Organising Data: Frequency Tables and Averages | 数据整理：频率表与平均数

📚 Organising Data: Frequency Tables and Averages | 数据整理：频率表与平均数

Data is all around us, and making sense of it starts with organisation. In KS3 Mathematics, we learn how to collect, sort and summarise data using frequency tables, and then describe the ‘average’ and spread with key statistics like the mode, median, mean and range. These skills help us answer real-world questions clearly and accurately.

数据无处不在，理解数据的第一步就是整理。在 KS3 数学中，我们学习如何使用频数表来收集、分类和汇总数据，然后用众数、中位数、平均数和极差等关键统计量来描述“平均水平”和分散程度。这些技能能帮助我们清晰准确地解答现实问题。

1. Starting with Raw Data | 从原始数据出发

Imagine you asked 20 classmates how many pets they own. You might end up with a list like: 2, 0, 1, 3, 1, 0, 2, 1, 4, 2, 1, 1, 0, 3, 2, 1, 0, 5, 1, 2. This unorganised collection is called raw data. It is difficult to spot patterns or answer questions directly from such a list.

想象一下，你问了 20 位同学各养了多少只宠物，可能会得到一串数字：2, 0, 1, 3, 1, 0, 2, 1, 4, 2, 1, 1, 0, 3, 2, 1, 0, 5, 1, 2。这种未整理的集合就叫原始数据。从这样的列表中很难直接发现规律或回答问题。

To make the data usable, we group identical values together and count how many times each value appears. This count is called the frequency, and when we place the results in a table, we create a frequency table. It provides a clear structure for further calculations.

为了让数据变得可用，我们把相同的数据值归到一起，并数出每个值出现了多少次。这个次数就叫频数，而把结果放到表格里就形成了频数表。它为后续的计算提供了清晰的结构。

2. Tally Marks – A Quick Way to Count | 计数符号——快速计数的方法

Before filling in frequencies, it is helpful to use tally marks. For each data value, we draw a vertical stroke. Every fifth stroke is drawn diagonally through the previous four, making a gate of five. This makes totalling much faster and less prone to mistakes.

在填入频数之前，使用计数符号会很有帮助。每出现一个数据值，我们就画一条竖线。每满五个，就用一条斜线划过前四条，形成一个“正”字状的计数。这让计数更快，也不容易出错。

For our pet data, as we read through the list we would put a tally mark next to the appropriate number of pets. Once all 20 responses have been tallied, we count the tallies to obtain the final frequency for each category.

对我们的宠物数据，在浏览列表时，我们在对应的宠物数量旁画上计数符号。当 20 个回答全部计完，数出每个类别的计数，就得到了最终的频数。

3. Building a Complete Frequency Table | 构建完整的频数表

A fully labelled frequency table must include three columns: the data value (or category), tally and frequency. The table below shows the organised pet data. Notice how the frequencies add up to 20, which checks that no responses have been missed.

一张完整标注的频数表必须包含三列：数据值（或类别）、计数和频数。下表展示了整理后的宠物数据。请注意频数加起来是 20，这可以核对是否有遗漏的回答。

Number of pets (数据值)	Tally (计数)	Frequency (频数)
0	||||	4
1	||~~||~~|	7
2	||~~	5
3	||	2
4	|	1
5	|	1
Total		20

A frequency table turns noisy raw data into a tidy summary. It is the foundation for drawing charts and calculating averages, which we will explore next.

频数表把嘈杂的原始数据变成了整洁的摘要。它是绘制图表和计算平均数的基础，我们接下来就会探索。

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4. Bar Charts – Displaying the Data Visually | 条形图——数据的可视化展示

Once we have a frequency table, we can draw a bar chart. For discrete data like ‘number of pets’, the bars do not touch, emphasising that the values are separate categories. The height of each bar represents its frequency.

