📚 Cumulative Frequency and Box Plots | 累积频率与箱形图
Cumulative frequency diagrams and box plots are powerful statistical tools that allow us to summarise large sets of data, identify key values such as the median and quartiles, and make meaningful comparisons between distributions. In the IGCSE Edexcel Mathematics syllabus, you are expected to construct cumulative frequency tables, draw smooth cumulative frequency curves, interpret graphs to find medians and interquartile ranges, and then use those values to build and analyse box plots. This article covers every essential concept, step‑by‑step method and common exam question type for this topic.
累积频率图和箱形图是功能强大的统计工具，能帮助我们概括大量数据、确定中位数和四分位数等关键值，并对不同分布进行有意义的比较。在 IGCSE Edexcel 数学课程中，你需要会构建累积频率表、绘制平滑的累积频率曲线、通过图像读出中位数和四分位距，并利用这些数值绘制和分析箱形图。本文全面覆盖这一主题的每一个核心概念、分步解题方法以及常见考试题型。
1. What is Cumulative Frequency? | 什么是累积频率？
Cumulative frequency is the running total of frequencies as you move through a data set arranged in ascending order. For grouped data, you add each class frequency to the sum of all previous frequencies. This running total tells you how many data values lie at or below the upper boundary of a particular class interval.
累积频率是指在按升序排列的数据集中，逐步累加得到的频率总和。对于分组数据，你将每个组距的频率与之前所有频率的和相加。这个累积总数告诉你位于某一组距上界及以下的数据值共有多少个。
To construct a cumulative frequency table, start with the first class interval. Its cumulative frequency is simply its frequency. For the next row, add the frequency of that class to the previous cumulative frequency. Continue until you reach the total frequency, which should equal the sum of all individual frequencies.
要构建累积频率表，从第一个组距开始，其累积频率就是该组的频率。对于下一行，将该组的频率与上一个累积频率相加。一直继续，直到最后一个累积频率等于总频率，即所有单独频率的总和。
2. Constructing a Cumulative Frequency Table | 构建累积频率表
Imagine we have the following grouped data showing the time (in minutes) 80 students spent on a test:
假设我们有如下分组数据，显示80名学生在一次测验中所用的时间（分钟）：
- 0 < t ≤ 10 : 5 students
- 10 < t ≤ 20 : 12 students
- 20 < t ≤ 30 : 25 students
- 30 < t ≤ 40 : 20 students
- 40 < t ≤ 50 : 13 students
- 50 < t ≤ 60 : 5 students
20 < t ≤ 30 we add 25 to get 42, and so on, until the final cumulative frequency is 80.
我们创建一个包含组距、频率和累积频率列的表格。0 < t ≤ 10 的累积频率是5；10 < t ≤ 20 加上12得到17；20 < t ≤ 30 加上25得到42，依此类推，最终累积频率为80。
| Interval | Frequency | Cumulative Frequency |
| 0 < t ≤ 10 | 5 | 5 |
| 10 < t ≤ 20 | 12 | 17 |
| 20 < t ≤ 30 | 25 | 42 |
| 30 < t ≤ 40 | 20 | 62 |
| 40 < t ≤ 50 | 13 | 75 |
| 50 < t ≤ 60 | 5 | 80 |
Always plot cumulative frequency against the upper class boundary, not the midpoint. Here the plotted points are (10, 5), (20, 17), (30, 42), (40, 62), (50, 75), (60, 80). We also include the point (0, 0) because no data falls below the lower boundary of the first interval.
绘制累积频率图时，始终用组距的上界而不是中点来描点。此处绘制的点坐标为 (10, 5), (20, 17), (30, 42), (40, 62), (50, 75), (60, 80)。我们还需要包含 (0,0) 点，因为没有数据位于第一个组距下界以下。
3. Drawing a Cumulative Frequency Curve | 绘制累积频率曲线
On graph paper, label the horizontal axis with the variable and the vertical axis with 'Cumulative Frequency', usually scaled from 0 to the total frequency. Plot the (upper boundary, cumulative frequency) points exactly, then join them with a smooth, increasing curve. It should never dip downwards and must pass through every plotted point neatly. Use a French curve or a steady hand; sharp corners are usually unacceptable unless there is a clear break in data.
