📚 Cumulative Frequency, Box Plots, and Histograms: Mastering Statistical Graphs for IGCSE Edexcel | 累积频率、箱形图与直方图：精通 IGCSE Edexcel 统计图表
In the Edexcel IGCSE Mathematics specification, statistical diagrams form a crucial bridge between raw data and meaningful interpretation. Three of the most powerful tools you will encounter are cumulative frequency curves, box plots, and histograms. These graphical methods allow you to summarise large data sets, identify key measures of location and spread, and make valid comparisons between distributions. This article walks you through the core concepts, step‑by‑step construction techniques, and exam‑ready strategies for each type of diagram, ensuring you can tackle any related question with confidence.
在 Edexcel IGCSE 数学大纲中，统计图表是连接原始数据与有意义解读的重要桥梁。你将接触到三种最有力的工具：累积频率曲线、箱形图和直方图。这些图形方法能帮助你概括大型数据集、找出关键的定位与离散程度指标，并对不同分布进行有效比较。本文带你逐一理解核心概念、分步掌握绘制技巧，并提供面向考试的应试策略，确保你可以自信应对任何相关题目。
1. The Purpose of Statistical Graphs in IGCSE | IGCSE 中统计图表的作用
Statistical graphs are not merely pictures to draw; they are instruments for analysis. In the Edexcel IGCSE exam, you are often asked to construct a diagram from a grouped frequency table and then use it to estimate the median, quartiles, interquartile range, or even the number of items above a certain threshold. Each graph type serves a distinct purpose: cumulative frequency curves excel at showing running totals and finding positional values, box plots provide a compact five‑number summary ideal for comparing spreads, and histograms deal with unequal class widths by representing frequency in terms of area rather than height alone.
统计图表不仅仅是画出来的图片，更是分析的工具。在 Edexcel IGCSE 考试中，你常常需要根据分组频数表绘制图形，然后利用图形估算中位数、四分位数、四分位距，甚至超过某个阈值的数据个数。每种图形都有独特的作用：累积频率曲线擅长展示累计总数并找出位置数值，箱形图提供简洁的五数概括，非常便于比较分布形态，而直方图通过用面积而非仅用高度表示频数，妥善处理了组距不等的情况。
2. Understanding Cumulative Frequency | 理解累积频率
Cumulative frequency is the running total of frequencies as you move from the lowest class interval to the highest. For grouped data, you first create a table with class intervals, frequencies, and an additional column for cumulative frequency. To calculate the cumulative frequency for a given row, add its frequency to the sum of all frequencies from previous rows. This process transforms a standard frequency distribution into a steadily rising sequence, enabling you to answer questions about 'how many up to this point' rather than just 'how many in this group'.
累积频率是指从最低组距到最高组距逐步累加的频数总和。对于分组数据，你首先要建立一个包含组距、频数和累积频率列的表格。要计算某一行的累积频率，只需将其频数与之前所有行的频数之和相加即可。这个过程将普通的频数分布转化为一个稳定上升的序列，使你能够回答“到此为止有多少个”而不仅仅是“这一组有多少个”。
Remember that the very first cumulative frequency equals the first frequency, and the final cumulative frequency must equal the total number of data values. This simple check can save you from careless errors in the exam. Cumulative frequency is always plotted at the upper boundary of each class interval, never at the midpoint.
请记住，第一个累积频率就等于第一个频数，而最后一个累积频率必须等于数据的总个数。这个简单的检验可以帮助你避免考试中的粗心错误。累积频率总是在每个组距的上限处描点，而不是在中点处。
3. Constructing a Cumulative Frequency Table Step by Step | 逐步构建累积频率表
50 < h ≤ 160 with frequency 12, add 12 to the previous 5 to get 17. Continue this addition until the final interval gives the total 80.
想象一个分组频数表，显示了 80 名学生的身高，组距如 140 < h ≤ 150、150 < h ≤ 160 等。首先，扩展表格，增加一列累积频率。对于组距 140 < h ≤ 150，如果频数是 5，则累积频率为 5。对于下一个组距 150 < h ≤ 160，频数为 12，将 12 加到前一个 5 上得到 17。继续这样累加，直到最后一个组距的累积频率等于总数 80。
A well‑ ry is 160. Plotting the point (160, 17) correctly anchors your cumulative frequency curve.
一个结构清晰的表格应始终列出带有明确上限的组距。上限就是你将在横轴上使用的数值。例如，在组距 150 < h ≤ 160 中，上限为 160。正确描点 (160, 17) 能够为你的累积频率曲线奠定准确的基础。
4. Drawing a Smooth Cumulative Frequency Curve | 绘制平滑的累积频率曲线
tive frequency). Crucially, the first point should be plotted at the lower boundary of the first interval with cumulative frequency 0, unless the first interval already starts at the minimum value. For a starting interval like 140 < h ≤ 150, you plot (140, 0) to show that before 140, nothing has been counted.