有了频数表，我们就可以绘制条形图。对于“宠物数量”这样的离散数据，条形之间不相接触，以此强调这些值是分开的类别。每个条形的高度代表了它的频数。

When drawing a bar chart, always label the horizontal axis with the data values and the vertical axis with the frequency. Choose a sensible scale so that your tallest bar fits comfortably. A pencil, ruler and sharp labels are essential for accuracy.

绘制条形图时，务必在横轴上标注数据值，在纵轴上标注频数。选择一个合理的刻度，让最高的条形也能舒适放置。使用铅笔、直尺和清晰的标注对准确性至关重要。

Bar charts allow us to instantly see which category is most common and how the data is spread. They are a powerful tool for comparing different groups and making the story behind the numbers come alive.

条形图能让我们瞬间看出哪个类别最常见，以及数据是如何分布的。它是比较不同组别、让数字背后故事生动起来的强大工具。

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5. The Mode – The Most Popular Choice | 众数——最受欢迎的选择

The mode is simply the value that appears with the highest frequency. In a frequency table, we can spot the mode by looking for the largest number in the frequency column. Using our pet data, the frequency of 1 pet is 7, which is the highest, so the mode is 1.

众数就是出现频数最高的那个值。在频数表中，我们可以通过查找频数列中最大的数来找到众数。根据我们的宠物数据，有 1 只宠物的频数是 7，是最高的，所以众数是 1。

It is possible for a data set to have no mode if all values occur equally, or to have more than one mode if two or more values share the highest frequency. These are called bimodal or multimodal distributions.

如果一个数据集的所有值出现次数相等，就可能没有众数；如果有两个或多个值并列最高频数，则可能有多个众数，这称为双峰或多峰分布。

The mode is especially useful for non-numerical data, such as the most popular colour or preferred fruit, where calculating a mean would be meaningless.

众数对非数值型数据特别有用，比如最受欢迎的颜色或最喜欢的水果，因为对这类数据计算平均数是没有意义的。

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6. The Median – The Middle Value | 中位数——中间的值

The median is the middle number when all the data values are arranged in order from smallest to largest. If there is an odd number of values, the median is the exact middle one. If there is an even number of values, the median is the average of the two middle numbers.

中位数是将所有数据值从小到大排列后，位于中间的那个数。如果有奇数个数据值，中位数就是正中间的那个；如果有偶数个，中位数就是中间两个数的平均数。

To find the median from a raw list, sort the data first. From a frequency table, we can list all values in order or use the cumulative frequency. For our pet data (20 values), the 10th and 11th values in order are both 1, so the median is (1 + 1) ÷ 2 = 1.

要从原始列表中找中位数，先给数据排序。从频数表中，我们可以按顺序列出所有的值，或者使用累计频数。对我们的宠物数据（20 个值），按顺序排列后第 10 和第 11 个值都是 1，所以中位数是 (1+1)÷2 = 1。

The median is robust against extreme values (outliers). For example, if one classmate owned 20 pets, the median would stay at 1, making it a better reflection of a ‘typical’ value than the mean in some situations.

中位数对极端值（异常值）有较强的抵抗力。例如，如果有一位同学养了 20 只宠物，中位数仍保持在 1，因此在某些情况下，它比平均数更能反映“典型”情况。

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7. The Mean – The Arithmetic Average | 平均数——算术平均值

The mean is what most people refer to as the ‘average’. It is calculated by adding up all the data values and then dividing by the number of values. In formula form, this is:

平均数就是大多数人所指的“平均值”。它的计算方法是：先求出所有数据值的总和，再除以数据值的个数。用公式表示为：

Mean = (sum of all values) ÷ (number of values)

When we have a frequency table, we multiply each data value by its frequency, sum these products, and then divide by the total frequency. For the pet data:

当我们有频数表时，我们将每个数据值乘以它的频数，求出这些乘积的总和，再除以总频数。以宠物数据为例：

Sum = (0×4) + (1×7) + (2×5) + (3×2) + (4×1) + (5×1) = 0 + 7 + 10 + 6 + 4 + 5 = 32

Total frequency = 4 + 7 + 5 + 2 + 1 + 1 = 20

Mean = 32 ÷ 20 = 1.6

So, on average, the 20 classmates own 1.6 pets each. The mean gives us a single number that summarises the whole data set, but it can be pulled by very high or very low values.

