Averages and Range from Frequency Tables | 频率表中的平均数与范围

📚 Averages and Range from Frequency Tables | 频率表中的平均数与范围

In KS3 mathematics, data often comes in lists, but when we have many repeated values, a frequency table is a much smarter way to organise information. Being able to read, interpret and calculate averages and the range from a frequency table is an essential skill – it helps you summarise data quickly and compare different sets fairly.

在KS3数学中，数据常常以列表形式出现，但当有很多重复数值时，频率表是整理信息更巧妙的方式。学会从频率表中读取、解释并计算平均数、中位数、众数和范围是一项基本技能——它能帮助你快速总结数据，并公平地比较不同数据集。

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1. From Raw Data to Frequency Table | 从原始数据到频率表

A frequency table shows each distinct data value alongside how many times it occurs (its frequency). The first column usually lists the values, often labelled ‘Score’, ‘Number’ or ‘x’. The second column gives the frequency ‘f’. Sometimes we add a third column to help with calculations, labelled ‘f × x’.

频率表列出了每个不同的数据值及其出现的次数（频率）。第一列通常列出数值，往往标记为“得分”“数量”或“x”。第二列给出频率“f”。有时我们添加第三列以便计算，标记为“f × x”。

For example, imagine the shoe sizes of 20 students: 4, 4, 5, 4, 6, 5, 5, 6, 4, 5, 6, 6, 5, 4, 5, 6, 4, 5, 6, 5. Instead of a long list, we can create a frequency table:

例如，假设20名学生的鞋码为：4, 4, 5, 4, 6, 5, 5, 6, 4, 5, 6, 6, 5, 4, 5, 6, 4, 5, 6, 5。我们可以不列长串，而是制作一个频率表：

Shoe size (x)	Frequency (f)
4	6
5	8
6	6

This compact table immediately shows that size 5 is the most common. The total frequency is 6 + 8 + 6 = 20, which matches the number of students.

这张紧凑的表格立刻显示出5码最常见。总频率为6 + 8 + 6 = 20，与学生人数相符。

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2. Calculating the Mean from a Frequency Table | 从频率表计算平均数

The mean (average) of a frequency table is found by multiplying each data value (x) by its frequency (f), adding all those products together, and then dividing by the total frequency. We often write this as:

频率表的平均数（平均值）是用每个数据值（x）乘以其频率（f），将所有乘积相加，然后除以总频率。我们通常写作：

Mean = (Σ f × x) ÷ (Σ f)

For the shoe size example, we add an f × x column:

对于鞋码的例子，我们添加一列 f × x：

Shoe size (x)	Frequency (f)	f × x
4	6	24
5	8	40
6	6	36
	Σ f = 20	Σ f×x = 100

Mean = 100 / 20 = 5. So the average shoe size is exactly 5.

平均数 = 100 / 20 = 5。因此平均鞋码正好是5。

Remember: the mean takes into account every single value, including repeated ones. It is useful when data is fairly symmetrical and has no extreme outliers.

记住：平均数考虑了每一个数值，包括重复值。当数据大致对称且没有极端异常值时，平均数很有用。

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3. Finding the Median from a Frequency Table | 从频率表找中位数

The median is the middle value when the data is arranged in order. With a frequency table, we find the position of the median using (total frequency + 1) / 2. This works for both odd and even total frequencies – for even totals, the formula gives a half-number position, meaning we take the average of the two middle values.

中位数是将数据按顺序排列后的中间值。使用频率表时，我们用（总频率 + 1）/ 2 找到中位数的位置。这对总频率为奇数和偶数的情况都适用——如果为偶数，公式会给出半整数位置，意味着我们要取中间两个值的平均数。

In the shoe size table, Σ f = 20, so position = (20 + 1) / 2 = 10.5. This tells us the median lies between the 10th and 11th data values when ordered.

在鞋码表中，Σ f = 20，因此位置 = (20 + 1) / 2 = 10.5。这表明中位数位于排序后第10个和第11个数据值之间。

We then use cumulative frequency: listing frequencies in order, size 4 covers positions 1 to 6, size 5 covers positions 7 to 14, and size 6 covers positions 15 to 20. The 10th and 11th values both fall in the size 5 group, so the median is 5.

