values.
这里 x̄ 表示均值（读作"x 拔"），Σ 表示"把所有的加起来"，x 代表每一个单独的数值，n 是数值的个数。
Worked example: find the mean of 4, 7, 9, 11, 14.
示例：求 4、7、9、11、14 的均值。
- Step 1: Add them. 4 + 7 + 9 + 11 + 14 = 45. 第一步：相加，4 + 7 + 9 + 11 + 14 = 45。
- Step 2: Count them. n = 5. 第二步：数个数，n = 5。
- Step 3: Divide. 45 ÷ 5 = 9. 第三步：相除，45 ÷ 5 = 9。
So the mean is 9. Always check that your answer lies between the smallest and largest values. If it does not, you have almost certainly made an arithmetic slip.
所以均值是 9。一定要检查答案是否落在最小值与最大值之间。如果不在，几乎可以肯定你算错了。
3. The Mean from a Frequency Table | 从频数表计算均值
When data are given in a frequency table, you cannot simply add up the listed values, because each value stands for several items. Instead you multiply each value by its frequency, add those products, then divide by the total frequency.
当数据以频数表的形式给出时，你不能直接把列出的数值相加，因为每个数值都代表若干个数据项。正确做法是把每个数值乘以它的频数，把这些乘积加起来，再除以总频数。
Mean = Σ(f × x) ÷ Σf
In this formula, x is the data value and f is its frequency. Σf is the total number of items in the data set.
在这个公式中，x 是数据值，f 是它的频数。Σf 就是数据集中数据项的总数。
| Value x | 数值 x | Frequency f | 频数 f | f × x |
| 0 | 3 | 0 |
| 1 | 5 | 5 |
| 2 | 7 | 14 |
| 3 | 4 | 12 |
| 4 | 1 | 4 |
| Total | 合计 | Σf = 20 | Σ(f × x) = 35 |
tice that the mean does not have to be one of the original data values, and it is quite normal for a mean to be a decimal.
因此均值是 35 ÷ 20 = 1.75。注意，均值不一定等于原始数据中的某个值，均值是小数是完全正常的。
A common exam question gives grouped data, for example 0-9, 10-19, 20-29. In that case you use the midpoint of each class as x. For the class 0-9 the midpoint is 4.5, and for 10-19 it is 14.5. The answer is then an estimate of the mean, because the exact values are unknown.
常见的考题还会给出分组数据，例如 0-9、10-19、20-29。这时应该用每组的组中值作为 x。0-9 组的组中值是 4.5，10-19 组是 14.5。这样算出的只是均值的估计值，因为具体的原始数值是未知的。
4. The Median: Finding the Middle Value | 中位数：找出中间位置的值
The median is the middle value of an ordered list. It splits the data into two equal halves, and it is not affected by extreme values, which makes it a good choice when the data contain outliers.
中位数是把数据排序后处于中间位置的值。它把数据分成相等的两半，并且不受极端值影响，因此当数据中存在离群值时，它是更好的选择。
There are two cases to remember:
需要记住两种情况：
- Odd number of values | 数值个数为奇数: the median is the single middle value. 中位数就是正中间那一个值。
- Even number of values | 数值个数为偶数: the median is the mean of the two middle values. 中位数是中间两个值的平均值。
7, 9, 11, 14 we have n = 5, so the position is (5 + 1) ÷ 2 = 3, and the third value is 9.
一个可靠的方法是使用位置公式。对于排序后的 n 个数值，中位数位于第 (n + 1) ÷ 2 个位置。对于 4、7、9、11、14，n = 5，位置是 (5 + 1) ÷ 2 = 3，第三个值是 9。
etween the third and fourth values, i.e. between 6 and 9. The median is (6 + 9) ÷ 2 = 7.5.
