Cumulative Frequency | 累积频率

📚 Cumulative Frequency | 累积频率

Cumulative frequency is a powerful tool in statistics that allows you to see how data accumulates across a distribution. By calculating a running total of frequencies, you can quickly estimate key measures such as the median, quartiles, and percentiles without needing access to every single data value.

累积频率是统计学中一个强大的工具，它让你能够看到数据在分布中是如何累积的。通过计算频率的累加总和，你可以快速估计关键指标，如中位数、四分位数和百分位数，而无需访问每一个单独的数据值。

1. What is Cumulative Frequency? | 什么是累积频率？

Cumulative frequency is the sum of all frequencies up to a given class or value. For a data set arranged in a frequency table, you start with the first frequency, then add each successive frequency to the running total. The final cumulative frequency always equals the total number of data points, n.

累积频率是指累加到某一给定组或数值的所有频率之和。对于排列在频率表中的数据集，你从第一个频率开始，然后将每个后续频率加到累加总和上。最终的累积频率总是等于数据点的总数 n。

Understanding cumulative frequency helps you answer questions like “How many students scored less than 60%?” or “What percentage of calls lasted under 5 minutes?”. Instead of tallying individual values, you use the cumulative frequency table or graph to read the answer directly.

理解累积频率有助于回答像“有多少学生的成绩低于60%？”或“有多少比例的通话持续不到5分钟？”这样的问题。你无需逐个统计数值，而是利用累积频率表或图直接读出答案。

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2. Building a Cumulative Frequency Table | 构建累积频率表

To build a cumulative frequency table from a grouped frequency distribution, you need the class intervals and their frequencies. Add an extra column for the upper class boundary of each interval – this is vital because the cumulative frequency for a class includes all data points up to and including the upper boundary. For intervals like 0–10, 10–20, the upper boundaries are 10, 20, etc., assuming data are continuous.

要从分组频率分布构建累积频率表，你需要组距和对应的频率。为每个区间添加上组界一栏——这一点至关重要，因为某一组的累积频率包括了直到且包含上组界在内的所有数据点。对于像0–10、10–20这样的区间，若数据是连续的，则上组界分别为10、20等。

Then add a cumulative frequency column. The first entry is the frequency of the first class. The second entry is the first cumulative frequency plus the second frequency, and so on. Always check that the last cumulative frequency matches the total frequency n.

然后添加累积频率列。第一个条目是第一组的频率；第二个条目是第一个累积频率加上第二组的频率，依此类推。始终检查最后一个累积频率是否等于总频率 n。

Example: exam scores of 35 students.

示例：35名学生的考试分数。

Score interval	Frequency	Upper boundary	Cumulative frequency
0–10	5	10	5
10–20	8	20	13
20–30	12	30	25
30–40	7	40	32
40–50	3	50	35

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3. Drawing the Cumulative Frequency Curve (Ogive) | 绘制累积频率曲线（肩形图）

Plot points using the upper boundary of each class against its cumulative frequency. For the first class, you also need a starting point: the lower boundary of the first class (here 0) has cumulative frequency 0. This ensures the curve begins at (0,0) if the lowest possible value is 0, or at the lowest class boundary with cf = 0. Then join the points with a smooth freehand curve – never with straight line segments unless the exam rubric specifically allows a polygon.

以每组的上组界为横坐标、对应的累积频率为纵坐标描点。对于第一组，你还需要一个起点：第一组的下组界（此处为0）对应的累积频率为0。这确保曲线始于(0,0)，如果最小可能值为0的话；否则从最小组的下界处cf=0开始。然后用光滑的手绘曲线连接各点——除非考卷评分细则明确允许使用多边形，否则切勿用直线段连接。

Label the horizontal axis with the variable name (e.g., Score) and the vertical axis with ‘Cumulative frequency’. Scale the axes so that the curve covers more than half of the graph paper. On the y‑axis, the maximum value should be at least the total frequency n. You do not need to draw bars; just the ogive.

横轴标注变量名称（如分数），纵轴标注“累积频率”。设定坐标轴比例，使曲线占据超过半幅坐标纸。纵轴最大值至少为总频率 n。无需绘制条形，只需绘制肩形图。

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4. Estimating the Median from the Ogive | 从肩形图估计中位数

The median is the value that splits the data into two equal halves. On the cumulative frequency curve, find the position of the median on the vertical axis: this is at n/2. Draw a horizontal line from n/2 across to the curve, then drop a vertical line down to the horizontal axis. The reading on the x‑axis is your estimate of the median.

