📚 Cumulative Frequency & Box Plots | 累积频数与箱线图
Cumulative frequency diagrams and box plots are powerful statistical tools used to summarise large data sets, identify key values, and compare distributions. In Edexcel IGCSE Mathematics, you are expected to construct cumulative frequency tables, draw smooth cumulative frequency curves, estimate quartiles and medians, and use this information to create box plots. Mastering these skills will help you interpret grouped data efficiently and answer questions on central tendency, spread, and skewedness with confidence.
累积频数图和箱线图是重要的统计工具，用于概括大型数据集、辨识关键数值并进行分布比较。在Edexcel IGCSE 数学考试中，你须学会构建累积频数表、绘制平滑的累积频数曲线、估算四分位数和中位数，并利用这些信息绘制箱线图。熟练掌握这些技能，能让高效解读分组数据，并自信地回答有关集中趋势、离散程度和偏态分布的题目。
1. What is Cumulative Frequency? | 什么是累积频数？
Cumulative frequency is the running total of frequencies as you move through ordered data. Instead of looking at how many items fall into each separate class or interval, you add up all frequencies up to the upper boundary of that interval. This gives the total number of observations that are less than or equal to that boundary value.
累积频数是顺着有序数据移动时频率的累加总和。它关注的不是每个独立组或区间中有多少项，而是将到达该区间上界之前的所有频率相加。这样得到的就是小于或等于该边界值的观测值总数。
mes 5 + 8 = 13. The final cumulative frequency must equal the total number of data values.
例如，若第一组 150 ≤ h < 155 cm 的频率为 5，则身高不超过 155 cm 的累积频数为 5。若下一组 155 ≤ h < 160 的频率为 8，则身高不超过 160 cm 的累积频数为 5 + 8 = 13。最后一个累积频数必须等于数据的总个数。
2. Constructing a Cumulative Frequency Table | 构建累积频数表
To create a cumulative frequency table, start by listing the class intervals and their frequencies. Add a third column for the upper class boundary of each interval, because cumulative frequency is plotted against these upper boundaries. Then, fill in the cumulative frequency column by adding each frequency to the sum of all previous frequencies. Always double-check that the final cumulative frequency matches the total frequency n.
要构建累积频数表，首先列出组区间及其频率。再增加一列记录每个区间的上组界，因为累积频数是以这些上界为横轴绘制的。然后，逐一累加频率，填入累积频数列。务必检查最终的累积频数是否等于总频数 n。
| Height, h (cm) | Frequency | Upper boundary | Cumulative frequency |
|---|---|---|---|
| 150 ≤ h < 155 | 5 | 155 | 5 |
| 155 ≤ h < 160 | 8 | 160 | 13 |
| 160 ≤ h < 165 | 15 | 165 | 28 |
| 165 ≤ h < 170 | 12 | 170 | 40 |
| 170 ≤ h < 180 | 10 | 180 | 50 |
l frequencies, we get a total of 50 observations. This systematic layout makes plotting straightforward.
在上表中，第一个区间的上界为 155 cm。经过第二个区间后的累积频数是 5 + 8 = 13。将所有频率相加后，得到总共 50 个观测值。这种系统化的表格布局使绘图变得清晰明了。
3. Drawing a Cumulative Frequency Curve | 绘制累积频数曲线
ulative frequency = 0. Then, plot (155, 5), (160, 13), (165, 28), (170, 40) and (180, 50).
将各点标于坐标系中，横轴为组上界，纵轴为对应的累积频数。第一个点放在第一个区间的下界处，累积频数为零。例如，在高度 150 cm 处，累积频数 = 0。然后标出 (155, 5)、(160, 13)、(165, 28)、(170, 40) 和 (180, 50)。
Join the points with a smooth curve that passes through all plotted points. Do not use straight line segments unless the data is discrete and you are drawing a step polygon. The curve should start at the first lower boundary on the horizontal axis and finish at the final upper boundary, reaching the total frequency on the vertical axis.
用一条平滑曲线连接各点，曲线必须经过所有标出的点。除非数据是离散的且你在绘制阶梯多边形，否则不要使用直线段连接。曲线应从横轴上首个下界处开始，并在最后一个上界处结束，纵轴达到总频数。
A well-drawn cumulative frequency curve is always increasing and has a characteristic S-shape if the data is normally distributed. For skewed data, the curve will be steeper in the region where frequencies are higher and flatter where frequencies are lower.
一条绘制良好的累积频数曲线总是上升的，如果数据呈正态分布，会呈现特有的 S 形。对于偏态数据，曲线在频率较高的区域更陡，在频率较低的区域则更为平缓。
4. Estimating Quartiles and Median | 估算四分位数和中位数
The median (Q₂) corresponds to the value below which 50% position is 50/2 = 25, so the median height is approximately 163 cm (read from the graph).