再看 2、5、6、9、10、13。这里 n = 6，位置是 (6 + 1) ÷ 2 = 3.5。也就是说中位数位于第三个值和第四个值之间，即 6 和 9 之间。中位数是 (6 + 9) ÷ 2 = 7.5。
The single most important step is to put the numbers in order first. Forgetting to sort the list is the classic mistake with the median.
最关键的一步是先把数字排好序。忘记排序是中位数计算中最典型的错误。
5. The Median from a Frequency Table | 频数表中的中位数
To find the median from a frequency table, work with cumulative frequency. Add the frequencies as you go down the table, then find the position of the median using the total frequency.
要从频数表求出中位数，需要用累积频数。沿着表格往下把频数依次相加，然后用总频数确定中位数的位置。
the median lies at position (20 + 1) ÷ 2 = 10.5, halfway between the 10th and 11th items.
沿用前面的表格，Σf = 20，所以中位数位于第 (20 + 1) ÷ 2 = 10.5 个位置，也就是第 10 个和第 11 个数据项之间。
| Value x | 数值 x | f | Cumulative frequency | 累积频数 |
| 0 | 3 | 3 |
| 1 | 5 | 8 |
| 2 | 7 | 15 |
| 3 | 4 | 19 |
| 4 | 1 | 20 |
h the 10th and the 11th items therefore fall in the group where x = 2, so the median is 2.
当 x = 1 时累积频数为 8，当 x = 2 时累积频数为 15。第 10 个和第 11 个数据项都落在 x = 2 这一组，所以中位数是 2。
For grouped data, the median lies in the class where the cumulative frequency first passes half of Σf, and you quote that class as the median class.
对于分组数据，中位数所在的那一组，就是累积频数首次超过 Σf 一半的那一组，你只需写出这个组作为中位数所在组。
6. The Mode: The Most Common Value | 众数：出现最多的值
The mode is the value or item that occurs most frequently. It is the only average that can be used for non-numerical data, such as favourite colours or types of pet.
众数是出现次数最多的数值或项目。它是唯一能用于非数值数据的平均数，比如最喜欢的颜色或宠物的种类。
In the list 3, 5, 5, 5, 7, 8, 8, the mode is 5 because it appears three times.
在数列 3、5、5、5、7、8、8 中，众数是 5，因为它出现了三次。
In a frequency table, the mode is simply the value with the largest frequency. In the table above, the largest frequency is 7, so the mode is 2.
在频数表中，众数就是频数最大的那个数值。在上面的表格中，最大频数是 7，所以众数是 2。
- No mode | 没有众数: if every value appears exactly once, there is no mode. 如果每个数值都恰好出现一次，就没有众数。
- Two modes | 两个众数: if two values tie for the highest frequency, the data are bimodal. 如果两个数值的频数并列最高，这组数据就是双众数。
- More than two modes | 多于两个众数: it is better to say there is no single mode. 更严谨的说法是：不存在唯一的众数。
The mode is especially useful for shops and businesses. A shoe shop cares most about the best-selling size, which is the mode, because that is what it should stock most of.
众数对商店和企业特别有用。鞋店最关心的是销量最好的尺码，也就是众数，因为那才是应该多备货的尺码。
7. The Range: Measuring Spread | 极差：衡量离散程度
The range is the difference between the largest and smallest values. It is quick to calculate and easy to understand, but it depends only on two values, so it can be misleading when there is an outlier.
极差是最大值与最小值之差。它计算快捷、含义直观，但它只依赖两个数值，所以当存在离群值时可能产生误导。
Range = largest value − smallest value
For 4, 7, 9, 11, 14 the range is 14 − 4 = 10.
对于 4、7、9、11、14，极差是 14 − 4 = 10。
Compare two classes with the same mean mark of 60. Class A has marks from 55 to 65, so the range is 10. Class B has marks from 20 to 100, so the range is 80. The means are identical, but the classes are very different, and the range is what reveals that difference.
