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 KS3 Mathematics, you will often work with sets of numbers collected from surveys, experiments or everyday situations. To make sense of these numbers, statisticians use measures of central tendency (averages) and the range. The three main averages are the mean, median and mode. Each one gives a ‘typical’ value but in a different way. The range tells you how spread out the data is. Understanding these concepts helps you interpret graphs, compare groups and solve real-world problems such as analysing test scores, weather data or sports performance.

在 KS3 数学中，你会经常处理从调查、实验或日常生活场景中收集到的数据集合。为了理解这些数字，统计学家会使用集中趋势量数（平均数）和极差。三个主要的平均数是平均数、中位数和众数。每一种都以不同的方式给出一个‘典型’值。极差则告诉你数据的分散程度。理解这些概念有助于你解读图表、比较不同组别，并解决现实世界中的问题，例如分析考试成绩、天气数据或运动表现。

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1. What Are Averages and Range? | 什么是平均数和极差？

An average is a single value that represents the centre of a data set. It is a way of summarising many numbers with one number. However, not all averages are the same. The mean uses all the data values and balances them out. The median is the middle value when the data is ordered. The mode is the value that appears most often. The range is not an average—it measures how far apart the highest and lowest values are. Together, averages and the range give a quick summary of a data set’s centre and spread.

平均数是一个能够代表数据集中心的单一数值。它是一种用一个数字来概括许多数字的方法。然而，并非所有平均数都一样。平均数会用到所有数据值并进行平衡。中位数是将数据排序后位于中间的值。众数是出现次数最多的值。极差不是平均数——它衡量的是最大值和最小值之间的差距。平均数和极差一起，可以快速概括数据集的中心和离散程度。

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2. The Mean: Calculation and Understanding | 平均数：计算与理解

The mean (often called the average) is found by adding up all the values and then dividing by the number of values. The formula is: Mean = sum of all data values ÷ number of data values. It is the most commonly used average because it includes every piece of data. However, the mean can be affected by very high or very low outliers. For example, if one person in a group earns a million pounds while others earn twenty thousand, the mean salary might suggest everyone earns more than they really do. In such cases, the median might be more useful.

平均数（通常被称为均值）的计算方法是：将所有数值相加，然后除以数值的个数。公式为：平均数 = 所有数据值的总和 ÷ 数据值的个数。这是最常用的平均数，因为它包含了每一个数据。然而，平均数可能会受到极高或极低异常值的影响。例如，如果一组人中有一人赚了一百万英镑，而其他人赚两万英镑，那么平均工资可能会让人觉得每个人的收入都比实际收入高。在这种情况下，中位数可能更有用。

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3. Worked Example: Finding the Mean | 计算实例：求平均数

Let us find the mean of these five test scores: 8, 6, 9, 7, 10. First, add the numbers: 8 + 6 + 9 + 7 + 10 = 40. Next, count how many values there are: 5. Then divide the total by 5: 40 ÷ 5 = 8. So the mean score is 8. Notice that the mean is not necessarily one of the original data values. In this set, 8 appears in the list, but often the mean will be a number not present in the data.

我们来求出这五个考试分数：8、6、9、7、10 的平均数。首先，把这些数字相加：8 + 6 + 9 + 7 + 10 = 40。接着，数出数据值的个数：5。然后用总和除以 5：40 ÷ 5 = 8。因此平均分是 8。注意，平均数不一定是原始数据值中的一个。在这个数据集中，8 出现在列表中，但很多时候平均数会是数据中不存在的数字。

Mean = (8 + 6 + 9 + 7 + 10) ÷ 5 = 40 ÷ 5 = 8

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4. Understanding the Mean in Context | 在情境中理解平均数

The mean is useful when the data is fairly symmetrical and there are no extreme values. It is often used for things like average temperature, average height or average marks. However, you must be careful when interpreting it. If a class has test scores of 0, 70, 75, 80, 85, the mean is (0+70+75+80+85) ÷ 5 = 62. This mean does not truly reflect the performance of most students because one very low score pulls it down. Always look at the data before deciding which average to use.