中位数 Q₂ 对应着 50% 数据所处的数值，它在累积频数轴上的位置为 n/2。从该值画一条水平线与曲线相交，再向下画垂线交于横轴，即可读得中位数。对于我们的例子 n = 50，中位数位置为 50/2 = 25，因此中位数身高大约为 163 cm（从图中读出）。
The lower quartile (Q₁) is the value that separates the lowest 25% pper quartile (Q₃) is at 3n/4 = 37.5, so the corresponding height is approximately 168 cm.
下四分位数 Q₁ 是分隔最低 25% 数据的值，它的累积频数位置是 n/4。当 n = 50 时，n/4 = 12.5，因此 Q₁ 在累积频数为 12.5 处读取，约为 159 cm。上四分位数 Q₃ 位于 3n/4 = 37.5，故对应身高约 168 cm。
Always remember that these are estimates because they rely on reading values from a smooth curve drawn through grouped data points. When an exam question asks for an estimate of the median or quartiles, it is this graphical reading they expect.
请始终记住这些只是估计值，因为它们依赖于从分组数据的平滑曲线上读取。当考题要求估算中位数或四分位数时，正是期望这样的图形读取方式。
5. Interquartile Range (IQR) | 四分位距
ween the upper and lower quartiles: IQR = Q₃ - Q₁. It measures the spread of the middle 50% making it a robust measure of dispersion. For the heights example, IQR = 168 - 159 = 9 cm.
四分位距是上四分位数与下四分位数之差：IQR = Q₃ - Q₁。它衡量中间 50% 数据的离散程度，且不受极端值影响，是一种稳健的离散度量。在身高例子中，IQR = 168 - 159 = 9 cm。
A smaller IQR indicates that the central half of the data is tightly clustered around the median, while a larger IQR suggests greater variability. When comparing two distributions, quoting both the median and IQR gives a clear picture of central tendency and spread without being skewed by outliers.
较小的 IQR 表示中央一半数据紧密聚集在中位数附近，而较大的 IQR 则意味着更大的变异性。在比较两个分布时，同时给出中位数和 IQR，可以清晰地呈现集中趋势和离散程度，而不会受到异常值的扭曲。
R. While this rule is useful, Edexcel IGCSE generally focuses on calculating and interpreting the IQR itself.
你还可以用 IQR 识别潜在的异常值。一种常见规则将小于 Q₁ - 1.5 × IQR 或大于 Q₃ + 1.5 × IQR 的数值定义为异常值。虽然此规则很有用，但 Edexcel IGCSE 一般侧重计算和解读 IQR 本身。
6. Percentiles and Other Quantiles | 百分位数及其他分位数
curve allows you to estimate any percentile. The pth percentile is the value below which p% s cumulative frequency on the vertical axis, move across to the curve, and read the corresponding data value on the horizontal axis.
除了四分位数，累积频数曲线还能让你估算任意百分位数。第 p 百分位数是 p% 数据所低于的那个数值，其位置在累积频数轴上的 (p/100) × n 处。例如，第 90 百分位数位于 0.90 × n 处。只需在纵轴上找到这一累积频数，水平移至曲线，再读取横轴上的对应数据值即可。
Similarly, deciles (tenths) and other quantiles can be found. The 40th percentile, for example, is at 0.4n. These measures help describe the shape and spread of a distribution in greater detail than quartiles alone.
同理，也可找到十分位数及其他分位数。例如第 40 百分位数位于 0.4n 处。这些度量能比单独使用四分位数更细致地描述分布的形状和离散度。
When an exam question asks for the percentage of data above a certain value, find that value on the horizontal axis, go up to the curve, then across to the cumulative frequency axis. Read the cumulative frequency and subtract from n; then express as a percentage of n.
当考题要求计算高于某数值的数据百分比时，先在横轴上找到该值，向上交于曲线，再平移至累积频数轴。读取累积频数并从 n 中减去，然后表示为 n 的百分比。
7. Box Plots (Box-and-Whisker Plots) | 箱线图
A box plot displays the five-number summary of a data set: the minimum value, lower quartile (Q₁), median (Q₂), upper quartile (Q₃), and maximum value. It provides a compact visual representation of the distribution's centre, spread, and potential skewness.
箱线图展示一组数据的五数概括：最小值、下四分位数 Q₁、中位数 Q₂、上四分位数 Q₃ 和最大值。它以一种紧凑的可视化方式呈现分布的中心、离散度和可能的偏态。
The box is drawn from Q₁ to Q₃, with a vertical line inside at the median. Whiskers extend from the box to the minimum and maximum values that are not outliers. If outliers are present, they are often marked with separate dots or crosses, and the whiskers may stop at the most extreme non-outlier values.
箱体从 Q₁ 画至 Q₃，内部在中位数处画一条竖线。须线从箱体延伸至非异常值的最小值和最大值。若存在异常值，通常以独立的点或叉号标出，而须线可能止于最远的非异常值。
Edexcel IGCSE typically uses simple box plots without outlier markings unless the question specifies otherwise. You must be able to draw and interpret box plots accurately, labelling the scale clearly.
