IGCSE CIE Maths: Statistics Key Points Review | IGCSE CIE 数学：统计考点精讲

📚 IGCSE CIE Maths: Statistics Key Points Review | IGCSE CIE 数学：统计考点精讲

Statistics is one of the most practical topics in the IGCSE CIE Mathematics syllabus, combining data handling, interpretation, and basic probability. This article provides a thorough review of the essential statistical concepts and techniques you need to master for both the Core and Extended papers. We break down each key area with clear explanations, worked examples, and the kind of notation you will see in real exam questions. By the end of this guide, you should feel confident when reading data tables, drawing cumulative frequency curves, finding quartiles, and calculating probabilities from experiments.

统计是 IGCSE CIE 数学大纲中最实用的课题之一，它结合了数据处理、图表解读和基础概率。本文全面回顾了在核心和扩展考试中都必须掌握的重要统计概念与技巧。我们逐一拆解每个关键领域，配以清晰的解释、实例以及你在真实考题中会遇到的表示方法。阅读完本指南后，你将能自信地阅读数据表、绘制累积频数曲线、求四分位数并根据实验计算概率。

1. Mean, Median, Mode and Range | 平均数、中位数、众数和范围

The mean is calculated by summing all data values and dividing by the number of values. For grouped data, we use midpoints of intervals and multiply by frequencies: mean = Σ(f × x) / Σf. The median is the middle value when data are ordered; for a frequency table, use cumulative frequency to locate the position (n+1)/2 for raw data, or n/2 for grouped data. The mode is the most frequently occurring value or class interval. The range = largest value – smallest value, measuring spread.

平均数的计算方式是将所有数据值相加再除以数据个数。对于分组数据，我们使用组中点乘以频数：平均数 = Σ(f × x) / Σf。中位数是将数据排序后的中间值；对频数表，利用累积频数确定位置，原始数据用 (n+1)/2，分组数据用 n/2。众数是出现次数最多的数值或组区间。范围 = 最大值 – 最小值，用于衡量分散程度。

Example: Data: 3, 7, 7, 8, 10 → Mean = (3+7+7+8+10)/5 = 7, Median = 7, Mode = 7, Range = 10 – 3 = 7.
示例：数据：3, 7, 7, 8, 10 → 平均数 = 7，中位数 = 7，众数 = 7，范围 = 7。

2. Quartiles and Percentiles | 四分位数与百分位数

The lower quartile (Q₁) is the median of the lower half of the data; the upper quartile (Q₃) is the median of the upper half. The interquartile range (IQR) = Q₃ – Q₁, which measures the spread of the middle 50% of the data. Percentiles divide the data into 100 equal parts; the kth percentile is the value below which k% of the data fall. For raw data, find the position (k/100) × (n+1) for small sets, or use cumulative frequency curves for grouped data.

下四分位数 (Q₁) 是数据下半部分的中位数；上四分位数 (Q₃) 是数据上半部分的中位数。四分位距 (IQR) = Q₃ – Q₁，衡量中间 50% 数据的分散程度。百分位数将数据分成 100 等份；第 k 个百分位数是指有 k% 的数据低于该值。对于原始数据，小数据集用位置 (k/100) × (n+1)，分组数据则需借助累积频数曲线。

Ordered data: 2, 4, 5, 6, 8, 9, 11

n=7, Q₁ position = (7+1)/4 = 2 → Q₁ = 4

Q₃ position = 3(7+1)/4 = 6 → Q₃ = 9

IQR = 9 – 4 = 5

3. Box-and-Whisker Plots | 箱线图

A box plot (or box-and-whisker diagram) displays the five‑number summary: minimum, Q₁, median, Q₃, maximum. The box spans from Q₁ to Q₃ with a line at the median; whiskers extend to the min and max (or to 1.5 × IQR beyond the quartiles for outlier identification). Box plots are excellent for comparing distributions side by side and for spotting skewness.