在坐标纸上，横轴标记变量，纵轴标记为“累积频率”，通常刻度从0到总频率。精确描出（上界，累积频率）各点，然后用一条平滑上升的曲线连接。曲线绝不能向下凹陷，必须平顺地通过每一个描点。使用曲线尺或手绘图时，应避免尖角，除非数据有明确断档。
A well‑drawn cumulative frequency curve has a characteristic stretched S‑shape (ogive) when the data is normally distributed. The curve starts shallow, becomes steeper around the densest part of the distribution, and then flattens again. For exam success, always use a sharp pencil, make points clearly visible with small crosses, and avoid joining points with straight line segments unless the question specifically asks for a cumulative frequency polygon.
当数据呈正态分布时，理想的累积频率曲线呈现特征的拉长S形（拱形曲线）。曲线起始平缓，在数据最密集的区域变陡，然后再次趋于平缓。考试中要使用尖头铅笔，用小的十字叉清晰标示各点，除非题目明确要求绘制累积频率多边形，否则不要用直线段连接各点。
4. Finding the Median from the Curve | 根据曲线求中位数
s is the median. For our data, the horizontal line meets the curve at about t = 28.5, so the median time is approximately 28.5 minutes.
中位数是将数据分成两等份的值。在累积频率曲线上，从纵轴上找到总频率的一半（80 ÷ 2 = 40）。由该点画一条水平线与曲线相交，再从交点向下画竖直线至横轴。横轴上的读数即为中位数。对于我们的数据，水平线与曲线相交于约 t = 28.5，因此中位数时间约为28.5分钟。
If the total frequency is odd, the position of the median is (n + 1)/2, but for large grouped data sets we normally still use n/2 as the position to read from the smooth curve. The difference is negligible and standard IGCSE marking allows the use of n/2 when drawing a curve.
如果总频率为奇数，中位数的位置是 (n+1)/2，但对于大型分组数据集，我们通常仍使用 n/2 作为从光滑曲线上读取的位置。差异微乎其微，IGCSE 标准评分允许在绘制曲线时使用 n/2。
5. Locating Quartiles and the Interquartile Range | 确定四分位数与四分位距
requency = 20 on the cumulative frequency axis. Draw a line across to the curve, then down – ₁ ≈ 20.5 minutes. The upper quartile (Q₃) is at (3/4) × 80 = 60, giving Q₃ ≈ 38.5 minutes.
下四分位数 (Q₁) 是穿过数据的四分之一处的值。在累积频率轴上找到位置 (1/4) × 总频率 = 20。画水平线与曲线相交，再向下读取即为 Q₁。根据我们的图像，Q₁ ≈ 20.5 分钟。上四分位数 (Q₃) 的位置在 (3/4) × 80 = 60，得 Q₃ ≈ 38.5 分钟。
nterquartile range (IQR) is the difference between the upper and lower quartiles: IQR = Q₃ – Q₁. Here IQR = 38.5 – 20.5 = 18.0 minutes. The IQR measures the spread of the middle 50% of data; it is a useful measure because it is unaffected by extreme values or outliers.
四分位距 (IQR) 是上四分位数与下四分位数的差：IQR = Q₃ – Q₁。此处 IQR = 38.5 – 20.5 = 18.0 分钟。IQR 衡量中间 50% 数据的分布范围，是一个有用的度量，因为它不受极端值或异常值的影响。
6. Constructing a Box Plot (Box‑and‑Whisker Diagram) | 绘制箱形图（盒须图）
A box plot uses five key values: the minimum value, lower quartile (Q₁), median, upper quartile (Q₃), and maximum value. These are often called the five‑number summary. To draw a box plot, you need a scaled horizontal axis below the box plot. Draw a rectangular box from Q₁ to Q₃, with a vertical line inside the box at the median. Then draw horizontal lines (whiskers) from the left side of the box to the minimum value, and from the right side of the box to the maximum value.