在坐标纸上，将横轴标上变量及其单位，纵轴标上累积频率并从 0 开始。选择一种在两个方向上都占据超过一半网格的刻度。用整齐的十字或圆点精确地在坐标 (上限, 累积频率) 处描出每个点。关键的是，第一个点应描在第一个组距的下限处，累积频率为 0，除非第一个组距已经起始于最小值。对于像 140 < h ≤ 150 这样的起始组距，你要描点 (140, 0) 来表示在 140 之前没有任何数据被计入。
Once all points are in place, join them with a smooth, freehand curve that passes through each point. Do not use a ruler to connect the dots with straight segments; examiners expect a gentle S‑shaped curve that rises slowly at first, steepens, and then levels off towards the end. A well‑drawn curve allows accurate readings of medians and quartiles.
所有点就位后，用一条平滑的手绘曲线将它们连接起来，并穿过每一个点。不要用直尺以直线段连接各点；考官期望的是一条缓缓上升的 S 形曲线：开始时上升缓慢，中间变陡，接近末尾时趋于平缓。绘制良好的曲线有助于准确读取中位数和四分位数。
5. Interpreting Medians, Quartiles, and Percentiles from the Curve | 从曲线解读中位数、四分位数和百分位数
To find the median, calculate half of the total cumulative frequency (n/2) and draw a horizontal line from that value on the vertical axis to the curve, then drop a vertical line to the horizontal axis. The reading on the horizontal axis is your estimate of the median. Similarly, the lower quartile (Q₁) corresponds to n/4, and the upper quartile (Q₃) corresponds to 3n/4. The interquartile range (IQR) is simply Q₃ − Q₁.
要找到中位数，先算出累积频率总数的一半 (n/2)，从纵轴上的这个数值画一条水平线连接到曲线，然后向下画一条垂直线到横轴。横轴上的读数就是你估算的中位数。类似地，下四分位数 (Q₁) 对应 n/4，上四分位数 (Q₃) 对应 3n/4。四分位距 (IQR) 就是 Q₃ − Q₁。
Edexcel IGCSE questions also ask for percentiles. The pth percentile is found by drawing from p% of the total frequency. For instance, the 90th percentile is obtained by locating 90% of n on the vertical axis. Always show your construction lines on the graph, as they earn marks even if your reading is slightly off due to graph imprecision.
Edexcel IGCSE 考题还会要求计算百分位数。第 p 百分位数是从频数总数的 p% 处画线得到的。例如，第 90 百分位数就是找到纵轴上 n 的 90% 来读取。一定要在图形上画出辅助作图线，因为即使你的读数因图形精度略有偏差，这些线也能为你赢得分数。
6. Introduction to Box Plots (Box‑and‑Whisker Diagrams) | 箱形图（盒须图）简介
A box plot displays a five‑number summary of a data set: the minimum value, lower quartile Q₁, median Q₂, upper quartile Q₃, and maximum value. The box spans from Q₁ to Q₃, with a vertical line inside marking the median. Whiskers extend from the box to the minimum and maximum values, provided there are no outliers. Box plots are especially useful for quickly visualising the central tendency, spread, and skewness of a distribution.
箱形图展示一个数据集的五数概括：最小值、下四分位数 Q₁、中位数 Q₂、上四分位数 Q₃ 以及最大值。矩形盒体从 Q₁ 延伸到 Q₃，盒内一条竖线标记中位数。须线从盒体延伸到最小值和最大值，前提是没有异常值。箱形图尤其适合快速可视化数据分布的集中趋势、散布程度和偏态。
To draw an accurate box plot, use a horizontal scale that covers the full range of the data. Draw the box with its left edge at Q₁ and right edge at Q₃. Mark the median clearly and draw the whiskers. The entire diagram should be drawn above a single number line, with all five values labelled if required. Scale selection again plays a vital role: a cramped scale makes interpretation difficult, while a well‑chosen scale earns method marks.
要绘制准确的箱形图，应使用一条涵盖整个数据范围的横轴刻度。盒体的左边缘位于 Q₁，右边缘位于 Q₃。清晰地标出中位数并画出须线。整个图形应绘制在一条数轴之上，并在需要时标出全部五个数值。刻度的选择同样至关重要：过于拥挤的刻度会让解读变得困难，而精心选择的刻度能为你赢得方法分。
7. Drawing and Interpreting Box Plots from Summary Data | 根据汇总数据绘制并解读箱形图
ring, say, 140 to 185. Mark small vertical ticks at each of the five positions. Construct the box from 155 to 168, with a vertical line segment at 162. Extend the left whisker from 155 down to 142, and the right whisker from 168 up to 180. This compact visual immediately tells you that the middle 50% lies between 155 and 168, with the median closer to Q₃, indicating a slight negative skew.
