Handling Data: Frequency Tables and Averages (p171) | 数据处理：频数表与平均数（p171）

📚 Handling Data: Frequency Tables and Averages (p171) | 数据处理：频数表与平均数（p171）

In many real-life situations, you collect data that contains repeated values. Instead of writing out a long list of numbers, you can organise the data into a frequency table. Page 171 of your Cambridge KS3 Mathematics book introduces how to construct and use frequency tables to calculate key statistics such as the mean, median, mode, and range. This article provides a complete revision guide to help you master these skills.

在许多实际情况下，你收集的数据会包含重复的值。与其写出一长串数字，不如将数据整理成频数表。你的剑桥KS3数学教材第171页介绍了如何构建并使用频数表来计算平均数、中位数、众数和极差等关键统计量。本文提供一份完整的复习指南，帮助你掌握这些技能。

1. What is a Frequency Table? | 什么是频数表？

A frequency table is a way of displaying data that shows how many times each value occurs. The first column lists the data values (or categories), and the second column lists the frequency – the number of times that value appears.

频数表是一种显示数据的方式，它展示每个值出现了多少次。第一列列出数据值（或类别），第二列列出频数——即该值出现的次数。

For example, imagine you survey 15 classmates about how many pets they own. The raw data might look like: 0, 1, 1, 0, 2, 1, 3, 0, 1, 2, 1, 0, 4, 1, 2. A frequency table organises this clearly.

例如，假设你调查了15名同学拥有多少只宠物。原始数据可能是：0, 1, 1, 0, 2, 1, 3, 0, 1, 2, 1, 0, 4, 1, 2。频数表可以清晰地整理这些数据。

Number of pets (x)	Frequency (f)
0	4
1	6
2	3
3	1
4	1
Total	15

This table tells you at a glance that 4 people have no pets, 6 people have 1 pet, and so on.

这张表让你一目了然：4人没有宠物，6人有1只宠物，以此类推。

---

2. Why Use Frequency Tables? | 为什么使用频数表？

Frequency tables save space and make it easier to spot patterns. Instead of scrolling through a long list, you immediately see which value occurs most often and how the data is distributed.

频数表节省空间，并且更容易发现规律。你无需翻看一长串列表，就能立刻看出哪个值出现得最频繁，以及数据的分布情况。

They are essential when you work with large sets of data, such as survey results, test scores, or measurements. By grouping information, you reduce the chance of miscounting.

当你处理大量数据时，比如调查结果、考试分数或测量数据，频数表就变得至关重要。通过将信息分组，你可以减少数错的可能性。

In addition, frequency tables form the basis for drawing charts like bar charts and pictograms, and they are the starting point for calculating averages.

此外，频数表是绘制条形图和象形图等图表的基础，也是计算各种平均数的起点。

---

3. Finding the Mode from a Frequency Table | 从频数表找众数

The mode is the value that appears most often. In a frequency table, you simply look down the frequency column and find the highest number. The data value next to that frequency is the mode.

众数是出现次数最多的值。在频数表中，你只需要观察频数那一列，找到最大的数字。该频数旁边的数据值就是众数。

Using the pets example, the frequencies are 4, 6, 3, 1, 1. The highest frequency is 6, which corresponds to 1 pet. Therefore, the mode is 1 pet.

用宠物的例子来说，频数分别是4, 6, 3, 1, 1。最高的频数是6，对应的是1只宠物。因此，众数是1只宠物。

If two values share the highest frequency, the data set is bimodal and you should list both values. The mode is especially useful for non-numerical data, such as favourite colours.

如果两个值共享最高频数，那么数据集是双众数的，你应该将两个值都列出来。众数对于非数值型数据（如最喜欢的颜色）尤其有用。

---

4. Finding the Median | 找出中位数

The median is the middle value when all data points are arranged in order. From a frequency table, you first need to find the total number of data values, n. Add up all the frequencies: 4+6+3+1+1 = 15.

中位数是将所有数据点按顺序排列后位于中间的那个值。根据频数表，你首先需要找出数据值的总数 n。将所有频数相加：4+6+3+1+1 = 15。

Since n = 15, the position of the median is (15 + 1) ÷ 2 = 8th value. Then work through the table adding frequencies until you reach or pass the 8th value.

因为 n = 15，中位数的位置是 (15 + 1) ÷ 2 = 第8个值。然后在频数表中逐个累加频数，直到达到或超过第8个值。

Cumulative frequencies: after 0 pets we have counted 4 values; after 1 pet we have 4+6 = 10 values. So the 5th to 10th values are all 1. The 8th value is therefore 1 pet. The median number of pets is 1.

累计频数：数完0只宠物后我们数了4个值；数完1只宠物后有4+6 = 10个值。所以第5到第10个值都是1。第8个值因此是1只宠物。宠物数量的中位数是1。

---

5. Calculating the Mean | 计算平均数

The mean is the sum of all data values divided by the number of values. To find the mean from a frequency table, you must multiply each data value by its frequency, add up those products, and then divide by the total frequency.

平均数是所有数据值的总和除以值的个数。要根据频数表求平均数，你必须将每个数据值乘以其频数，把这些乘积相加，然后除以总频数。

Key formula:

关键公式：

Mean = (Σ f × x) ÷ (Σ f)

Here f × x means frequency multiplied by the corresponding data value. Extend the table by adding an extra column for f × x.

