📚 Averages and Range | 平均数与范围
In Key Stage 3 Mathematics, understanding how to summarise a set of numbers is essential. Averages and the range help us describe the centre and spread of data. Whether you are analysing test scores, temperatures, or sports results, these tools allow you to make sense of the numbers quickly and accurately. This article covers the three main averages — mean, median, and mode — along with the range, and shows you how to calculate them for both simple lists and frequency tables.
在 KS3 数学中，理解如何概括一组数据至关重要。平均数与范围帮助我们描述数据的中心趋势和离散程度。无论是分析考试成绩、气温变化还是运动比赛结果，这些工具都能让你快速准确地理解数字背后的含义。本文将介绍三种主要的平均数——算术平均数、中位数和众数，以及范围，并展示如何对简单列表和频数表进行计算。
1. What is an Average? | 什么是平均数？
An average is a single value that represents the typical or central value of a dataset. It gives us a quick summary of a large set of numbers without having to look at every individual data point. In KS3, we study three types of average: the mean, the median, and the mode. Each one describes the centre of the data in a different way, and knowing when to use each is a valuable skill.
平均数是一个代表数据集中典型或中心值的单一数值。它让我们无需逐一检查每个数据点，就能快速了解大量数字的概况。在 KS3 阶段，我们学习三种平均数：算术平均数、中位数和众数。每一种都以不同的方式描述数据的中心，知道何时使用哪一种是一项重要的技能。
2. The Mean | 算术平均数
The mean is what most people think of as the average. To find the mean, you add up all the values and then divide the total by the number of values. This gives a sense of the 'fair share' if all values were distributed equally. The mean takes every data point into account, which makes it very useful when the data does not have extreme outliers.
算术平均数是大多数人提到平均数时所指的概念。要计算算术平均数，需将所有数值相加，再将总和除以数值的个数。这就像将所有数值重新平均分配后每人应得的份额。算术平均数考虑了每一个数据点，因此在数据没有极端异常值时非常有用。
3. Calculating the Mean | 计算算术平均数
he number of values, which is 5. So the mean = 30 ÷ 5 = 6. We can write this as a formula:
Mean = (Sum of all values) ÷ (Number of values)
假设我们有一组数据：4, 8, 6, 5, 7。首先求和：4 + 8 + 6 + 5 + 7 = 30。然后除以数值的个数 5。因此算术平均数 = 30 ÷ 5 = 6。我们可以将此写为公式：
算术平均数 = (所有数值之和) ÷ (数值的个数)
When working with larger sets, careful addition and accurate division are essential. Using a calculator is often allowed in Cambridge assessments, but you must show your method clearly.
处理较大的数据集时，仔细做加法和精确的除法至关重要。在剑桥考试中通常允许使用计算器，但必须清晰地展示解题步骤。
4. The Median | 中位数
The median is the middle value when the data is arranged in order from smallest to largest. If there is an even number of values, the median is the mean of the two middle numbers. The median is not affected by extremely high or low values, which makes it a better representative when the data contains outliers.
中位数是将数据从小到大排序后位于中间位置的数值。如果有偶数个数值，则中位数是中间两个数的算术平均数。中位数不受极高或极低值的影响，因此当数据包含异常值时，它能更好地代表数据的中心。
5. Finding the Median | 找中位数
always order the data first. A common mistake is to pick a middle number from an unordered list.
对于数据集 3, 7, 9, 12, 15，中位数是 9，因为它正好位于正中间。对于偶数个数据，如 2, 4, 6, 8，中间的两个数是 4 和 6，因此中位数 = (4 + 6) ÷ 2 = 5。请务必先将数据排序。一个常见错误是从未排序的列表中挑选中间的数。
6. The Mode | 众数
The mode is the value that appears most often in a dataset. A set may have one mode, more than one mode (bimodal or multimodal), or no mode at all if all values occur only once. The mode is especially useful for non-numerical data, such as finding the most popular colour or favourite subject.
众数是数据集中出现次数最多的值。一组数据可能有一个众数、多个众数（双众数或多众数），或者如果所有值都只出现一次，则没有众数。众数对于非数值数据特别有用，例如找出最受欢迎的颜色或最喜欢的科目。
7. The Range | 范围
lest value. A larger range means the data is more varied; a smaller range indicates that the values are more consistent. The range is not an average — it describes dispersion, not central tendency.