然后我们利用累积频率：按顺序列出，4码覆盖位置1到6，5码覆盖位置7到14，6码覆盖位置15到20。第10和第11个值都落在5码组内，所以中位数为5。

The median is not affected by extreme values, making it a better measure of centre when the data is skewed.

中位数不受极端值影响，因此在数据偏态分布时是更好的中心度量。

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4. Identifying the Mode from a Frequency Table | 从频率表识别众数

The mode is simply the value with the highest frequency. In the shoe size table, frequency of size 5 is 8, which is the largest, so the mode is 5.

众数就是频率最高的数值。在鞋码表中，5码的频率为8，是最大的，因此众数为5。

It is possible to have more than one mode (bimodal or multimodal) if two or more values share the highest frequency. In a frequency table, the mode is easy to spot – just look for the biggest number in the frequency column.

如果有两个或更多值共享最高频率，就可能有一个以上的众数（双众数或多众数）。在频率表中，众数很容易辨认——只需在频率列中寻找最大的数字。

Mode is particularly useful for categorical data, like favourite colours or types of pet, where calculating a mean is impossible.

众数对于类别数据特别有用，比如最喜欢的颜色或宠物类型，而此时计算平均数是不可能的。

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5. Working Out the Range | 计算范围

The range measures how spread out the data is. It is found by subtracting the smallest data value from the largest data value. In a frequency table, simply take the maximum x and minimum x.

范围衡量数据的分散程度。它由最大数据值减去最小数据值得到。在频率表中，只需取x的最大值和最小值。

For the shoe size data, the smallest size is 4 and the largest is 6, so the range = 6 – 4 = 2. A larger range indicates greater variability; a smaller range shows the data is more consistent.

对于鞋码数据，最小码是4，最大码是6，因此范围 = 6 – 4 = 2。范围越大表明变异性越大；范围越小表明数据越一致。

Always check that you are using the values themselves, not the frequencies! A common mistake is to subtract the smallest frequency from the largest frequency – that gives a meaningless number.

务必检查你使用的是数值本身，而不是频率！一个常见的错误是用最大频率减去最小频率——那样会得到无意义的数字。

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6. Worked Example with Grouped Data | 分组数据的实例

Sometimes, especially with continuous data, we use grouped frequency tables. Here each ‘class interval’ covers a range of values. To estimate the mean, we use the midpoint of each interval. The median is found from a cumulative frequency graph, but for smaller sets we can locate the interval containing the median.

有时，尤其是连续数据，我们使用分组频率表。此时每个“组距”覆盖一个范围的值。要估算平均数，我们用每个区间的中点值。中位数需从累积频率图中找到，但对于较小的数据集，我们可以定位包含中位数的区间。

Consider the heights of 30 plants grouped as: 0 ≤ h < 10 (f=5), 10 ≤ h < 20 (f=12), 20 ≤ h < 30 (f=8), 30 ≤ h < 40 (f=5).

考虑30株植物的高度分组为：0 ≤ h < 10 (f=5), 10 ≤ h < 20 (f=12), 20 ≤ h < 30 (f=8), 30 ≤ h < 40 (f=5)。

Midpoints: 5, 15, 25, 35. Then f × midpoint: 5×5=25, 12×15=180, 8×25=200, 5×35=175. Sum = 580. Total f = 30. Estimated mean = 580 / 30 ≈ 19.3.

中点值：5, 15, 25, 35。然后 f × 中点：5×5=25, 12×15=180, 8×25=200, 5×35=175。总和 = 580。总频率 = 30。估算平均数 = 580 / 30 ≈ 19.3。

The modal class is the interval with the highest frequency: 10 ≤ h < 20. The class containing the median: position (30+1)/2 = 15.5; cumulative frequencies: 5, 17, 25, 30. The 15.5th value lies in the second class, so median class is 10 ≤ h < 20.

众数所在组是频率最高的区间：10 ≤ h < 20。包含中位数的组：位置 (30+1)/2 = 15.5；累积频率：5, 17, 25, 30。第15.5个值位于第二组，因此中位数所在组为 10 ≤ h < 20。

The range is still maximum possible value minus minimum possible value: 40 – 0 = 40.