这里的 f × x 是指频数乘以相应的数据值。通过增加一列 f × x 来扩展表格。

x	f	f × x
0	4	0
1	6	6
2	3	6
3	1	3
4	1	4
Total	15	19

Sum of f × x = 0 + 6 + 6 + 3 + 4 = 19. Mean = 19 ÷ 15 ≈ 1.27 pets (to two decimal places).

f × x 的总和 = 0 + 6 + 6 + 3 + 4 = 19。平均数 = 19 ÷ 15 ≈ 1.27 只宠物（保留两位小数）。

---

6. Mean from a Frequency Table – Step by Step | 从频数表逐步计算平均数

Follow these steps every time you calculate the mean from a frequency table:

每次从频数表计算平均数时，请遵循以下步骤：

Step 1: Create a third column headed f × x. Multiply each x value by its frequency and write the result in this column.

第1步：创建第三列，标题为 f × x。将每个 x 值乘以其频数，并将结果写在此列中。

Step 2: Find the total of the f × x column. This is the sum of all data values.

第2步：求出 f × x 列的总和。这就是所有数据值的总和。

Step 3: Find the total frequency (Σ f). This is the number of data items.

第3步：求出总频数（Σ f）。这就是数据项的个数。

Step 4: Divide the sum of f × x by the total frequency. Round your answer sensibly if needed.

第4步：用 f × x 的总和除以总频数。如有需要，合理地对答案进行四舍五入。

---

7. Using a Frequency Table with Grouped Data | 分组频数表

When data is spread over a wide range, we often group values into intervals. For example, test marks out of 100 might be grouped as 0-10, 11-20, 21-30, and so on.

当数据分布在很宽的范围内时，我们通常将数值分组为区间。例如，满分为100的考试分数可能被分为0-10、11-20、21-30等组。

In a grouped frequency table, you do not know the exact values anymore. You only know how many fall into each class interval. This means you can only estimate the mean.

在分组频数表中，你不再知道确切的值。你只知道每个组区间内有多少个数据。这意味着你只能估算平均数。

Test score (%)	Frequency (f)
0 – 20	3
21 – 40	7
41 – 60	8
61 – 80	5
81 – 100	2

Grouped tables are particularly useful when you have a large number of different values and you want to see the overall shape of the distribution.

当你拥有大量不同的数值，并且想了解分布的整体形态时，分组表格特别有用。

---

8. Estimating the Mean from Grouped Data | 从分组数据估算平均数

To estimate the mean for grouped data, you assume that all values in an interval are equal to the midpoint of that interval. The midpoint is (lower bound + upper bound) ÷ 2.

要从分组数据估算平均数，你假设某个区间内的所有值都等于该区间的中点值。中点 =（下限 + 上限）÷ 2。

For the interval 0-20, the midpoint is (0+20) ÷ 2 = 10. For 21-40, it is (21+40) ÷ 2 = 30.5. Then you multiply each midpoint by its frequency, sum these up, and divide by total frequency.

对于区间 0-20，中点是 (0+20) ÷ 2 = 10。对于 21-40，中点是 (21+40) ÷ 2 = 30.5。然后将每个中点乘以其频数，求和，再除以总频数。

Calculations for the table above: midpoints are 10, 30.5, 50.5, 70.5, 90.5. Total f = 25. Sum of f × midpoint = 3×10 + 7×30.5 + 8×50.5 + 5×70.5 + 2×90.5 = 30 + 213.5 + 404 + 352.5 + 181 = 1181. Estimated mean = 1181 ÷ 25 = 47.24.

上表的计算：中点分别是10, 30.5, 50.5, 70.5, 90.5。总频数 f = 25。f × 中点的总和 = 3×10 + 7×30.5 + 8×50.5 + 5×70.5 + 2×90.5 = 30 + 213.5 + 404 + 352.5 + 181 = 1181。估算平均数 = 1181 ÷ 25 = 47.24。

Remember that this is only an estimate, not the exact mean, because we do not know the original raw data.

请记住，这只是一个估计值，而不是精确的平均数，因为我们不知道原始的原始数据。

---

9. Finding the Range | 找出极差

The range is the difference between the largest and the smallest data values. From a frequency table, simply identify the highest and lowest x values and subtract.

极差是最大值与最小值之间的差。从频数表中，只需找出最高和最低的 x 值并相减即可。

In our pets frequency table, the smallest number of pets is 0 and the largest is 4. Range = 4 – 0 = 4 pets.

在我们的宠物频数表中，最小的宠物数量是0，最大的是4。极差 = 4 – 0 = 4只宠物。

The range gives you a quick sense of how spread out the data is. However, it is sensitive to extreme outliers – one very high or very low value can make the range very large.

极差让你快速了解数据的分散程度。然而，它对极端异常值很敏感——一个非常高或非常低的值都可能使极差变得非常大。

---

10. Common Mistakes to Avoid | 常见错误

Mistake 1: Confusing the data value with its frequency. Always check which column is being used for the calculation.

错误1：混淆数据值与其频数。务必检查你在计算中使用的是哪一列。

Mistake 2: Forgetting to divide by total frequency when finding the mean. Use the formula mean = (Σ f × x) ÷ (Σ f), not just Σ f × x

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

更多咨询请联系16621398022（同微信）