中位数是将数据平分为两半的值。在累积频率曲线上，在纵轴上找到中位数的位置：即 n/2 处。从 n/2 处画一条水平线交曲线于一点，再向下作垂线至横轴。横轴上的读数便是你对中位数的估计值。

For our example with n = 35, the median position is at 17.5. On the curve, a horizontal line from 17.5 might intersect the ogive at around 24. So the estimated median score is 24. This geometric approach is quick and visualises the concept clearly.

在我们的例子中，n = 35，中位数位置在17.5。在曲线上，从17.5处引水平线可能交曲线于约24处，因此估计的中位数分数为24。这种几何方法快捷，并能清晰地可视化概念。

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5. Estimating Quartiles and Other Percentiles | 估计四分位数及其他百分位数

The same principle applies to any percentile. The lower quartile Q₁ is at n/4, the upper quartile Q₃ at 3n/4. The p‑th percentile is at (p/100)×n. Simply locate that cumulative frequency value on the y‑axis, trace to the curve, and read off the corresponding x‑value.

同样的原理适用于任何百分位数。下四分位数 Q₁ 的位置在 n/4，上四分位数 Q₃ 在 3n/4。第 p 百分位数的位置在 (p/100)×n。只需在纵轴上找到该累积频率值，引至曲线，再读出对应的 x 值即可。

For our data, Q₁ position = 35/4 = 8.75. Reading from the curve might give around 15. Q₃ position = 3×35/4 = 26.25, giving roughly 33. The interquartile range (IQR) = Q₃ – Q₁ provides a measure of spread.

在我们的数据中，Q₁ 位置 = 35/4 = 8.75，从曲线读出的值大约为15。Q₃ 位置 = 3×35/4 = 26.25，读出的值大约为33。四分位距 (IQR) = Q₃ – Q₁，用以度量离散程度。

The 90th percentile position would be (90/100)×35 = 31.5. Horizontal line from 31.5 leads to an x‑value of about 43, meaning 90% of students scored 43 or less.

第90百分位数的位置为 (90/100)×35 = 31.5。从31.5引水平线得到的 x 值约为43，意味着90%的学生得分不高于43分。

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6. Using Linear Interpolation for a More Precise Median | 使用线性插值求更精确的中位数

Graphical readings are approximate. Examiners often expect you to use linear interpolation for grouped data. This method assumes the data are evenly spread within the median class. The formula is derived from similar triangles on the ogive.

图形读取是近似的。考官通常期望你对分组数据使用线性插值。该方法假设中位数组内的数据均匀分布。该公式源于累积频率曲线上的相似三角形。

Median = L + ((n/2 – F) / f) × w

Where L = lower boundary of the median class, F = cumulative frequency before the median class, f = frequency of the median class, w = class width.

其中 L = 中位数组的下组界，F = 中位数组之前的累积频率，f = 中位数组的频率，w = 组距。

First, find the class that contains n/2. In our example, n/2 = 17.5 falls in the 20–30 class. So L = 20, F = 13, f = 12, w = 10. Substituting:

首先，找出包含 n/2 的组。在我们的示例中，n/2 = 17.5 落在 20–30 组内。因此 L = 20, F = 13, f = 12, w = 10。代入：

Median = 20 + ((17.5 – 13) / 12) × 10 = 20 + (4.5/12)×10 = 20 + 3.75 = 23.75

Interpolation gives 23.75, which is more accurate than the rough graph estimate of 24. The same technique works for quartiles: use n/4 for Q₁ and 3n/4 for Q₃, locating the appropriate class and applying the formula with the corresponding values.

插值法得到23.75，这比图上粗略估计的24更准确。同样的方法也适用于四分位数：对 Q₁ 使用 n/4，对 Q₃ 使用 3n/4，找到正确的组并代入相应数值进行运算。

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7. Reading Other Information from the Curve | 从曲线读取其他信息

You can also use the ogive in reverse: to find how many observations lie below a given value, go from that value on the x‑axis up to the curve, then across to the y‑axis. This is useful for interpreting statements like “60% of the light bulbs last fewer than 800 hours”.

你也可以反向使用累积频率曲线：要找出低于某一给定值的观测数，从横轴上的该值向上引至曲线，再水平延伸至纵轴。这对于解释诸如“60%的灯泡使用寿命少于800小时”之类的陈述非常有用。

Similarly, to estimate the pass mark above which a certain percentage of students fall, find the cumulative frequency corresponding to the complementary percentage, read back to the x‑axis, and that gives the mark. For example, if you want the mark exceeded by 20% of students, locate the 80th percentile on the y‑axis and read the corresponding score.