当数据大致对称并且没有极端值时，平均数很有用。它经常用于平均温度、平均身高或平均分数等情况。但是在解读时必须小心。如果一个班级的考试成绩为 0、70、75、80、85，平均数是 (0+70+75+80+85) ÷ 5 = 62。这个平均数并不能真实反映大多数学生的表现，因为一个非常低的分数拉低了它。在决定使用哪种平均数之前，一定要先查看数据。

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5. The Median: The Middle Value | 中位数：中间值

The median is the middle number when the data is arranged in order from smallest to largest. If there is an odd number of values, the median is simply the central value. If there is an even number of values, the median is the mean of the two middle numbers. The median is not affected by outliers, making it a better average when data is skewed. For example, house prices often use the median because a few expensive houses can make the mean misleading.

中位数是将数据从小到大排列后位于中间的那个数。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平均数。中位数不受异常值的影响，因此在数据有偏斜时是一个更好的平均数指标。例如，房价通常会使用中位数，因为少数昂贵的房子可能会使平均数产生误导。

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6. Finding the Median: Step-by-Step | 计算中位数的步骤

To find the median, first list the numbers in ascending order. For the data set 12, 7, 9, 15, 6, first order them: 6, 7, 9, 12, 15. With 5 numbers, the median is the third value: 9. For an even set like 4, 8, 1, 3, order them: 1, 3, 4, 8. The two middle numbers are 3 and 4. Add them and divide by 2: (3 + 4) ÷ 2 = 3.5. The median is 3.5. Remember to always sort the data first, otherwise you will get a wrong answer.

求中位数时，先将数字按升序排列。对于数据集 12、7、9、15、6，先排序为：6、7、9、12、15。当有 5 个数字时，中位数是第三个值：9。对于偶数个数据的情况，比如 4、8、1、3，排序为：1、3、4、8。中间两个数是 3 和 4。把它们相加并除以 2：(3 + 4) ÷ 2 = 3.5。中位数是 3.5。记住一定要先排序，否则你会得到错误答案。

Median (odd set) = middle ordered value

Median (even set) = (middle two values) ÷ 2

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7. The Mode: The Most Frequent | 众数：最频繁出现的值

The mode (or modal value) is the number that appears most often in a data set. A set can have one mode, more than one mode (bimodal or multimodal), or no mode at all if all values appear with the same frequency. The mode is the only average that can be used for non-numeric data, such as finding the most popular colour or favourite subject. It is very easy to spot, but it may not always be useful, especially if the data has many different values with the same frequency.

众数（或模态值）是数据集中出现次数最多的那个数。一个数据集可以有一个众数、多个众数（双众数或多众数），或者如果所有值都出现相同次数，就可能没有众数。众数是唯一可以用于非数值数据的平均数，例如找出最流行的颜色或最喜爱的科目。它很容易被发现，但并不总是有用，尤其是在数据有很多不同值且出现频率都相同时。

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8. Finding the Mode: Tips | 求众数的小贴士

To find the mode, look for the value that repeats the most. In the list 5, 3, 8, 3, 9, 3, 5, the number 3 appears three times, while others appear once or twice. Therefore, the mode is 3. If the data is given in a frequency table, the mode is the value with the highest frequency. Always check for multiple modes. For example, in 2, 2, 4, 4, 6, 6, there is no clear mode unless two numbers tie for the highest frequency, then both are modes. Understanding frequency tables helps you quickly identify the modal class interval in grouped data, though that is beyond KS3 level, the idea is introduced.

要找出众数，就寻找重复次数最多的值。在列表 5、3、8、3、9、3、5 中，数字 3 出现了三次，而其他数字出现一或两次。因此，众数是 3。如果数据是以频数表的形式给出，众数就是频数最高的那个值。一定要检查是否存在多个众数。例如在 2、2、4、4、6、6 中，如果不出现两个数字并列最高频数的情况，就没有明显的众数，但如果并列，那么两者都是众数。理解频数表有助于你在分组数据中快速识别众数所在的组，虽然这超出了 KS3 的水平，但概念已经引入。

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9. The Range: Spread of Data | 极差：数据的分散程度

The range is the difference between the largest and smallest values in a data set. It is calculated simply: Range = largest value – smallest value. The range gives an idea of how spread out the data is. A small range means the data is very consistent, while a large range shows more variability. For example, the temperatures 18°C, 20°C, 19°C, 21°C have a range of 21 – 18 = 3°C, indicating stable weather. But unlike averages, the range does not give a central value; it is a measure of dispersion.

