📚 A-Level Mathematics: Cumulative Frequency Curves and Applications | A-Level 数学：累积频率曲线及应用
Cumulative frequency curves, also known as ogives, are one of the most powerful graphical tools in statistics. They allow us to estimate key measures such as the median, quartiles, and percentiles directly from a graph, making them a staple of every A-Level statistics exam paper.
累积频率曲线（又称拱形图）是统计学中最强大的图形工具之一。它使我们能够直接从图形中估计中位数、四分位数和百分位数等关键统计量，因此成为每一份 A-Level 统计学试卷中的常客。
1. What Is Cumulative Frequency? | 什么是累积频率？
Cumulative frequency is the running total of the frequencies of all data values up to a given point. For each class interval in a grouped frequency table, we add the frequency of that class to the total of all previous classes.
累积频率是截止到某个给定数据点为止的所有数据频数的累计总和。对于分组频数表中的每一个组区间，我们将该组的频数加到之前所有组的频数总和之上。
The result is a monotonically increasing sequence that tells us how many observations lie below or equal to each upper class boundary.
所得结果是一个单调递增的序列，它告诉我们有多少观测值小于或等于每个组上限。
2. Constructing the Cumulative Frequency Table | 构建累积频率表
To build a cumulative frequency table, follow these steps:
构建累积频率表，需遵循以下步骤：
Identify the class intervals and their corresponding frequencies.
确定组区间及其对应的频数。
For each row, add the current class frequency to the sum of all previous frequencies.
对于每一行，将当前组的频数加到之前所有频数的总和上。
Write the cumulative frequency against the upper class boundary of each interval.
将累积频率写在每个区间的组上限处。
Consider the following grouped frequency table of marks scored by 50 students:
考虑以下 50 名学生得分的分组频数表：
| Marks | 分数 | Frequency | 频数 | Cumulative Frequency | 累积频率 |
|---|---|---|
| 0 ≤ x < 10 | 4 | 4 |
| 10 ≤ x < 20 | 10 | 14 |
| 20 ≤ x < 30 | 16 | 30 |
| 30 ≤ x < 40 | 14 | 44 |
| 40 ≤ x < 50 | 6 | 50 |
3. Plotting the Cumulative Frequency Curve | 绘制累积频率曲线
When plotting a cumulative frequency curve, the upper class boundary is placed on the x-axis and the cumulative frequency is placed on the y-axis. A key convention is that the curve must start from the lower boundary of the first class interval at a cumulative frequency of zero.
绘制累积频率曲线时，x 轴表示组上限，y 轴表示累积频率。一个重要约定是：曲线必须从第一个组区间的下限开始，此时累积频率为零。
Plot points at each upper class boundary with its corresponding cumulative frequency.
在每个组上限处描点，对应其累积频率。
Start the curve at the lowest class boundary with cumulative frequency 0.
从最低组边界、累积频率为 0 处开始绘制曲线。
Join the points with a smooth, ascending curve (do not draw straight line segments unless instructed).
用平滑的上升曲线连接各点（除非题目另有要求，否则不要画直线段）。
The resulting S-shaped curve is called an ogive. It always rises from left to right and never decreases.
所得到的 S 形曲线称为拱形图。它总是从左向右上升，且永远不会下降。
4. Finding the Median | 求中位数
The median is the middle value of a data set. For a cumulative frequency curve, the median is found by locating the value n/2 on the y-axis, where n is the total number of observations.
中位数是一组数据中的中间值。对于累积频率曲线，中位数通过在 y 轴上找到 n/2 的位置来确定，其中 n 是观测值的总数。
Median position = n/2
From this point, draw a horizontal line across to the curve, then draw a vertical line down to the x-axis. The x-coordinate where the vertical line meets the x-axis is the median.
从该点画一条水平线直至与曲线相交，然后画一条垂线向下至 x 轴。垂线与 x 轴相交处的 x 坐标即为中位数。
5. Quartiles and the Interquartile Range | 四分位数与四分位距
Quartiles divide the data into four equal parts. The lower quartile Q₁ is located at position n/4, and the upper quartile Q₃ is located at position 3n/4 on the cumulative frequency axis.
四分位数将数据分成四个相等的部分。下四分位数 Q₁ 位于累积频率轴上的 n/4 处，上四分位数 Q₃ 位于 3n/4 处。
Q₁ position = n/4, Q₃ position = 3n/4
The interquartile range (IQR) is the difference between the upper and lower quartiles:
四分位距（IQR）是上四分位数与下四分位数之差：
IQR = Q₃ − Q₁
The IQR measures the spread of the middle 50% of the data and is preferred over the range because it is not affected by extreme values or outliers.
四分位距衡量中间 50% 数据的离散程度，由于不受极端值或异常值的影响，它比极差更受青睐。
6. Box-and-Whisker Plots | 箱线图
Once the median, quartiles, and extreme values are known, we can construct a box-and-whisker plot. This graphical representation is drawn parallel to the x-axis and consists of a box spanning from Q₁ to Q₃, with a vertical line at the median.
一旦求出中位数、四分位数和极值，我们就可以绘制箱线图。这种图形表示是平行于 x 轴绘制的，由一个从 Q₁ 延伸到 Q₃ 的箱体构成，并在中位数处画一条竖线。
The left whisker extends from the minimum value to Q₁.