箱线图展示五数概括：最小值、Q₁、中位数、Q₃、最大值。箱体从 Q₁ 延伸到 Q₃，中间标示中位数；触须延伸至最小值与最大值（或至距离四分位数 1.5 倍 IQR 处用于标记异常值）。箱线图非常适用于并排比较分布情况以及识别偏态。

If median is closer to Q₁, the data is positively skewed (skewed to the right).
如果中位数更靠近 Q₁，数据呈正偏态（右偏）。
If median is closer to Q₃, the data is negatively skewed (skewed to the left).
如果中位数更靠近 Q₃，数据呈负偏态（左偏）。

4. Histograms with Unequal Class Widths | 不等宽直方图

In a histogram, the area of each bar is proportional to the frequency. When class widths are unequal, you must calculate frequency density: frequency density = frequency / class width. The vertical axis represents frequency density, not frequency. This ensures that the area of a bar accurately reflects the number of observations in that interval. Always check the class boundaries and use the exact widths.

在直方图中，每个条形的面积与频数成正比。当组距不相等时，必须计算频数密度：频数密度 = 频数 / 组距。纵轴表示频数密度而非频数。这确保条形的面积准确反映该区间的观测值数量。务必检查组界并使用精确的组距。

Frequency density = Frequency ÷ Class width

For example, a class 10 ≤ x < 20 has width 10; if its frequency is 25, frequency density = 25/10 = 2.5. On the histogram, draw the bar from 10 to 20 on the horizontal axis with a height of 2.5 on the vertical axis.

例如，组 10 ≤ x < 20 的组距为 10；若其频数为 25，频数密度 = 25/10 = 2.5。在直方图上，水平轴上从 10 到 20 绘制条形，纵轴上高度为 2.5。

5. Frequency Polygons | 频数折线图

A frequency polygon is drawn by plotting the frequencies at the midpoints of each class interval and joining the points with straight lines. It often begins and ends on the horizontal axis at the midpoints of the adjacent empty intervals. Frequency polygons are useful for comparing two sets of data on the same axes because overlapping outlines are easy to distinguish.

频数折线图通过将各组的组中点上的频数标出并用直线连接各点来绘制。它通常始于和结束于相邻空组的组中点所在的水平轴。频数折线图便于在同一坐标轴上比较两组数据，因为重叠的轮廓线容易区分。

Midpoint = (lower bound + upper bound) / 2
组中点 = (下限 + 上限) / 2
Always plot against the midpoint, not the class boundaries.
总是对组中点作图，而非组界。

6. Cumulative Frequency Graphs | 累积频数曲线

A cumulative frequency table adds up frequencies as you move from the smallest class to the largest. The cumulative frequency is plotted against the upper class boundary of each interval. Join the points with a smooth curve (not a series of straight lines). The graph can then be used to estimate medians, quartiles, and percentiles. For the median, find the value at 50% of the total frequency; for Q₁, use 25%; for Q₃, use 75%.

累积频数表在从最小类向最大类推进时将频数累加。累积频数绘制在每组的组上界上。用平滑曲线连接各点（而非折线）。然后可利用该图估算中位数、四分位数和百分位数。中位数取总频数 50% 对应的值；Q₁ 取 25%；Q₃ 取 75%。

Total frequency N → median at N/2, Q₁ at N/4, Q₃ at 3N/4.
总频数 N → 中位数在 N/2，Q₁ 在 N/4，Q₃ 在 3N/4。

7. Interpreting Statistical Diagrams | 统计图表的解读

Exam questions often require you to read information from bar charts, pie charts, stem-and-leaf diagrams, and back-to-back stem-and-leaf diagrams. Key skills include finding the mode from bar charts (tallest bar), reading percentages from pie charts (angle ÷ 360° × 100), and identifying the median, range, and distribution shape from stem-and-leaf plots. A back-to-back stem-and-leaf allows comparison of two data sets sharing the same stem.

试题通常要求你从条形图、饼图、茎叶图和背靠背茎叶图中提取信息。关键技能包括：从条形图中查找众数（最高的条形），从饼图中读取百分比（角度 ÷ 360° × 100），以及从茎叶图中识别中位数、范围和分布形状。背靠背茎叶图可比较共享同一茎的两组数据。

In a stem-and-leaf, the ‘stem’ runs down the middle, ‘leaves’ go left for one set, right for the other.
在背靠背茎叶图中，“茎”位于中央，“叶子”一组向左，一组向右。

8. Basic Probability Concepts | 基本概率概念

Probability is a measure of how likely an event is to occur, given by: P(event) = number of favourable outcomes / total number of equally likely outcomes. Probability values lie between 0 and 1, inclusive. The sum of probabilities of all possible outcomes of an experiment is 1. The probability that an event does NOT occur is 1 – P(event).