假设给出的数据是 Q₁ = 155，中位数 = 162，Q₃ = 168，最小值 = 142，最大值 = 180。画一条横轴，范围大约是 140 到 185。在这五个位置各标记一条细竖线。构建从 155 到 168 的盒体，并在 162 处画一条竖线段。将左须线从 155 向下延伸到 142，右须线从 168 向上延伸到 180。这个简洁的视觉图形立刻告诉你，中间 50% 的数据落在 155 到 168 之间，且中位数更靠近 Q₃，表明分布略呈负偏态。
Box plots excel at comparing two or more distributions side by side. When Edexcel asks you to compare, always comment on a measure of central tendency (median) and a measure of spread (interquartile range or range). Mention skewness if visible, and relate your statements to the context, for example, 'on average, males are taller, and their heights are more spread out'.
箱形图在并列比较多组分布时表现优异。当 Edexcel 要求你进行比较时，务必对集中趋势指标（中位数）和离散程度指标（四分位距或全距）进行评论。如果偏态明显，也要提及，并将你的陈述联系到实际语境，例如“平均而言，男性身高更高，而且他们的身高分布更分散”。
8. Comparing Distributions Using Box Plots | 使用箱形图比较分布
Consider two box plots drawn on the same scale: one for the test scores of Class A and another for Class B. Class A's box plot shows a higher median, a shorter box, and symmetrical whiskers. Class B's plot reveals a lower median, a longer box, and a right‑skewed whisker. From these observations, you can conclude that Class A performed better on average and more consistently, while Class B had more variability and possibly a few very high outliers pulling the upper whisker.
考虑两条绘制在同一刻度上的箱形图：一条是 A 班的考试成绩，另一条是 B 班的。A 班的箱形图中位数更高、盒体更短、须线对称。B 班的箱形图显示中位数较低、盒体较长、须线右偏。从这些观察中可以得出结论：A 班平均表现更好且成绩更稳定，而 B 班差异更大，可能存在几个非常高的异常值拉长了上须线。
Always anchor your comparison in exact values where possible. Instead of simply saying 'the median is higher', state 'the median score of Class A is 72 compared to 58 for Class B, a difference of 14 marks'. Such precision demonstrates full engagement with the data and secures the highest marks in 'compare' questions.
在可能的情况下，始终用具体数值支撑你的比较。不要简单地说“中位数更高”，而要说“A 班的中位数分数为 72，而 B 班为 58，相差 14 分”。这种精确性展示了与数据的充分互动，能在“比较”类题目中助你拿下满分。
9. Introducing Histograms with Unequal Class Widths | 引入组距不等的直方图
Unlike bar charts, histograms are used exclusively for continuous data grouped into intervals, and their defining feature is that the area of each bar is proportional to the frequency it represents. When class widths are unequal, you cannot simply use the frequency as the bar height. Instead, you must first calculate frequency density using the formula: frequency density = frequency ÷ class width. The vertical axis then represents frequency density, not raw frequency.
与条形图不同，直方图专门用于按区间分组的连续数据，其本质特征是每个矩形的面积与其所代表的频数成比例。当组距宽度不等时，不能简单地用频数作为柱高。你必须首先使用公式计算频率密度：频率密度 = 频数 ÷ 组距。此时纵轴表示频率密度，而非原始频数。
This concept is one of the most commonly tested in Edexcel IGCSE and a frequent source of error. Students often forget to divide by class width and simply plot frequencies, which leads to a distorted representation. Remember, a wide interval with a large frequency may have a relatively low frequency density, while a narrow interval with a small frequency can have a very high density. The histogram tells the true story through area.
这一概念是 Edexcel IGCSE 中最常考的知识点之一，也是常见的失分点。学生经常忘记除以组距，直接绘制频数，导致图形失真。请记住，一个组距很宽且频数很大的区间，其频率密度可能相对较低，而一个组距很窄且频数较小的区间，其频率密度却可能很高。直方图是通过面积来讲述真实情况的。
10. Calculating Frequency Density and Drawing Histograms | 计算频率密度并绘制直方图
h time t and the vertical axis with frequency density. For the first bar, draw a rectangle from t = 0 to t = 5 with height 4. The second bar spans 5 to 15 with height 3. The third bar goes from 15 to 30 with height 3. Check that the area of each bar equals its frequency: 5×4 = 20, 10×3 = 30, 15×3 = 45.