范围衡量数据的离散程度。它的计算方法是最大值减去最小值。范围 = 最大值 − 最小值。范围越大表示数据越分散；范围越小说明数值越集中。范围不是平均数——它描述的是离散程度，而非中心趋势。
8. Choosing the Right Average | 选择合适的平均数
Each average tells a different story about the data. Use the mean when you want to include every value and the data is fairly symmetric with no extreme outliers. Choose the median when there are outliers or skewed data, as it remains stable. The mode is best when you need the most frequent category or when working with non-numeric information. In real-life situations, like reporting house prices, the median is often preferred because a few very expensive houses can distort the mean.
每种平均数都从不同角度描述数据。当你想包含每个数值且数据大致对称、没有极端异常值时，使用算术平均数。当存在异常值或数据偏斜时选择中位数，因为它保持稳定。当需要最常见的类别或处理非数值信息时，众数是最佳选择。在现实生活中，比如报告房价时，通常倾向于使用中位数，因为少数非常昂贵的房子会扭曲算术平均数。
9. Averages from Frequency Tables | 从频数表中计算平均数
Sometimes data is presented in a frequency table instead of a raw list. To find the mean from a frequency table, multiply each value by its frequency, sum these products, and then divide by the total frequency. For the median and mode, the table can help locate the middle position or the highest frequency quickly.
有时数据会以频数表的形式呈现，而不是原始列表。要从频数表中计算算术平均数，需将每个值乘以其频数，将这些乘积相加，再除以总频数。对于中位数和众数，表格可以帮助快速定位中间位置或最高频数。
Example table:
| Value (x) | Frequency (f) | f × x |
|---|---|---|
| 2 | 3 | 6 |
| 4 | 5 | 20 |
| 6 | 2 | 12 |
total frequency = 3 + 5 + 2 = 10. Mean = 38 ÷ 10 = 3.8. The mode is 4 (highest frequency).
f × x 的总和 = 6 + 20 + 12 = 38，总频数 = 3 + 5 + 2 = 10。算术平均数 = 38 ÷ 10 = 3.8。众数是 4（频数最高）。
10. Outliers and Their Effect | 异常值及其影响
pical value well. The median is 2, which is far more typical. Always check for outliers before deciding which average to use.
异常值是指远大于或远小于数据集中其他值的数值。异常值会对算术平均数产生很大影响，将其拉向极端方向。中位数和众数受影响较小。例如，在数据集 1, 2, 2, 3, 20 中，算术平均数为 (1+2+2+3+20) ÷ 5 = 5.6，这并不能很好地代表典型值。中位数为 2，更具代表性。在决定使用哪种平均数之前，务必检查是否存在异常值。
11. Practice Tips for KS3 | KS3 练习建议
When revising averages and range, always work through a variety of questions: simple lists, frequency tables, and word problems. Pay attention to keywords in Cambridge assessments: 'compare the averages and range' often means you should calculate both central tendency and spread, then make a comparative statement. Show all your working steps — even if the final answer is wrong, you can earn method marks.
在复习平均数与范围时，要练习各种题型：简单列表、频数表和应用题。注意剑桥考试中的关键词：“比较平均数和范围”通常意味着你需要同时计算中心趋势和离散程度，然后做出比较性陈述。展示所有解题步骤——即使最终答案错误，你仍可获得过程分。
12. Summary | 总结
- Mean: sum of values divided by count; sensitive to outliers. | 算术平均数：数值之和除以个数；对异常值敏感。
- Median: middle value when sorted; robust against outliers. | 中位数：排序后位于中间的值；不易受异常值影响。
- Mode: most frequent value; useful for categories. | 众数：出现次数最多的值；适用于分类数据。
- Range: difference between highest and lowest; measures spread. | 范围：最大值与最小值之差；衡量离散程度。
Mastering these concepts will give you a solid foundation for data handling topics all the way through IGCSE and beyond.
掌握这些概念将为你在 IGCSE 及更高阶段的数据处理专题打下坚实基础。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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