📚 Mastering Averages: Mean, Median, Mode and Range | 掌握平均数：均值、中位数、众数和极差
In KS3 Mathematics, understanding how to summarise a set of numbers using averages is a fundamental skill. Whether you are analysing test scores, measuring temperatures, or comparing sports statistics, three key measures of central tendency – the mean, median, and mode – together with the range, help you describe data in a meaningful way. This article will guide you through each concept step by step, with clear examples and common pitfalls to avoid.
在KS3数学中，理解如何使用平均数来概括一组数字是一项基本技能。无论你是在分析考试分数、测量温度还是比较体育统计数据，三个关键的中心趋势度量——均值、中位数和众数，以及极差，都能帮助你以有意义的方式描述数据。本文将带你逐步学习每个概念，并通过清晰的示例和常见错误提醒，让你轻松掌握。
1. Understanding the Mean | 理解均值
first add them: 4 + 8 + 6 + 5 + 7 = 30. Since there are 5 numbers, divide 30 by 5 to get a mean of 6.
均值是最常见的平均数类型。要计算一组数字的均值，你需要将所有数值相加，然后将总和除以数值的个数。它有时被称为算术平均数，因为它涉及基本的算术运算。例如，计算4、8、6、5和7的均值：先将它们相加：4 + 8 + 6 + 5 + 7 = 30。因为有5个数字，所以将30除以5，得到均值为6。
The mean is very useful when all the data points are relatively close to each other, but it can be misleading if there are extreme values (outliers). For instance, if one person's income is ten times higher than everyone else's in a group, the mean income will be pulled upward and no longer represent a 'typical' value.
当所有数据点都相对接近时，均值非常有用，但如果存在极端值（离群值），均值就可能产生误导。例如，如果一群人中有一个人的收入是其他人的十倍，那么平均收入就会被拉高，无法代表“典型”值。
In formula form, the mean is often written as:
Mean = (Sum of all values) ÷ (Number of values)
公式形式通常写作：
均值 = （所有数值的总和）÷（数值的个数）
2. Finding the Median | 求中位数
65, 170. The median is 160 cm, the third value. Now take the set: 12, 14, 15, 17. Since there are four numbers, the two middle numbers are 14 and 15. Their mean is (14 + 15) ÷ 2 = 14.5, so the median is 14.5.
中位数是将一组数据从小到大排列后处于中间位置的数值。如果数值的个数是奇数，中位数就是正中间的那个数；如果是偶数，中位数就是中间两个数的均值。以身高（厘米）为例：150、155、160、165、170。中位数是160厘米，即第三个数值。再看数据集：12、14、15、17。因为有四个数字，中间的两个数是14和15。它们的均值是 (14 + 15) ÷ 2 = 14.5，所以中位数为14.5。
The median is particularly useful when the data contains outliers. For example, if a class test score set is {3, 7, 8, 9, 10}, the median is 8, which gives a better sense of the typical performance than the mean (which is 7.4). If an outlier score of 30 is added, the median becomes 8.5, while the mean jumps to 11.17 – the median remains much more stable.
中位数在数据包含离群值时特别有用。例如，一个班级的考试分数为{3, 7, 8, 9, 10}，中位数是8，这比均值（7.4）更能体现典型表现。如果再加入一个离群分数30，中位数变为8.5，而均值跳升至11.17——中位数保持稳定得多。
Always remember to order the data first – failing to sort is one of the most common mistakes students make when finding the median.
务必记住先对数据进行排序——忘记排序是学生在求中位数时最常犯的错误之一。
3. Identifying the Mode | 确定众数
The mode is the value that appears most often in a data set. A set of numbers can have one mode, more than one mode (bimodal or multimodal), or no mode at all if no number repeats. For instance, in the set {2, 3, 4, 4, 5, 5, 7, 9}, the number 4 appears twice and 5 appears twice, so there are two modes: 4 and 5. The set {1, 2, 3} has no mode because each number occurs only once.
众数是数据集中出现次数最多的数值。一组数字可以有一个众数、多个众数（双众数或多众数），或者如果没有数字重复出现，则没有众数。例如，在数据集{2, 3, 4, 4, 5, 5, 7, 9}中，4出现了两次，5也出现了两次，所以众数为4和5。而{1, 2, 3}没有众数，因为每个数字只出现一次。
The mode is especially useful for categorical data, such as favourite colours or car models, where calculating a mean or median is not possible. It tells you the most popular choice at a glance. However, for numerical data, the mode can sometimes be at the extreme end and not represent the centre well.
众数对于类别数据尤其有用，比如最喜欢的颜色或汽车型号，因为这些数据无法计算均值或中位数。众数可以让你一眼看出最受欢迎的选择。不过，对于数值数据，众数有时可能处于极端位置，不能很好地代表中心。
Mode is the only average that can be used with non-numeric data, making it unique among the measures of central tendency.
众数是中心趋势度量中唯一可用于非数值数据的平均数，这使其独一无二。
4. Calculating the Range | 计算极差
ts are closely bunched together; a large range indicates greater variability. The range is often reported alongside the mean, median or mode to give a fuller picture of the data set.