类似地，要估计超过一定百分比学生的通过分数线，可以找到对应互补百分比的累积频率，回读到横轴，即得该分数。例如，若要知道被20%学生超过的分数，则定位纵轴上的第80百分位数，读出相应的得分。

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8. Comparing Distributions Using Ogives | 使用累积频率曲线比较分布

By plotting cumulative frequency curves for two or more data sets on the same axes, you can compare their central tendency and spread. A curve lying further to the right indicates higher values; a steeper curve implies that data are more concentrated, while a flatter ogive suggests greater variability.

通过在同一坐标轴上绘制两个或多个数据集的累积频率曲线，你可以比较它们的集中趋势和离散程度。位置更靠右的曲线表示数值更大；更陡峭的曲线意味着数据更集中，而较平缓的曲线则表明变异性更大。

For instance, comparing test scores of two classes: if Class B’s ogive is right of Class A’s ogive, Class B generally performed better. If Class A’s ogive is steeper in the middle, their scores are more tightly clustered around the median. You can also compare quartiles and ranges directly from the curves.

例如，比较两个班的考试分数：如果B班的累积频率曲线位于A班右侧，则B班的总体表现更好。如果A班曲线中部更陡，说明其分数更紧密地聚集在中位数周围。你也可以直接从曲线上比较四分位数和全距。

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9. Cumulative Frequency and Box Plots | 累积频率与箱线图

The five‑number summary (minimum, Q₁, median, Q₃, maximum) can be extracted from a cumulative frequency graph and used to draw a box plot. The minimum is read at the very start of the ogive (cf = 0), and the maximum at the final point (cf = n). Sometimes the extreme values are taken as the lowest and highest boundaries of the data range.

五数概括（最小值、Q₁、中位数、Q₃、最大值）可以从累积频率图中提取，并用于绘制箱线图。最小值在曲线起点处读取（cf = 0），最大值在终点处读取（cf = n）。有时极值取数据范围的最低下界和最高上界。

In our example, the minimum is 0 (or the first lower boundary), Q₁ ≈ 15, median ≈ 23.75 (by interpolation), Q₃ ≈ 33, maximum = 50. These five values directly feed into constructing the box plot, reinforcing the connection between these two visual summaries.

在我们的例子中，最小值为0（或第一个下界），Q₁ ≈ 15，中位数 ≈ 23.75（插值法），Q₃ ≈ 33，最大值 = 50。这五个数值可直接用于构建箱线图，从而加强了这两种可视化汇总之间的联系。

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10. Common Mistakes and How to Avoid Them | 常见错误及如何避免

A frequent mistake is to use the midpoint of a class interval instead of the upper boundary when plotting the ogive. This shifts the entire curve and gives incorrect estimates. Always check that you have listed upper boundaries precisely and used them as x‑coordinates.

一个常见错误是在绘制累积频率曲线时使用组中值而非上组界。这会使整条曲线偏移，得到错误的估计值。务必检查你是否准确列出了上组界并用作 x 坐标。

Another pitfall is miscounting cumulative frequency, especially when copying from a frequency column. Re‑add after the table is complete and ensure the final total matches your n. Also, many students forget to include the starting point (lowest boundary, 0) on the graph, which can chop off the lower end of the curve.

另一个陷阱是累积频率计数错误，特别是从频率列抄写时。填完表后重新累加，确保最后的总和与 n 一致。此外，许多学生忘记在图上标出起点（最低下界，0），这可能会截去曲线的下端。

When interpolating, selecting the wrong class for the median (or quartile) is common. Use the cumulative frequency to identify the class where the positional value first exceeds n/2, not just the class with the highest frequency. Also ensure you use the correct lower boundary and class width for that specific interval.

在进行插值时，选错中位数（或四分位数）所在的组也很常见。应利用累积频率来确定位置值首次超过 n/2 的组，而不只是频率最高的组。同时，要确保对该特定区间使用正确的下组界和组距。

Finally, on the graph, draw a smooth curve through points, not a series of straight segments, unless specifically instructed. A roughened curve can be acceptable if it captures the trend. Always use a sharp pencil and read axes carefully, double‑checking that you haven’t transposed n/2 or n/4 values.

最后，在图上一律用光滑曲线连接各点，非明确指示不得用一系列直线段。只要能反映趋势，稍显粗糙的曲线也可接受。始终使用尖细的铅笔，并仔细读取坐标轴，再次确认你没有把 n/2 或 n/4 的数值弄混。

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Published by TutorHao | Mathematics Revision Series | aleveler.com

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