箱形图使用五个关键值：最小值、下四分位数 (Q₁)、中位数、上四分位数 (Q₃) 和最大值。这些值常被称为五数概括。要绘制箱形图，需要在图下方画出带刻度的横轴。从 Q₁ 到 Q₃ 画一个长方形盒子，在盒内中位数位置画一条竖线。然后从盒子的左侧画水平线（须）到最小值，从盒子右侧画水平线到最大值。
our example, suppose the minimum time is 3 minutes and the maximum is 58 minutes. The five‑ x stretches from 20.5 to 38.5, with the median line at 28.5. Whiskers extend to 3 and 58. Always label the axis and give the box plot a title.
对于我们这个例子，假设最小时间为3分钟，最大时间为58分钟。五数概括为：最小值 = 3，Q₁ = 20.5，中位数 = 28.5，Q₃ = 38.5，最大值 = 58。盒子的范围从20.5到38.5，中位数线在28.5处。须延伸到3和58。务必给坐标轴加上标签，并为箱形图写上标题。
7. Interpreting Box Plots – Skewness and Spread | 解读箱形图 – 偏度与分布范围
The shape of a box plot reveals the symmetry or skewness of the data distribution. If the median line lies approximately in the centre of the box, and the whiskers are roughly equal in length, the data is roughly symmetric. If the median is closer to Q₁, or the right whisker is longer than the left, the data is positively skewed (skewed to the right). If the median is closer to Q₃ or the left whisker is longer, the data is negatively skewed.
箱形图的形状可以揭示数据分布的对称性或偏度。如果中位数线大致位于盒子中间，且两条须的长度大致相等，则数据大致对称。如果中位数更靠近 Q₁，或者右侧须比左侧须长，数据呈正偏态（右偏）。如果中位数更靠近 Q₃ 或左侧须更长，数据呈负偏态。
The width of the box (the IQR) indicates the spread of the central half of the data. A narrow box implies that the middle 50% of values are tightly clustered around the median, while a wide box suggests greater variability. By placing two box plots on the same scale, you can easily compare the central tendency and spread of two different data sets.
盒子的宽度（即 IQR）反映中间一半数据的分布广度。窄盒子表示中间 50% 的数值紧密聚集在中位数周围，宽盒子则表示变异性较大。将两个箱形图放在同一标尺上，你可以轻松比较两组数据的集中趋势和离散程度。
8. Comparing Distributions Using Box Plots | 使用箱形图比较分布
When asked to compare two box plots, always comment on an average (median) and a measure of spread (IQR or range). For example: 'The median of data set A is higher than that of data set B, suggesting that on average, values in A are larger.' Then add: 'The IQR of A is 12 while that of B is 20, so the middle 50% of B is more spread out than A.' Mention skewness if a clear difference is visible.
当被要求比较两个箱形图时，务必对平均值（中位数）和离散程度（IQR 或极差）进行评述。例如：“数据集 A 的中位数高于数据集 B，表明平均而言 A 的值更大。”然后补充：“A 的 IQR 为12，而 B 的 IQR 为20，因此 B 中间 50% 的数据比 A 更分散。”如果出现明显的偏态差异，也要提及。
Never just list numbers; always interpret what the statistics tell you about the real‑world situation. If the data are test scores, a higher median and smaller IQR might indicate better performance and more consistency. Use comparative phrases such as 'is higher/lower than', 'more/less consistent', and 'greater/smaller variability'.
切勿只罗列数字；一定要解释这些统计量所反映的真实情况。如果数据是测验成绩，中位数更高且 IQR 更小可能表明表现更好且更稳定。使用比较性的短语，如“高于/低于”、“更一致/更不一致”和“变异性更大/更小”。
9. Dealing with Outliers and Extreme Values | 处理异常值和极端值
e outliers as values more than 1.5 × IQR below Q₁ or above Q₃. These boundaries are called 'fences' . Any data point outside the fences is marked with a dot or asterisk, and the whisker is drawn to the most extreme non‑outlier value.
异常值是指与其他数值距离异常远的数据点。在箱形图中，我们常将小于 Q₁ – 1.5×IQR 或大于 Q₃ + 1.5×IQR 的值定义为异常值。这些界线称为“围栏”。任何落在围栏外的数据点都用点或星号标出，然后将须画到最极端的非异常值处。
In IGCSE Edexcel, you are rarely required to calculate fences, but you may be asked to identify an outlier from a box plot where it is already shown with a special symbol. Knowing that outliers can indicate experimental error or genuinely unusual results helps you comment on the reliability of data.