极差不是平均数，而是衡量数据分散程度的量度。它就是最大值与最小值之间的差。例如，数据集{3, 7, 8, 12, 15}的极差是15 - 3 = 12。极差小表示数据点紧密聚集在一起；极差大表示变异性更大。极差通常与均值、中位数或众数一同报告，以便更全面地呈现数据集。
One weakness of the range is that it only uses the two most extreme values, so a single outlier can make the range very large and not truly representative of the overall spread. Despite this, it is quick to calculate and easy to understand, making it a useful first step in data analysis.
极差的一个缺点是它只使用两个最极端的值，因此一个离群值就可能使极差变得非常大，无法真正代表整体离散程度。尽管如此，它计算快速且易于理解，是数据分析的有用第一步。
5. Choosing the Right Average | 选择合适的平均数
Choosing between the mean, median, and mode depends on the type of data and the presence of outliers. For symmetric data without outliers, the mean is usually the best choice because it uses all the data values. For skewed data or data with outliers, the median is more robust and gives a better central value. The mode is best when you need the most frequent value or when dealing with categorical data.
在均值、中位数和众数之间进行选择取决于数据的类型和是否存在离群值。对于没有离群值的对称数据，均值通常是最佳选择，因为它利用了所有数据值。对于偏斜数据或有离群值的数据，中位数更稳健，能给出更好的中心值。当你需要最常见的值或处理类别数据时，众数是最佳选择。
A good habit is to calculate all three averages first, then decide which one tells the most honest story. For example, in a salary survey, the median salary is often reported because a few extremely high earners would distort the mean. On the other hand, a shoe shop might use the mode to determine which shoe size to stock the most.
一个好习惯是首先计算出所有三个平均数，然后判断哪一个能最真实地反映情况。例如，在薪资调查中，通常报告中位数工资，因为少数极高收入者会扭曲均值。另一方面，鞋店可能使用众数来决定哪种鞋码需要最多备货。
6. Step-by-Step Worked Example | 逐步求解示例
Let us work through a complete example to see how these concepts link together. Here is a set of eight students' marks out of 50 in a science quiz: 18, 22, 45, 22, 36, 22, 40, 35. First, arrange the marks in ascending order: 18, 22, 22, 22, 35, 36, 40, 45.
我们来完成一个完整的示例，看看这些概念如何相互关联。以下是八名学生在一场科学测验中的成绩（满分50分）：18、22、45、22、36、22、40、35。首先，将分数按升序排列：18、22、22、22、35、36、40、45。
- Mean: = 18 + 22 + 22 + 22 + 35 + 36 + 40 + 45 = 240. Number of values = 8. Mean = 240 ÷ 8 = 30.
- 均值：总和 = 18 + 22 + 22 + 22 + 35 + 36 + 40 + 45 = 240。数值个数 = 8。均值 = 240 ÷ 8 = 30。
- Median: bers, the two middle values are the 4th and 5th: 22 and 35. Median = (22 + 35) ÷ 2 = 28.5.
- 中位数：有八个已排序的数字，中间两个是第4和第5个数：22和35。中位数 = (22 + 35) ÷ 2 = 28.5。
- Mode: The number 22 appears three times, more than any other value, so the mode is 22.
- 众数：数字22出现了三次，多于任何其他数值，所以众数是22。
- Range: Maximum 45 minus minimum 18 gives a range of 27.
- 极差：最大值45减去最小值18，得到极差为27。
From this, you can see that although the mean is 30, the median is 28.5 and the mode is 22, which reflects that most students scored around the low 20s. The range of 27 shows quite a wide spread of marks.
从这里可以看出，尽管均值是30，但中位数是28.5，众数是22，这反映出大多数学生的分数在20分左右。27分的极差则表明分数分布相当广泛。
7. Common Mistakes and How to Avoid Them | 常见错误及如何避免
Forgetting to sort data for the median. Always rewrite the list in ascending order before locating the middle. A quick check: count the number of values, then count across from both ends to ensure you have the correct middle.
求中位数时忘记排序。在确定中间位置之前，始终将数据列表按升序重新排列。快速检查方法：数出数值的总个数，然后从两端向中间数，确保你找到了正确的中间位置。
Dividing incorrectly when finding the mean. Double-check your addition and division. A helpful tip is to estimate the mean roughly before calculating to see if your final answer makes sense.
求均值时除法出错。仔细检查加法和除法。一个有用的技巧是在计算前先大致估算均值，看最终答案是否合理。
Confusing the mode with the highest frequency. The mode is the value that occurs most frequently, not the frequency itself. In the earlier example, the mode is 22, not 3 (which is the frequency).
混淆众数和最高频次。众数是出现最频繁的值，而不是频次本身。在前面的示例中，众数是22，而不是3（3是频次）。
Forgetting the range uses only two numbers. You do not need to do anything with the other values. Simply subtract the smallest from the largest.
