📚 Cumulative Frequency Graphs | 累积频率图
Cumulative frequency graphs, also known as ogives, are a powerful tool in statistics for visualising how data accumulates over a range. They are particularly useful for estimating medians, quartiles and percentiles without needing the raw data. In IB Mathematics, understanding how to construct and interpret these graphs is essential for data analysis.
累积频率图（又称累积频数曲线）是统计学中用于展示数据在某一范围内累积情况的有力工具。它特别适用于在无需原始数据的情况下估计中位数、四分位数和百分位数。在 IB 数学中，掌握如何构建和解读这些图形对于数据分析至关重要。
1. What is Cumulative Frequency? | 什么是累积频率？
Cumulative frequency is the running total of frequencies up to a particular class interval or data point. It tells you how many observations fall below the upper boundary of that interval.
累积频率是截至某一特定组距或数据点的频率累计总和。它告诉你该组距上界以下有多少个观测值。
s 5+8=13, meaning 13 observations are less than or equal to 30 (assuming continuous data).
例如，如果组距 10-20 的频率是 5，20-30 的频率是 8，那么截至 30 的累积频率是 5+8=13，意味着有 13 个观测值小于或等于 30（假设是连续数据）。
2. Constructing a Cumulative Frequency Table | 构建累积频率表
To construct a cumulative frequency table, start with a frequency distribution table showing class intervals and their frequencies. Add a third column for cumulative frequency. Begin with the first interval: the cumulative frequency equals that interval's frequency. For each subsequent interval, add its frequency to the previous cumulative frequency.
要构建累积频率表，首先需要有一个显示组距及其频率的频率分布表。然后添加第三列累积频率。从第一个组距开始：累积频率等于该组距的频率。对于每个后续组距，将其频率加到前一个累积频率上。
Here is an example table for the test scores of 30 students:
以下是针对 30 名学生考试分数的示例表格：
| Score Interval | Frequency (f) | Cumulative Frequency |
|---|---|---|
| 0-10 | 2 | 2 |
| 10-20 | 5 | 7 |
| 20-30 | 8 | 15 |
| 30-40 | 7 | 22 |
| 40-50 | 5 | 27 |
| 50-60 | 3 | 30 |
The final cumulative frequency always equals the total number of observations.
最终的累积频率总是等于观测值的总数。
3. Drawing a Cumulative Frequency Curve (Ogive) | 绘制累积频率曲线（累积频数图）
Plot the cumulative frequency against the upper boundary of each class interval. Use a scatter plot with smooth lines or a line graph. The x-axis represents the upper class boundaries, and the y-axis represents cumulative frequency. Start the graph at the lower boundary of the first class with a cumulative frequency of zero (unless data starts from zero). This ensures the curve starts from the origin.
以每组距的上界为横坐标，累积频率为纵坐标绘制散点图，并用平滑曲线连接。x 轴代表组距上界，y 轴代表累积频率。通常从第一个组距的下界开始，累积频率为 0（除非数据从零开始），这样曲线就能从原点开始延伸。
Points are plotted at (upper boundary, cumulative frequency). Then join the points with a smooth curve. Do not use a straight line between points; the ogive is a cumulative distribution, which is typically monotonically increasing and smooth if the underlying distribution is continuous.
将点绘制在 (上界, 累积频率) 处，然后用平滑的曲线连接各点。不要在点之间使用直线段；累积频数曲线代表的是累积分布，如果底层分布是连续的，它应呈单调递增且光滑的形态。
4. Key Features of the Graph | 图形关键特征
The ogive is an S-shaped curve if the data is normally distributed. It always starts at or near zero on the y-axis and ends at the total frequency. The steepness of the curve indicates the density of data
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