在 IGCSE Edexcel 中，很少要求你计算围栏，但可能要求你从已用特殊符号标示了异常值的箱形图中识别出异常点。了解异常值可能表示实验误差或真正不寻常的结果，有助于你评论数据的可靠性。
10. Constructing a Cumulative Frequency Table from a Frequency Table with Gaps | 根据含间断的频率表构建累积频率表
Sometimes class intervals have gaps, such as 1–10, 11–20, 21–30, etc. In such cases, the upper boundary for plotting is 10.5, 20.5, 30.5 to account for the continuous scale. Always look carefully at how the data is grouped: if it is discrete and recorded to the nearest whole number, the upper boundary is halfway between the upper limit of one class and the lower limit of the next.
有时组距之间存在间断，例如 1–10, 11–20, 21–30 等。这种情况下，绘图用的上界是 10.5, 20.5, 30.5，以反映连续尺度。一定要仔细察看数据的分组方式：如果数据是离散的并记录到最近的整数，那么上界就是某一组上限与下一组下限之间的半途值。
The cumulative frequency is always zero at the start of the first class. If the first class starts at 1, the lower boundary is 0.5, and the point (0.5, 0) is plotted. Some students forget this point, which can affect the shape of the curve and the reading of the median. Plotting (0,0) is only correct if the lower boundary of the first interval is exactly 0.
累积频率在第一个组距开始时总是为零。如果第一个组距从1开始，下界为0.5，则描点 (0.5, 0)。有些学生忘记这一点，这会影响曲线的形状和中位数的读取。只有当第一个组距的下界恰好为0时，(0,0) 点才是正确的。
11. Common Exam Mistakes and How to Avoid Them | 常见考试错误及如何避免
One frequent mistake is plotting cumulative frequency against the midpoint instead of the upper boundary. This produces a curve that gives incorrect values for median and quartiles. Always double‑check that your x‑coordinate is the end of the interval. Another error is drawing a bar chart instead of a smooth curve – cumulative frequency diagrams are always line graphs.
一个常见错误是将累积频率的中点而非上界用来描点。这样做出的曲线给出的中位数和四分位数不正确。务必再次检查你的 x 坐标是否为组距的终点。另一个错误是画成条形图而不是光滑曲线——累积频率图始终是折线图（曲线图）。
Avoid calculating the IQR by simply subtracting the positions on the vertical axis; you must read the corresponding values on the horizontal axis. Also, when drawing a box plot, ensure the whiskers are drawn horizontally and the box is clearly marked. Ensure the scale of the box plot is accurate; a common pitfall is a squeezed scale that distorts the representation.
避免只通过纵轴位置相减来计算 IQR；你必须读取横轴上的相应值。此外，绘制箱形图时，确保须是水平画出，盒子标示清楚。确保箱形图的比例准确；一个常见的陷阱是刻度压缩导致图形呈现失真。
12. Summary and Key Takeaways for Exam Success | 要总结与考试成功要点
Cumulative frequency graphs and box plots are linked: the graph provides the quartiles needed for the box plot. Mastering these skills demands careful table construction, accurate point plotting, smooth curve drawing, and precise interpretation of statistical measures. Practice with real exam questions under timed conditions, paying close attention to the word 'compare' – which always requires both a measure of central tendency and a measure of spread.
累积频率图和箱形图是相互关联的：累积频率图提供绘制箱形图所需的四分位数。掌握这些技能需要仔细构建表格、准确描点、绘制光滑曲线，并能精确解读统计测量值。在限时条件下练习真题，对“比较”一词要特别留意——它始终要求同时提及集中趋势的测量值和离散程度的测量值。
always plot against upper boundaries; median = n/2 reading; Q₁ = n/4; Q₃ = 3n/4; IQR = Q₃ – this topic becomes one of the most straightforward and rewarding parts of the IGCSE paper.
记住：累积频率始终是累加和；始终依据上界描点；中位数 = n/2 的读数；Q₁ = n/4；Q₃ = 3n/4；IQR = Q₃ – Q₁；箱形图概括五个关键值。通过持续练习，这一主题将成为 IGCSE 试卷中最直接、最易得分的部分之一。
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