Edexcel IGCSE 通常采用不带异常值标记的简易箱线图，除非题目另作说明。你必须能够准确绘制和解读箱线图，并清晰地标出刻度。
8. Drawing a Box Plot from a Cumulative Frequency Diagram | 从累积频数图绘制箱线图
Once you have obtained the median, Q₁ and Q₃ from the cumulative frequency curve, and you know the minimum and maximum values from the data (or from the class boundaries), you can construct the box plot. In grouped data, the minimum is the lower boundary of the first interval and the maximum is the upper boundary of the last interval.
一旦你从累积频数曲线获得了中位数、Q₁ 和 Q₃，并知晓数据的最小值与最大值（或从组边界得知），即可绘制箱线图。在分组数据中，最小值为第一个区间的下界，最大值为最后一个区间的上界。
Draw a suitable horizontal scale covering the range of the data. Above this axis, draw the box from Q₁ to Q₃ and mark the median inside. Then draw whiskers from Q₁ to the minimum and from Q₃ to the maximum. The box plot does not show frequencies, only positions along the data scale.
画一条覆盖数据范围的合适水平尺度。在此轴上方，从 Q₁ 到 Q₃ 画出箱体，并在内部标注中位数。然后从 Q₁ 到最小值以及从 Q₃ 到最大值画出须线。箱线图不显示频率，只显示在数据尺度上的位置。
Sometimes you may be asked to compare two box plots drawn on the same scale. Ensure both are clearly labelled and aligned so that medians, IQRs, and ranges can be easily compared.
有时你可能被要求比较同一尺度下绘制的两个箱线图。应确保两者均标注清晰并对齐，以便比较中位数、IQR 和全距。
9. Comparing Distributions Using Box Plots | 使用箱线图比较分布
When comparing two data sets using box plots, comment on the median as a measure of central tendency, the interquartile range or the overall range for spread, and the shape or skewness of the distribution. For example, 'The median of Set A is higher than that of Set B, so on average, Set A values are larger.'
当使用箱线图比较两组数据时，应评论中位数作为集中趋势的度量、四分位距或全距作为离散度的度量，以及分布的形状或偏态。比如“数据集 A 的中位数高于数据集 B，因此平均而言 A 的数值更大”。
Discuss spread: 'Set B has a larger interquartile range, indicating greater variability in the middle 50% of the data.' If the whiskers are asymmetric, comment on skew: 'Set A is positively skewed because the upper whisker is longer than the lower whisker, suggesting a tail towards higher values.'
讨论离散度：“数据集 B 的四分位距更大，表明中间 50% 数据的变异性更大。”若须线不对称，则评论偏态：“数据集 A 呈正偏态，因为上部须线比下部须线更长，意味着数据向高端延伸。”
Always support comparisons with numerical values read from the graph, such as medians of 163 and 158, IQRs of 9 and 14. This demonstrates precise interpretation and earns full marks in Edexcel IGCSE statistics questions.
比较时务必引用从图中读取的具体数值，如中位数 163 与 158、IQR 9 与 14。这体现了解读的精确性，能在 Edexcel IGCSE 统计题目中获得满分。
10. Common Mistakes and Tips | 常见错误与技巧
A frequent error is plotting cumulative frequency against class midpoints instead of upper class boundaries. This leads to an incorrect curve and inaccurate quartile estimates. Always use the upper boundary. In discrete data with simple values, you may use the exact data point as the boundary, but the principle remains: cumulative frequency corresponds to 'up to and including' that value.
一个常见错误是把累积频数绘制在组中点而非上组界上，这会导致曲线错误和四分位数估计不准确。务必使用上组界。对于简单取值的离散数据，可将确切数据点作为边界，但原则不变：累积频数对应“不超过且包含”该值。
Another mistake is joining points with straight lines. A smooth curve is expected for grouped continuous data. If a step polygon is required for discrete data, the question will guide you. Do not forget to start the curve at the lower boundary of the first interval with cumulative frequency zero.
另一个错误是用直线连接各点。对于连续分组数据，需画平滑曲线。若离散数据要求阶梯多边形，题目会明确指引。切记曲线应从第一个区间的下界、累积频数为零处开始。
When reading values from the curve, use a ruler and draw light horizontal and vertical lines to improve accuracy. Be careful with scaling the axes: a poorly chosen scale can make readings less precise. Always label axes as 'Cumulative frequency' and the variable with its unit.
从曲线上读取数值时，应使用直尺并轻画水平线与垂线以提高精度。注意坐标轴的尺度：选择不当的尺度会降低读数精度。始终将纵轴标记为“累积频数”，横轴标记为变量及其单位。
Finally, practice interpreting box plots in context, not just drawing them. Edexcel IGCSE often puts box plots alongside questions about whether one group performed better, or if there is more consistency. Contextual answers, such as 'Class A has a higher median score and smaller IQR, suggesting better and more consistent performance,' show high-level understanding.
最后，要练习在具体情境中解读箱线图，而不仅仅是绘制它们。Edexcel IGCSE 常将箱线图与“哪个小组表现更好”或“哪个更稳定”等问题结合。结合情境的回答，如“A 班中位数较高且 IQR 较小，表明表现更好且更稳定”，能展现高阶理解力。
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