因此，平均来说，这 20 位同学每人拥有 1.6 只宠物。平均数用一个数字概括了整个数据集，但它可能会被非常高或非常低的值拉偏。

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8. The Range – How Spread Out Is the Data? | 极差——数据分散程度如何？

The range tells us how far apart the smallest and largest values are. It is found by subtracting the smallest data value from the largest.

极差告诉我们最小值和最大值之间相差多远。它的计算方法是：最大值减去最小值。

Range = largest value − smallest value

In our pet example, the largest number of pets is 5 and the smallest is 0. So the range is 5 − 0 = 5. A large range indicates the data is spread out; a small range shows the data is tightly packed together.

在我们的宠物例子中，最大宠物数是 5，最小是 0，所以极差是 5 − 0 = 5。极差大表明数据分布分散；极差小则说明数据比较集中。

Because the range only uses two numbers, it can be easily distorted by a single outlier. That is why scientists and statisticians also look at other measures of spread, such as the interquartile range, which you will meet later in your studies.

由于极差只使用两个数字，它很容易被单个异常值扭曲。因此科学家和统计学家还会看其他分散程度的指标，比如四分位距，在后续的学习中你会接触到。

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9. Choosing the Right Average | 选择合适的平均数

Each average tells a different story about the data. The mode tells us about the most frequent case, the median shows the centre when data is ordered, and the mean balances all values mathematically. Knowing which one to use depends on the context and the presence of outliers.

每种平均数讲述数据的不同一面。众数告诉我们最常见的情况，中位数展示排序后数据的中心，平均数则在数学上平衡所有数值。知道用哪个取决于上下文以及是否存在异常值。

For instance, if a shoe shop wants to stock the most popular shoe size, the mode is the best choice. If a teacher wants to report a typical test score without being influenced by one extremely low mark, the median might be fairest. If a smoothie company needs to estimate total fruit needed, the mean consumption per person is most useful.

例如，鞋店想储备最畅销的鞋码，众数是最佳选择。老师想报告一个不受某个极低分影响的典型考试分数，中位数可能最公平。如果一家果昔公司需要估算总共需要的水果量，人均消费平均数最有用。

Always ask yourself what each average represents and whether it could be misleading before drawing a conclusion. A set of data could have all three averages quite different from each other, so careful reporting is essential.

在得出结论之前，一定要问问自己每种平均数代表什么，以及它是否可能产生误导。一个数据集的三类平均数可能彼此相差很大，因此谨慎地报告至关重要。

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10. Common Mistakes and How to Avoid Them | 常见错误及避免方法

A very frequent error is confusing the mean with the mode or median. Make sure you can define each clearly and work through the correct steps. Another pitfall is miscounting tallies when constructing a frequency table – always double-check that the sum of frequencies equals the total number of data items.

一个非常常见的错误是把平均数与众数或中位数混淆。务必能清晰定义每一个概念并按照正确的步骤计算。另一个陷阱是构建频数表时计错了数——务必复查频数之和是否等于数据项的总数。

When drawing bar charts, some students forget to leave gaps between bars for discrete data, or they use a scale that compresses the tallest bar so much that differences become invisible. Always plan your vertical axis scale carefully and use a sharp pencil.

在绘制条形图时，有些同学忘记为离散数据在条形之间留出间隙，或者使用的刻度把最高的条形压缩得太厉害，导致差异难以看见。务必仔细规划纵轴刻度，并使用削尖的铅笔。

For the mean, a classic mistake is forgetting to multiply each data value by its frequency before adding, or dividing by the wrong total. Write down each multiplication clearly; it saves marks in exams.