左侧须从最小值延伸到 Q₁。
The box represents the interquartile range.
箱体代表四分位距。
The right whisker extends from Q₃ to the maximum value.
右侧须从 Q₃ 延伸到最大值。
Box plots are excellent for visualising the shape of a distribution and for comparing two or more data sets side by side.
箱线图非常适合可视化分布形状，也适合并排比较两个或多个数据集。
7. Comparing Data Sets | 比较数据集
Cumulative frequency curves and box plots allow us to compare two or more distributions meaningfully. When comparing, we focus on two main features:
累积频率曲线和箱线图使我们能够有意义地比较两个或多个分布。比较时，我们主要关注两个特征：
Central tendency: a higher median indicates that the data set is generally larger in value.
集中趋势：中位数越高，表明该数据集的值总体越大。
Spread: a larger IQR indicates that the middle 50% of the data is more spread out, suggesting lower consistency.
离散程度：四分位距越大，表明中间 50% 的数据越分散，即一致性越低。
In an exam answer, always quote the actual values of the medians and IQRs found, and then state what they imply in the context of the question.
在考试作答中，务必引用求出的中位数和四分位距的具体数值，然后说明它们在题目情境中意味着什么。
8. Percentiles: A Broader View | 百分位数：更广阔的视角
A percentile is the value below which a given percentage of observations fall. The p-th percentile is found at position pn/100 on the cumulative frequency axis.
百分位数是这样一个值，低于它的观测值占给定百分比。第 p 百分位数位于累积频率轴上的 pn/100 处。
Position of p-th percentile = pn/100
n 90n/100 = 0.9n. This is useful in real-world applications such as determining the top 10% of students in an examination or the highest-paying 10% of jobs in a salary survey.
例如，第 90 百分位数位于 90n/100 = 0.9n 处。这在现实应用中有重要价值，比如确定考试成绩排名前 10% 的学生，或在薪资调查中找出收入最高的前 10% 的职位。
9. Worked Example: Exam-Style Question | 例题：考试风格题目
Using the marks table from Section 2, estimate the median, quartiles, and interquartile range.
利用第 2 节中的分数表，估计中位数、四分位数和四分位距。
Step 1 | 第一步: tion is 50/2 = 25. The Q₁ position is 50/4 = 12.5, and the Q₃ position is 3 × 50/4 = 37.5.
总频数 n = 50，因此中位数位置为 50/2 = 25。Q₁ 的位置为 50/4 = 12.5，Q₃ 的位置为 3 × 50/4 = 37.5。
Step 2 | 第二步: Reading from a smoothly drawn ogive:
从平滑绘制的拱形图上读取：
Median ≈ 25th value corresponds to a mark of about 29.
中位数 ≈ 第 25 个值对应的分数约为 29。
Q₁ ≈ 12.5th value corresponds to a mark of about 19.
Q₁ ≈ 第 12.5 个值对应的分数约为 19。
Q₃ ≈ 37.5th value corresponds to a mark of about 36.
Q₃ ≈ 第 37.5 个值对应的分数约为 36。
Step 3 | 第三步: Calculate the interquartile range:
计算四分位距：
IQR = Q₃ − Q₁ = 36 − 19 = 17 marks
When drawing the ogive in the exam, use a large, accurate graph and always show your reading lines on the graph, as this is where the method marks are awarded.
在考试中绘制拱形图时，应选用足够大的坐标纸以保证精度，并且务必在图上面画出读取线，因为方法分正是在这些地方评给的。
10. Common Pitfalls and Exam Tips | 常见失误与考试技巧
Students frequently lose marks on cumulative frequency questions due to avoidable errors. Keep the following advice in mind:
学生在累积频率题目中经常因为可以避免的错误而失分。请牢记以下建议：
Always plot cumulative frequency against the upper class boundary, not the midpoint.
务必用组上限而不是组中点来对应累积频率描点。
Remember to start the curve from the lower boundary of the first class with cumulative frequency 0.
记住曲线要从第一个组的下限、累积频率为 0 处开始。
Use n/2, n/4, and 3n/4 for the median, Q₁, and Q₃ respectively. Do not confuse these positions.
分别使用 n/2、n/4 和 3n/4 来定位中位数、Q₁ 和 Q₃，切勿混淆这些位置。
When reading values from the graph, draw the horizontal line first, then the vertical line down to the x-axis. This process must be visible.
从图上读取数值时，先画水平线，再画垂线至 x 轴。这一过程必须清晰可见。
Quote units in your final answer, such as "marks" or "kg" or "seconds".
在最终答案中注明单位，如"分"、"千克"或"秒"。
11. Summary | 总结
Cumulative frequency curves are a compact yet powerful way to analyse grouped data. From a single graph, we can extract the median, quartiles, percentiles, and interquartile range, and use these to compare distributions. Mastering the construction of the table and the curve, along with accurate graph reading, will secure full marks on this topic in exams.
累积频率曲线是分析分组数据的一种紧凑而强大的方法。仅凭一张图，我们就可以提取中位数、四分位数、百分位数和四分位距，并利用这些来比较分布。掌握频数表和曲线的构建，加上准确的读图能力，将帮助你在考试中稳稳拿下这一主题的满分。
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