假设一个分组频数表包含区间 0 < t ≤ 5（频数 20）、5 < t ≤ 15（频数 30）以及 15 < t ≤ 30（频数 45）。首先计算组距：分别为 5、10 和 15。然后计算频率密度：20 ÷ 5 = 4，30 ÷ 10 = 3，45 ÷ 15 = 3。在坐标纸上，横轴标为时间 t，纵轴标为频率密度。第一个柱体从 t = 0 到 t = 5 绘制一个高度为 4 的矩形。第二个柱体跨度 5 到 15，高度为 3。第三个柱体从 15 到 30，高度为 3。请检验每个矩形的面积是否等于其频数：5×4 = 20，10×3 = 30，15×3 = 45。
Always use a ruler for drawing histograms and ensure bars touch one another because data is continuous. Gaps between bars imply discrete data, which is inappropriate for a histogram. Label both axes with units and provide a clear, linear scale. Small inaccuracies in bar widths or heights can lose precision marks, so double‑check all divisions before plotting.
绘制直方图时务必使用直尺，并确保柱体彼此紧贴，因为数据是连续的。柱体之间有间隙会暗示数据是离散的，这对于直方图来说是不合适的。为两个坐标轴标上单位，并提供清晰、线性的刻度。柱体宽度或高度的任何微小不准确都可能导致精确度失分，因此在描点前务必复算所有除法。
11. Interpreting Histograms and Finding Missing Frequencies | 解读直方图与求取缺失频数
If an exam question provides an incomplete histogram and a partially blank frequency table, you can use area to find missing values. For instance, given one complete bar with known frequency and class width, calculate its frequency density to establish the scale. Then, for any other bar, measure the class width, multiply by its drawn frequency density to obtain the frequency. This area‑frequency relationship is reversible and is a powerful problem‑solving tool.
如果考题给出一张不完整的直方图和一张部分空白的频数表，你可以利用面积来求出缺失值。例如，给定一个已知频数和组距的完整柱体，先计算它的频率密度以确定比例尺。然后，对于任何其他柱体，量出组距宽度，乘以其所绘制的频率密度即可得到频数。这种面积与频数的关系是可逆的，是一个强大的解题工具。
In higher‑tier Edexcel IGCSE, you may encounter questions asking you to estimate the number of items within a sub‑interval of a histogram bar. Because the histogram shows frequency distributed uniformly within each bar, you assume the frequency is spread evenly across the class width. To estimate the frequency for a part of a bar, calculate the fraction of the width that falls within your sub‑interval and multiply by the total frequency of that bar.
在 Edexcel IGCSE 的高阶级别中，你可能会遇到要求估算直方图某柱体子区间内数据个数的问题。由于直方图显示频数在每个柱体内均匀分布，你可以假设频数在组距宽度内均匀铺开。要估算柱体一部分的频数，先计算落入子区间的宽度占整柱宽度的比例，再乘以该柱体的总频数。
12. Common Pitfalls and Proven Exam Strategies | 常见误区与行之有效的应试策略
One frequent mistake is confusing the median with the mean when reading from a cumulative frequency graph — only positional averages are readable. Another is forgetting to plot the (lower boundary, 0) point, which leaves the curve disconnected from the axis. In box plots, drawing whiskers through the box or omitting the scale entirely are costly errors. In histograms, the most damaging error is using frequency instead of frequency density as the vertical axis, which fundamentally misrepresents the data and can cause you to lose a significant chunk of marks.
一个常见错误是在读取累积频率图时混淆中位数和均值——只有位置平均数可从图中读取。另一个是忘记描出（下限, 0）这一点，导致曲线与坐标轴脱节。在箱形图中，须线穿过盒体或完全省略刻度都是代价高昂的错误。在直方图中，破坏性最大的错误是使用频数而非频率密度作为纵轴，这从根本上扭曲了数据，可能导致你丢失大块分数。
Before any exam, practice drawing and interpreting all three diagram types under timed conditions. Create a checklist: for cumulative frequency — upper boundaries, smooth curve, reading lines; for box plots — five‑number summary, labelled scale, comparisons; for histograms — frequency density, consistent scale, area check. Approach each question methodically, always relating the visual representation back to the context of the problem. With consistent practice and attention to detail, these statistical graphs become your most reliable marks on the Edexcel IGCSE paper.
在考试前，请计时练习三种图表的绘制与解读。制作一个检查清单：累积频率——上限、平滑曲线、读数辅助线；箱形图——五数概括、标注刻度、比较；直方图——频率密度、一致刻度、面积检验。有条不紊地解答每一道题，始终将视觉呈现与问题背景联系起来。通过持续练习和对细节的关注，这些统计图表将成为你在 Edexcel IGCSE 试卷上最可靠的得分点。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图