范围仍然是最大可能值减去最小可能值：40 – 0 = 40。

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7. Comparing Two Data Sets Using Averages and Range | 用平均数和范围比较两个数据集

Exam questions often ask you to compare two frequency tables and decide which one shows higher average performance or more consistency. Use the mean (or median) to comment on the typical value, and the range to comment on spread.

考试题目常要求你比较两个频率表，判断哪个显示更高的平均表现或更大的稳定性。用平均数（或中位数）来评述典型值，用范围来评述离散程度。

For example, if Class A’s test scores have a median of 72 and range 18, while Class B has a median of 68 and range 30, we can say: ‘Class A performed better on average (higher median) and was more consistent (smaller range).’ Always back up your comparison with numbers from the tables.

例如，若A班的测试分数中位数为72、范围18，而B班中位数68、范围30，我们可以说：“A班平均表现更好（中位数更高），且成绩更稳定（范围更小）。”务必用表格中的数字支持你的比较。

When data is skewed, the median is a more reliable measure than the mean. If one class has a few extremely high scores, the mean might be misleadingly high.

当数据呈偏态分布时，中位数比平均数更可靠。如果一个班级有几个极高的分数，平均数可能会被误导性地拉高。

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8. Avoiding Common Pitfalls | 避免常见错误

One of the biggest mistakes is forgetting to multiply by the frequency when calculating the mean. Adding up the x-values only once will give a different, wrong answer.

最大的错误之一是在计算平均数时忘记乘以频率。仅将x值相加一次将得到完全不同的错误答案。

Another error is misidentifying the median position. Always use (Σ f + 1) / 2, not Σ f / 2. If you use Σ f / 2 with an even total, you will land on the lower of the two middle numbers without averaging – losing marks.

另一个错误是错误识别中位数位置。始终使用 (Σ f + 1) / 2，而非 Σ f / 2。若总数为偶数却用了 Σ f / 2，你就会落在中间两数中较低的那个而没有取平均——这样会丢分。

For range, students sometimes subtract the smallest frequency from the largest frequency, or report the difference between the highest and lowest frequency. This shows a misunderstanding of what range measures.

至于范围，学生有时会用最大频率减去最小频率，或者报告最高频率与最低频率之差。这显示出对范围测量内容的理解有误。

Finally, when reading values from a grouped table, never use the class boundaries as the data values for the mean – always use the midpoint.

最后，当从分组表中读取数值时，绝不要用组界作为计算平均数的数据值——一定要用中点值。

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9. Tips for Success in Exams | 考试成功技巧

Always draw an extra ‘f × x’ column and show your totals clearly. Even if you make an arithmetic slip, you can earn method marks. Label your columns and write the formula for mean in words or symbols before substituting numbers.

务必额外画一列“f × x”并清晰地展示总和。即便你计算小有失误，也能得到方法分。给各列做标记，并在代入数字前先用文字或符号写出平均数的公式。

Use cumulative frequency to find the median confidently. Write a small running total next to the frequency column; this prevents skipping values. When the position is .5, remember to average the two surrounding numbers.

自信地使用累积频率寻找中位数。在频率列旁边写一个小型的累积总数；这能防止跳过数值。当位置为0.5时，记住要对相邻两数求平均。

Practise interpreting what the statistics tell you in context. A higher mean indicates a higher typical value; a smaller range shows greater consistency. Use comparative language in your answers.

练习在语境中解释统计量告诉你什么。较高的平均数表明典型值较高；较小的范围显示更大的稳定性。在答案中使用比较性的语言。

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10. Summary and Quick Reference | 总结与快速参考

From any frequency table you can find the four key summaries:

从任何频率表中，你都可以找到四个关键总结量：

• Mean: Σ(f × x) / Σ f
• Median: value at position (Σ f + 1)/2
• Mode: value with highest frequency
• Range: largest x – smallest x

• 平均数：Σ(f × x) / Σ f
• 中位数：位于 (Σ f + 1)/2 位置的值
• 众数：最高频率对应的数值
• 范围：最大x – 最小x

Keep these definitions at your fingertips, and always show your working. With a clear method and careful arithmetic, frequency table questions become reliable marks in any KS3 assessment.

将这些定义牢记于心，并始终展示你的解题步骤。方法明确、计算仔细，频率表问题在任何KS3评估中都能成为可靠的得分点。

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Published by TutorHao | Mathematics Revision Series | aleveler.com

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