关于平均数，一个典型的错误是在求总和时忘记将每个数据值乘以它的频数，或者除以了错误的总数。把每次乘法都清楚地写下来，这在考试中能保住分数。

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11. Worked Example – From Table to Interpretation | 实例解析——从表格到解读

Let us work through a fresh example. Fifteen pupils recorded the number of books they read in a month: 3, 4, 2, 3, 5, 3, 4, 2, 3, 6, 3, 4, 2, 5, 3. We can construct a frequency table, find the averages and range, and then interpret the results.

我们来看一个新例子。15 名学生记录了他们一个月内阅读的书籍数量：3, 4, 2, 3, 5, 3, 4, 2, 3, 6, 3, 4, 2, 5, 3。我们可以构建频数表，求出各种平均数和极差，然后解读结果。

Books (x)	Tally	Frequency (f)	f × x
2	III	3	6
3	~~||||~~|	6	18
4	III	3	12
5	II	2	10
6	I	1	6
Total		15	52

Mode: The highest frequency is 6, which corresponds to 3 books. So the mode is 3 books.

众数：最高频数是 6，对应 3 本书，所以众数是 3 本书。

Median: With 15 values, the 8th value is the middle. Listing in order gives six 3’s in the centre; the 8th value is 3. Median = 3 books.

中位数：一共有 15 个值，第 8 个是中间值。按顺序排列后，中间是 6 个 3；第 8 个值是 3。中位数 = 3 本书。

Mean: Sum of (f × x) = 52, total frequency = 15, so mean = 52 ÷ 15 ≈ 3.47 books.

平均数：f × x 的总和是 52，总频数是 15，所以平均数 = 52 ÷ 15 ≈ 3.47 本书。

Range: Largest value (6) − smallest value (2) = 4 books.

极差：最大值 (6) − 最小值 (2) = 4 本书。

The averages tell us that a typical pupil read around 3 books, but the slightly higher mean suggests there were a few pupils reading more, pulling the mean upward. The range of 4 shows some variation in reading habits.

平均数告诉我们，典型的学生大约读了 3 本书，但平均数略高，说明有几位学生读得更多，拉高了平均值。极差为 4，表明阅读习惯存在一定的差异。

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12. Key Points for Revision | 复习要点

Here is a summary to help you remember the essentials: always organise data with a clear frequency table; use tallies to avoid counting errors; choose the mode, median or mean according to the question’s needs; and never forget to divide correctly when working out the mean. Practise drawing neat bar charts with labelled axes and equal scales.

下面是帮助你记住关键点的总结：始终用清晰的频数表整理数据；使用计数符号避免数错；根据问题的需要选择众数、中位数或平均数；计算平均数时绝不要忘记正确进行除法。多加练习绘制坐标轴标注清晰、刻度均匀的条形图。

• Frequency table – three columns: value, tally, frequency.
• Mode – the value with the highest frequency.
• Median – middle value when ordered; average of two middle if even number.
• Mean – total of (value × frequency) divided by total frequency.
• Range – largest minus smallest; shows spread.
• Bar chart – gaps between bars for discrete data; label axes clearly.

• 频数表——包含三列：数据值、计数、频数。
• 众数——频数最高的那个值。
• 中位数——排序后中间的值；偶数个时为中间两个数的平均数。
• 平均数——（数值 × 频数）的总和除以总频数。
• 极差——最大值减最小值；表示分散程度。
• 条形图——离散数据的条形间要有间隙；清楚地标注坐标轴。

Mastering these core ideas will give you confidence not only in handling data but also in interpreting information you meet in science, geography and everyday life. Keep practising with different data sets, and soon you will find that working with numbers becomes second nature.

掌握这些核心概念，不仅能让你自信地处理数据，还能帮助你解读在科学、地理和日常生活中遇到的信息。用不同的数据集持续练习，很快你就会发现与数字打交道会变得自然而然。

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Published by TutorHao | Mathematics Revision Series | aleveler.com

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