📚 Understanding and Applying Cumulative Frequency Graphs | IB数学：累积频率图的理解与应用
A cumulative frequency graph is a powerful tool in descriptive statistics. It allows you to visualise how frequencies accumulate across ordered data, making it straightforward to estimate medians, quartiles, and percentiles.
累积频率图是描述性统计学中的一个强大工具。它使你能够直观地看到频数沿有序数据累积的过程，从而轻松估算中位数、四分位数和百分位数。
1. What is a Cumulative Frequency Graph? | 什么是累积频率图？
A cumulative frequency graph is a plot of cumulative frequency against the upper boundary of each class interval. The curve, often drawn as a smooth ‘S’ shape, rises from zero to the total number of data points.
累积频率图是以每个组区间的上边界为横坐标、累积频数为纵坐标绘制的图形。曲线通常呈光滑的“S”形，从零逐渐上升至数据点总数。
The key idea is to answer questions like: ‘How many values are less than or equal to a certain number?’ The cumulative frequency at any point gives exactly that count.
其核心思想是回答“有多少个数值小于或等于某个数？”这类问题。任意一点处的累积频数就给出了这个数量。
2. Constructing a Cumulative Frequency Table | 构建累积频率表
To draw the graph, you first need a grouped frequency table. For each class interval, calculate the cumulative frequency by adding the frequencies of all intervals up to and including that interval.
要绘制图形，首先需要一张分组频数表。对于每个组区间，将该区间及之前所有区间的频数相加，得到累积频数。
Consider the following data set:
考虑以下数据集：
| Height (cm) | Frequency | Cumulative Frequency |
|---|---|---|
| 140 < h ≤ 150 | 5 | 5 |
| 150 < h ≤ 160 | 12 | 17 |
| 160 < h ≤ 170 | 20 | 37 |
| 170 < h ≤ 180 | 8 | 45 |
Notice that we use the upper boundary of each class (150, 160, 170, 180) as the x-coordinate.
注意，我们使用每个区间的上边界（150、160、170、180）作为横坐标。
3. Plotting the Cumulative Frequency Graph | 绘制累积频率图
Plot points at (upper boundary, cumulative frequency). Then join them with a smooth curve or straight-line segments. The first point is often placed at the lower boundary of the first class with cumulative frequency 0.
以（上边界，累积频数）为坐标描点，然后用平滑曲线或直线段连接各点。通常将第一个点设在第一组的下边界处，累积频数为0。
For the example above, plot (140,0), (150,5), (160,17), (170,37), (180,45). Join the points with a smooth curve.
对于上面的例子，描点（140,0）、（150,5）、（160,17）、（170,37）、（180,45），并用平滑曲线连接。
Cumulative frequency = f₁ + f₂ + … + fₖ
where fₖ is the frequency of the k-th class.
其中 fₖ 是第 k 组的频数。
4. Finding the Median | 求中位数
The median is the middle value of the data set. On a cumulative frequency graph, locate the point on the y-axis at n/2, draw a horizontal line to the curve, then drop a vertical line to the x-axis.
中位数是数据集中的中间值。在累积频率图上，先在纵轴上找到 n/2 的位置，画水平线交曲线于一点，再从该点作垂线到横轴。
If n = 45, then the median position is 45/2 = 22.5. Find 22.5 on the cumulative frequency axis and read the corresponding height.
如果 n = 45，则中位数位置是 45/2 = 22.5。在累积频率轴上找到 22.5，然后读取对应的高度。
This value is an estimate because the original data are grouped. Always describe it as an estimate when classes are used.
这个值是估计值，因为原始数据是分组的。当使用组区间时，应始终将其描述为估计值。
5. Quartiles and Interquartile Range | 四分位数与四分位距
The lower quartile (Q₁) is the value below which one quarter of the data lie. Its position is n/4. The upper quartile (Q₃) lies below three quarters of the data, at position 3n/4.
下四分位数（Q₁）是低于四分之一数据所对应的值，其位置为 n/4。上四分位数（Q₃）是低于四分之三数据所对应的值，其位置为 3n/4。
To find Q₁: locate n/4 on the y-axis, read across to the curve, then down to the x-axis. Repeat for Q₃ using 3n/4.
求 Q₁：在纵轴上找到 n/4，平移到曲线，再向下读到横轴。对 Q₃ 重复同样的步骤，但使用 3n/4。
The interquartile range (IQR) is the difference between Q₃ and Q₁.
四分位距（IQR）是 Q₃ 与 Q₁ 之差。
IQR = Q₃ − Q₁
The IQR is a robust measure of spread that is not affected by extreme values.
四分位距是衡量离散程度的稳健指标，不受极端值的影响。
6. Percentiles | 百分位数
A percentile is the value below which a given percentage of the data falls. The k-th percentile corresponds to a cumulative frequency of (k/100) × n.
百分位数是指低于某一给定百分比数据所对应的值。第 k 个百分位数对应的累积频数为 (k/100) × n。
For example, the 90th percentile is found at 0.90n on the cumulative frequency axis. This tells you that 90% of the data lie at or below that value.