极差是一组数据中最大值与最小值之差。它的计算方法很简单：极差 = 最大值 – 最小值。极差可以让你了解数据的离散程度。极差小意味着数据很一致，而极差大则显示变化较大。例如，气温 18°C、20°C、19°C、21°C 的极差是 21 – 18 = 3°C，表明天气稳定。但与平均数不同，极差不提供中心值；它是一种离散程度的量数。

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10. Choosing the Right Average | 选择正确的平均数

Each average has strengths and weaknesses. Use the mean when the data has no outliers and you want a value that involves all numbers. Use the median when there are extreme values or the distribution is skewed, because it is more representative. Use the mode when data is non-numeric or you want the most popular category. Sometimes you need to calculate all three to give a complete picture. Adding the range helps describe how consistent the data is. For example, in comparing two football players’ goal-scoring records, the mean and median might be similar but a bigger range suggests less consistency.

每种平均数都有优点和缺陷。当数据没有异常值并且你希望得到一个涵盖所有数字的值时，使用平均数。当存在极端值或分布偏斜时，使用中位数，因为它更具代表性。当数据是非数值型或你需要找出最流行类别时，使用众数。有时你需要计算出所有三种平均数才能给出完整的图景。加上极差有助于描述数据的一致性。例如，在比较两名足球运动员的进球记录时，平均数和中位数可能相似，但更大的极差表明稳定性较差。

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11. Common Mistakes to Avoid | 需要避免的常见错误

Students often forget to order the data before finding the median, leading to an incorrect middle value. Another mistake is dividing incorrectly when calculating the mean, especially with decimals. When finding the mode, some write ‘no mode’ without checking if each number appears the same number of times. For the range, always subtract the smallest from the largest; never forget to use subtraction. Also, be careful with zero: zero is a data value and should be included in the calculation of mean, median, and range. Misinterpreting frequency tables is another common pitfall—make sure you use the frequencies correctly.

学生们经常忘记在找中位数之前对数据进行排序，从而导致找到错误的中间值。另一个错误是在计算平均数时除法执行错误，尤其是涉及小数时。在找众数时，有些人没有检查每个数字的出现次数是否相同就写‘无众数’，这是不对的。对于极差，总是用最大值减去最小值；千万不要忘记使用减法。此外，对零要小心：零是一个数据值，应该被包括在平均数、中位数和极差的计算中。误解频率表是另一个常见的陷阱——确保你正确使用了频数。

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12. Practice Questions and Summary | 练习题和总结

Try these quick questions to test your understanding: (a) Find the mean, median, mode and range of 4, 7, 2, 7, 10. (b) A shop records daily sales over a week: £120, £150, £110, £130, £140. What is the mean sales per day? (c) The heights of five plants in cm are 14, 18, 15, 18, 20. Which average best represents the data? Always remember: mean is the balance point, median is the middle, mode is the most frequent, and range shows the spread. Master these four tools and you’ll have a solid foundation for handling data confidently in KS3 and beyond.

试试这些快速问题来检验你的理解：(a) 求 4、7、2、7、10 的平均数、中位数、众数和极差。(b) 一家商店记录了一周的日销售额：£120、£150、£110、£130、£140。每天的日平均销售额是多少？(c) 五株植物的高度（单位：cm）为 14、18、15、18、20。哪种平均数最能代表这组数据？永远记住：平均数是平衡点，中位数是中间值，众数是最常见的值，极差显示离散程度。掌握这四个工具，你就能为自信地处理数据打下坚实的基础，无论是 KS3 还是更远的学习。

Average	计算方式 / How to find	受异常值影响？/ Affected by outliers?
平均数 (Mean)	总和 ÷ 个数 / Sum ÷ count	是 / Yes
中位数 (Median)	排序后中间值 / Middle ordered value	否 / No
众数 (Mode)	出现次数最多的值 / Most frequent value	否 / No
极差 (Range)	最大值 – 最小值 / Highest – Lowest	是 / Yes

For the plant heights in (c): mean = (14+18+15+18+20)÷5 = 17 cm, median = 18 cm, mode = 18 cm. Here the mode or median might be better if we think 15 cm is slightly low. Practice more with your own data sets to become fluent.

对于 (c) 中的植物高度：平均数 = (14+18+15+18+20)÷5 = 17 cm，中位数 = 18 cm，众数 = 18 cm。如果我们认为 15 cm 稍低，众数或中位数可能更好。用你自己的数据集多练习，以达到熟练。

Published by TutorHao | KS3 Mathematics Revision Series | aleveler.com

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