比较两个平均分都是 60 分的班级。A 班分数在 55 到 65 之间，极差是 10；B 班分数在 20 到 100 之间，极差是 80。两个班的均值完全相同，但实际情况差别很大，正是极差揭示了这一点。
The range is always zero or positive. If you ever calculate a negative range, you have subtracted in the wrong order.
极差永远大于或等于零。如果你算出了负数极差，说明你减法的顺序弄反了。
8. Choosing the Right Average | 选择合适的平均数
Different averages suit different data sets. Exam questions often ask you to justify a choice, so you should be able to explain your reasoning in one clear sentence.
不同的平均数适合不同的数据集。考试经常要求你说明选择理由，因此你要能用一句清楚的话解释自己的判断。
| Average | 平均数 | Best used when | 最适用的情形 | Weakness | 缺点 |
| Mean | 均值 | The data are fairly evenly spread and every value matters. 数据分布比较均匀，且每个值都重要。 | Affected by extreme values. 受极端值影响。 |
| Median | 中位数 | There are outliers or the data are skewed, e.g. house prices or wages. 存在离群值或数据偏斜时，例如房价或工资。 | Ignores the actual size of most values. 忽略了大多数数值的实际大小。 |
| Mode | 众数 | The data are categorical, or you want the most popular item. 数据是分类数据，或你想知道最受欢迎的项目。 | May not exist, or there may be several. 可能不存在，也可能有多个。 |
A useful sentence for the exam is: 'The median is better here because the very high value would make the mean misleading.' This shows the examiner you understand why, not just how.
考试中一个很实用的句子是："这里用中位数更合适，因为那个特别大的值会让均值产生误导。"这能向阅卷人表明你不仅会算，而且懂得为什么。
9. Comparing Two Data Sets | 比较两组数据
When a question gives you two data sets and asks you to compare them, you must compare an average and the range, and you must do it in context. A comparison that says 'the mean is bigger' earns few marks; one that says 'students in class A scored higher on average and were more consistent' earns full marks.
当题目给出两组数据要求比较时，你必须比较一个平均数以及极差，而且要结合具体情境。只写"均值更大"得分很低；写成"A 班学生平均得分更高，且成绩更稳定"才能拿满分。
A good comparison follows three steps:
一个完整的比较包含三步：
- Compare on average | 比较平均水平: e.g. the mean of A is 62 and of B is 58, so A is higher on average. 例如 A 的均值是 62，B 是 58，所以 A 的平均水平更高。
- Compare on spread | 比较离散程度: e.g. the range of A is 10 and of B is 35, so A is more consistent. 例如 A 的极差是 10，B 是 35，所以 A 更稳定。
- Write a conclusion in context | 结合情境下结论: e.g. A's training programme produced higher and more reliable results. 例如 A 的训练方案带来了更高且更稳定的成绩。
If one data set is higher on average but also more spread out, say so directly. Real data rarely give a perfectly clean answer, and examiners reward honest, precise statements.
如果一组数据平均水平更高但波动也更大，就直接说出来。真实数据很少给出完美的结论，阅卷人反而欣赏诚实而准确的表述。
10. Worked Example: A Full Question | 例题精讲：一道完整题目
Here is a typical KS3 question that combines everything in this guide. The number of goals scored in 12 matches was: 1, 3, 0, 2, 2, 5, 2, 1, 4, 2, 3, 1. Find the mean, median, mode and range.
下面是一道综合了本指南所有内容的典型 KS3 题目。某队 12 场比赛的进球数为：1、3、0、2、2、5、2、1、4、2、3、1。求均值、中位数、众数和极差。
2 + 1 + 4 + 2 + 3 + 1 = 26, and n = 12, so the mean is 26 ÷ 12 = 2.17 to 2 decimal places.
第一步，求均值。总和为 1 + 3 + 0 + 2 + 2 + 5 + 2 + 1 + 4 + 2 + 3 + 1 = 26，n = 12，所以均值是 26 ÷ 12 = 2.17（保留两位小数）。
, the median is the mean of the 6th and 7th values, which are 2 and 2. So the median is 2.