例如，第90百分位数位于累积频率轴上的 0.90n 处。这告诉我们90%的数据小于或等于该值。
Pₖ = value at cumulative frequency = (k/100) × n
Percentiles are commonly used in educational testing and health growth charts.
百分位数常用于教育测评和健康生长曲线中。
7. From Cumulative Frequency to Box Plot | 从累积频率图到箱线图
Once you have the five-number summary — minimum, Q₁, median, Q₃, maximum — you can draw a box plot. The cumulative frequency graph helps you estimate the middle three numbers.
一旦你有了五数概括——最小值、Q₁、中位数、Q₃、最大值——就可以绘制箱线图。累积频率图帮助你估计中间三个数。
A box plot shows the spread and skewness of the data clearly. The box extends from Q₁ to Q₃, with a line at the median.
箱线图清晰地展示了数据的分布和偏态。箱子从 Q₁ 延伸到 Q₃，中间线表示中位数。
To sketch the box plot, mark the five values on a number line. Draw a box from Q₁ to Q₃ and draw ‘whiskers’ to the minimum and maximum.
要绘制箱线图，在数轴上标出这五个值。画一个从 Q₁ 到 Q₃ 的箱子，并画出延伸到最小值和最大值的“须”。
8. Common Problems and Solution Steps | 常见题型与解题步骤
IB exam questions often provide a cumulative frequency graph and ask you to estimate the median, quartiles, or percentage of data above a certain value. Follow these steps:
IB考试题常给出累积频率图，要求你估计中位数、四分位数或高于某个值的数据百分比。请遵循以下步骤：
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Step 1: Identify the total frequency n from the top of the graph.
步骤1： 从图形顶端确定总频数 n。
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Step 2: Take the appropriate fraction of n (½n for median, ¼n for Q₁, ¾n for Q₃).
步骤2： 取 n 的相应比例（中位数取½n，Q₁取¼n，Q₃取¾n）。
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Step 3: Draw a horizontal line from that y-value to the curve, then a vertical line to the x-axis. Read the x-value.
步骤3： 从该纵坐标值画水平线至曲线，再画垂线至横轴，读出横坐标值。
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Step 4: For ‘how many’ questions, read the cumulative frequency from the graph first, then subtract to find the number in a range.
步骤4： 对于“有多少个”类型的问题，先从图上读出累积频数，再相减求出某个范围内数据的个数。
Always show your construction lines on the graph in the exam. This helps the examiner award method marks.
考试时务必在图上画出辅助线，这有助于考官给方法分。
9. Real-World Applications | 实际应用
Cumulative frequency graphs are widely used in business to analyse income distributions, in meteorology to study rainfall data, and in sports to compare athlete performance.
累积频率图在商业中广泛用于分析收入分布，在气象学中用于研究降雨数据，在体育中用于比较运动员表现。
For example, a school may use a cumulative frequency graph to determine the percentage of students scoring below a university entrance threshold. This helps administrators decide where to provide extra support.
例如，一所学校可以用累积频率图来确定低于大学录取分数线的学生比例。这有助于管理者决定在哪些方面提供额外支持。
The graph also makes it easy to compare two data sets by plotting both curves on the same axes.
该图还能将两条曲线绘制在同一坐标轴上，从而方便地比较两组数据。
10. Common Mistakes and Tips | 常见错误与注意事项
A common mistake is plotting the cumulative frequency at the midpoint of each class instead of the upper boundary. Always use the upper class boundary.
一个常见错误是使用各组区间的中点而非上边界来绘制累积频数。务必使用上边界。
Another mistake is confusing cumulative frequency with frequency. Remember that cumulative frequency never decreases as x increases.
另一个错误是混淆频数与累积频数。记住，随着 x 增大，累积频数永远不会减小。
When estimating values, do not extrapolate beyond the data range. Also, when the question asks for the number of items less than a value, read the cumulative frequency directly; for ‘more than’, subtract from n.
估计值时，不要外推到数据范围之外。此外，当问题要求小于某值的数据个数时，直接读取累积频数；“大于”某值时，则用 n 减去该累积频数。
Finally, be precise about whether you are dealing with discrete or continuous data. For continuous data, use the upper boundary; for discrete data, you may need to use the lower boundary of the next class.
最后，要明确处理的是离散数据还是连续数据。对于连续数据，使用上边界；对于离散数据，可能需要使用下一组的下边界。
11. Summary | 总结
A cumulative frequency graph turns a grouped frequency table into a visual tool for estimating medians, quartiles, and percentiles. It connects directly to box plots and is a frequent topic in IB exams.
累积频率图将分组频数表转化为一种可视化工具，用于估计中位数、四分位数和百分位数。它与箱线图直接相关，也是IB考试中的常见考点。
Master the steps of building the table, reading the curve, and interpreting the results. With practice, cumulative frequency graphs become a reliable source of marks.
掌握构建表格、读取曲线和解释结果的步骤。通过练习，累积频率图将成为一个可靠的得分点。
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