Cumulative Frequency | 累积频率

📚 Cumulative Frequency | 累积频率

Cumulative frequency is a powerful tool in Statistics for summarising grouped data. It shows the number of observations that lie below a given upper class boundary, allowing us to estimate the median, quartiles, percentiles, and to compare distributions visually.

累积频率是统计学中总结分组数据的有力工具。它显示小于某一给定组上边界的观测值数量，使我们能够估计中位数、四分位数、百分位数，并直观比较分布。

1. What is cumulative frequency? | 什么是累积频率？

In a frequency table, each class has a frequency — the number of data values in that class. Cumulative frequency is the running total of these frequencies as you move from the lowest class to the highest class. For grouped continuous data, cumulative frequency is always associated with the upper class boundary of each class.

在频率表中，每个组都有一个频率，即该组内数据的个数。累积频率是从最低组到最高组移动时这些频率的累计总和。对于分组连续数据，累积频率总是与每一组的上边界相关联。

For example, if the first three class frequencies are 4, 11 and 15, then the cumulative frequencies are 4, 4+11=15 and 15+15=30. The third cumulative frequency of 30 means that 30 observations have values less than or equal to the upper boundary of the third class.

例如，如果前三个组的频率分别是 4、11 和 15，那么累积频率分别为 4、4+11=15 和 15+15=30。第三个累积频率为 30，表示有 30 个观测值的数值小于或等于第三组的上边界。

2. Constructing a cumulative frequency table | 构建累积频率表

To construct a cumulative frequency table, start with the grouped frequency table. Add a third column labelled ‘Cumulative frequency’. Write the first frequency as the first cumulative frequency. Then add each successive frequency to the previous cumulative total.

要构建累积频率表，从分组频率表开始。添加第三列，标记为 “累积频率”。将第一个频率作为第一个累积频率，然后将每个后续频率加到前一个累积总数上。

For continuous data, you must also record the upper class boundary of each interval. These boundaries are used as the horizontal coordinates when plotting. For example, if a class is written as 10 ≤ x < 20, its upper boundary is 20. If a class is written as 10-19, and the data are rounded to the nearest integer, the true upper boundary is 19.5, but in many exam questions the intervals are already continuous and the upper endpoint is used directly.

对于连续数据，还必须记录每个区间的上边界。这些边界在绘图时用作水平坐标。例如，如果一个组写成 10 ≤ x < 20，则其上边界为 20。如果一个组写成 10-19，且数据四舍五入到最近整数，则真实上边界为 19.5，但在许多考试题中区间已经是连续的，直接使用上端点即可。

Always check that the final cumulative frequency equals the total number of observations, often denoted n. This is a useful check for arithmetic errors.

始终检查最后一个累积频率是否等于观测值总数，通常记为 n。这是检查算术错误的有效方法。

3. Plotting a cumulative frequency curve | 绘制累积频率曲线

A cumulative frequency curve is plotted with cumulative frequency on the vertical axis and the variable on the horizontal axis. For each class, plot a point at (upper class boundary, cumulative frequency). Do not plot at the class midpoint — this is one of the most common errors.

累积频率曲线的绘制方法是将累积频率放在纵轴上，变量放在横轴上。对于每个组，在（组上边界，累积频率）处描点。不要使用组中点描点，这是最常见的错误之一。

If the first class has a lower boundary, you should also plot the starting point (lower boundary of the first class, 0). This anchors the curve at zero cumulative frequency. After plotting all points, join them with a smooth S-shaped curve, not with straight line segments.

如果第一个组有下边界，还应该绘制起点（第一个组的下边界，0）。这将曲线固定在累积频率为零的位置。绘制所有点后，用平滑的 S 形曲线连接它们，而不是用直线段连接。

The curve is sometimes called an ogive. It should be monotonically increasing — cumulative frequency never decreases. If your curve dips or goes backwards, you have made an error.

该曲线有时称为累积频率曲线。它应该是单调递增的，累积频率绝不会下降。如果你的曲线出现下沉或倒退，就说明出错了。

4. Estimating the median | 估计中位数

Median position = n ÷ 2

To estimate the median from a cumulative frequency curve, first calculate the position of the median: n/2, where n is the total frequency. Locate n/2 on the vertical cumulative frequency axis. Draw a horizontal line from this value to the curve, then draw a vertical line down to the horizontal axis. The point where it meets the horizontal axis is the estimated median.

要从累积频率曲线估计中位数，首先计算中位数的位置：n/2，其中 n 是总频率。在纵轴的累积频率轴上找到 n/2。从该值画一条水平线到曲线，然后画一条垂直线向下到横轴。垂直线与横轴的交点就是估计的中位数。

For grouped continuous data, this estimate assumes that the data values are evenly spread within each class interval. It is an estimate, not necessarily an exact value, because the original raw data are not known.

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