忘记极差只使用两个数字。你不需要对其他数值进行任何运算，只需用最大值减去最小值即可。
8. Using Averages in Real-Life Contexts | 在现实情境中使用平均数
Averages appear everywhere in daily life. Weather reports give mean monthly temperatures, sports statistics use batting averages and points per game (mean), and schools analyse test scores using medians to compare classes. When you read that the 'average household income' has increased, it is often the median that is quoted to avoid distortion from the very rich. The mode helps businesses decide on the most popular product size or colour.
平均数在日常生活中无处不在。天气预报提供月均温度，体育统计使用击球率或场均得分（均值），学校在比较班级时使用中位数来分析考试成绩。当你读到“平均家庭收入”增长时，通常引用的是中位数，以避免极高收入家庭的扭曲。众数则帮助企业决定最畅销的产品尺寸或颜色。
Understanding these concepts also helps you critically evaluate statistics presented in the news. Ask yourself: which average is being used? Could there be outliers affecting the mean? What does the range tell us about the consistency of the data?
理解这些概念也有助于你批判性地评价新闻中呈现的统计数据。问问自己：他们使用的是哪种平均数？是否存在影响均值的离群值？极差告诉我们数据的稳定性如何？
9. Practice Questions to Test Your Skills | 练习题检验你的能力
Try these questions to strengthen your understanding. Solutions are provided at the end of the article.
试试下面的题目来强化理解。答案在文章末尾。
- Find the mean, median, mode and range of: 8, 10, 12, 15, 15, 16, 20, 22.
- 找出以下数据的均值、中位数、众数和极差：8, 10, 12, 15, 15, 16, 20, 22。
- A runner’s times (in minutes) for five races are: 24, 26, 28, 30, 45. Which average best represents her typical performance? Explain.
- 一位跑步运动员五场比赛的时间（分钟）为：24、26、28、30、45。哪个平均数最能代表她的典型表现？请解释。
- The shoe sizes of ten students are: 5, 7, 6, 5, 8, 5, 9, 7, 6, 5. What is the mode? Why is it useful here?
- 十名学生的鞋码为：5、7、6、5、8、5、9、7、6、5。众数是多少？为什么这里使用众数有用？
10. Solutions to Practice Questions | 练习题解答
Question 1: Ordered data: 8, 10, 12, 15, 15, 16, 20, 22.
Mean = (8+10+12+15+15+16+20+22) ÷ 8 = 118 ÷ 8 = 14.75.
Median: there are 8 numbers; middle two are 15 and 15, so median = 15.
Mode: 15 appears twice, so mode = 15.
Range: 22 – 8 = 14.
第1题：排序后数据：8, 10, 12, 15, 15, 16, 20, 22。
均值 = (8+10+12+15+15+16+20+22) ÷ 8 = 118 ÷ 8 = 14.75。
中位数：有8个数字，中间两个是15和15，所以中位数为15。
众数：15出现两次，所以众数为15。
极差：22 – 8 = 14。
Question 2: The median would be best. The time 45 is an outlier and pulls the mean (30.6) upward; the median is 28, which more accurately reflects her typical race time.
第2题：中位数最合适。时间45是一个离群值，将均值（30.6）拉高；中位数是28，更准确地反映了她通常的比赛时间。
Question 3: The mode is 5. It is useful because shoe size is discrete and the shop needs to know the most frequently demanded size to manage stock.
第3题：众数是5。它有用是因为鞋码是离散的，商店需要知道最常见的需求尺码来管理库存。
11. Using Technology to Check Your Work | 使用技术检查作业
You can use spreadsheets like Excel or Google Sheets to calculate averages quickly. For the mean, use the AVERAGE function; for median, MEDIAN; for mode, MODE.SNGL; and subtract MIN from MAX for the range. However, it’s essential to understand the manual methods so you can interpret the results correctly and spot errors.
你可以使用像Excel或Google Sheets这样的电子表格软件快速计算平均数。均值使用AVERAGE函数；中位数使用MEDIAN；众数使用MODE.SNGL；极差则用MAX减去MIN。然而，理解手动计算方法很关键，这样你才能正确解读结果并发现错误。
In exams, you might not have access to technology, so practising by hand builds both speed and confidence. Always show your working clearly – marks are often awarded for method even if the final answer is slightly off.
在考试中，你可能无法使用技术工具，因此手动练习可以提升速度和信心。务必清晰地展示解题步骤——即使最终答案略有偏差，方法步骤通常也能得分。
12. Summary and Key Takeaways | 总结与要点回顾
- The mean is the sum of values divided by the count – best for symmetric data.
- 均值是所有值的总和除以个数——最适合对称数据。
- The median is the middle ordered value – best when outliers are present.
- 中位数是排序后的中间值——存在离群值时最合适。
- The mode is the most frequent value – best for categorical data.
- 众数是最频繁出现的值——最适合类别数据。
- The range measures spread: maximum minus minimum.
- 极差衡量离散度：最大值减最小值。
- Always sort data before finding the median, and check for multiple modes.
- 在求中位数前务必排序，并检查是否存在多个众数。
- Consider the context to choose the most appropriate average.
- 根据实际情境选择最合适的平均数。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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