第二步，求中位数。把数据排序：0、1、1、1、2、2、2、2、3、3、4、5。n = 12，中位数是第 6 个和第 7 个值的平均，两个都是 2，所以中位数是 2。
Step 3, the mode. The value 2 appears four times, more than any other, so the mode is 2.
第三步，求众数。数值 2 出现了四次，多于其他任何数值，所以众数是 2。
tep 4, the range. The largest value is 5 and the smallest is 0, so the range is 5 − 0 = 5.
第四步，求极差。最大值是 5，最小值是 0，所以极差是 5 − 0 = 5。
Finally, interpret. The mean of 2.17 goals per match is above the median of 2, which suggests that the 5-goal match is pulling the mean upwards. The range of 5 shows that the team's scoring was quite variable.
最后进行解读。场均 2.17 球高于中位数 2，说明那场进 5 球的比赛把均值拉高了。极差为 5 说明该队的进球表现波动比较大。
11. Common Mistakes and Exam Tips | 常见错误与考试技巧
Most lost marks in this topic come from a small number of repeated slips. Check this list before every test.
这个专题丢分的原因大多集中在少数几个反复出现的失误上。每次测验前都对照下面这份清单检查一遍。
- Forgetting to order the data before finding the median | 求中位数前忘记排序. This is the number one error. 这是头号错误。
- Dividing by the wrong number | 除以了错误的个数. For a frequency table, divide by Σf, not by the number of rows. 在频数表中要除以 Σf，而不是表格的行数。
- Confusing the range with an average | 把极差误当成平均数. The range describes spread only. 极差只描述离散程度。
- Reporting the mode as a frequency | 把频数当作众数. The mode is the value, not how many times it occurs. 众数是那个数值，不是它出现的次数。
- Rounding too early | 过早四舍五入. Keep full accuracy until the final answer, then round as the question asks. 中间过程保留完整精度，最后再按题目要求取近似值。
- Giving a bare number with no context | 只给一个数字而不结合情境. Always write a short sentence in words. 一定要用一句话把结论表述出来。
One final tip on checking: the mean, median and mode should all lie between the smallest and largest values. If any of them does not, go back and check your working before moving on.
最后一条检查技巧：均值、中位数和众数都应该位于最小值与最大值之间。如果有任何一个不在这个范围内，先回头检查计算过程再继续往下做。
12. Quick Reference Summary | 速查总结
Keep this summary in your notes. It covers every formula you need for averages and range at KS3 level.
把这份总结记在笔记里。它涵盖了 KS3 阶段平均数与极差所需的全部公式。
| Measure | 度量 | Formula or rule | 公式或规则 |
| Mean (list) | 均值（数列） | Σx ÷ n |
| Mean (frequency table) | 均值（频数表） | Σ(f × x) ÷ Σf |
| Median (position) | 中位数（位置） | (n + 1) ÷ 2 after ordering the data | 排序后取第 (n + 1) ÷ 2 个位置 |
| Mode | 众数 | The value with the highest frequency | 频数最大的那个数值 |
| Range | 极差 | Largest − smallest | 最大值 − 最小值 |
| Grouped data midpoint | 分组数据组中值 | (lower bound + upper bound) ÷ 2 | （组下限 + 组上限）÷ 2 |
Remember the two key ideas. An average tells you where the centre of the data is, and the range tells you how spread out the data are. Together they give a compact and useful picture of any data set, whether it is presented as a simple list or as a frequency table.
记住两个核心概念：平均数告诉你数据的中心在哪里，极差告诉你数据有多分散。两者结合，就能对任何一组数据给出简洁而有用的描述，无论它是简单的数列还是频数表。
Practise with real data whenever you can, such as sports scores, weather records or your own test marks. The more familiar the numbers feel, the easier the exam questions become.
尽可能用真实数据练习，比如体育比分、天气记录或你自己的测验成绩。
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