概率是衡量事件发生可能性的指标，公式为：P(事件) = 有利结果数 / 等可能结果总数。概率值介于 0 到 1 之间。一个实验所有可能结果的概率之和为 1。事件不发生的概率为 1 – P(事件)。

Rolling a fair six-sided die: P(rolling a 4) = 1/6.
投掷一枚均匀六面骰子：P(掷出 4) = 1/6。
P(not rolling a 4) = 5/6.
P(未掷出 4) = 5/6。

9. Probability from Experiments and Expected Frequency | 实验概率与期望频数

When outcomes are not equally likely, we use experimental probability (relative frequency): relative frequency = number of times event occurred / total number of trials. Expected frequency predicts how often an event would happen in a given number of trials using theoretical probability: expected frequency = probability × number of trials. The more trials, the closer the relative frequency tends to be to the theoretical probability.

当结果不等可能时，我们使用实验概率（相对频率）：相对频率 = 事件发生次数 / 试验总次数。期望频数利用理论概率预测在给定试验次数中事件发生的次数：期望频数 = 概率 × 试验次数。试验次数越多，相对频率越趋近理论概率。

Coin tossed 200 times, heads obtained 95 times: relative frequency = 95/200 = 0.475.
掷硬币 200 次，正面 95 次：相对频率 = 95/200 = 0.475。
Expected heads in 500 tosses = 0.5 × 500 = 250.
掷 500 次的期望正面数 = 0.5 × 500 = 250。

10. Scatter Graphs and Correlation | 散点图与相关性

A scatter graph shows the relationship between two variables. If points tend to slope upwards from left to right, the correlation is positive. If they slope downwards, the correlation is negative. Correlation can be strong or weak, and it is described as positive, negative, or no correlation. Remember that correlation does not imply causation.

散点图展示两个变量之间的关系。如果点从左到右大致向上倾斜，则相关性为正。如果向下倾斜，则为负相关。相关性可以有强有弱，并可描述为正、负或无相关。记住，相关性并不意味着因果关系。

Strong positive: points lie close to a straight upward line.
强正相关：点紧贴一条上升直线。
Weak negative: points loosely spread around a downward slope.
弱负相关：点松散分布在下降趋势周围。

11. Line of Best Fit and Predictions | 最佳拟合线与预测

When a scatter graph shows correlation, you can draw a line of best fit that passes as close to as many points as possible, with roughly equal numbers of points above and below the line. This line is used to make predictions: reading a value from the line within the range of existing data is interpolation (reliable); reading outside is extrapolation (unreliable). Always answer within the context of the data, and if the correlation is zero, do not draw a line of best fit.

当散点图显示相关关系时，你可以绘制一条最佳拟合线，使其尽可能接近多数点，线上方和线下方的点数大致相等。这条线用于预测：在现有数据范围内从线上读取数值为内插（可靠）；在范围外读取则为外推（不可靠）。始终在数据背景下作答，如果相关为零，则不要绘制最佳拟合线。

Line should follow the trend; avoid forcing through the origin unless justified.
线条应跟随趋势；除非有理由，否则不要强制通过原点。

12. Sampling Methods | 抽样方法

In statistics, a population is the whole group of interest; a sample is a subset used to make inferences about the population. Common sampling techniques at IGCSE include: random sampling (every member has equal chance), stratified sampling (population divided into strata and a random sample taken proportionally from each), and systematic sampling (selecting every kth member after a random start). Bias must be avoided to ensure the sample is representative. A questionnaire should use clear, unbiased questions with appropriate response boxes.

在统计学中，总体是我们感兴趣的整个群体；样本是用来推断总体特征的一个子集。IGCSE 阶段常见抽样方法包括：随机抽样（每个成员被选中的机会相等）、分层抽样（总体分为若干层，从每层中按比例随机抽取样本）以及系统抽样（随机起点后每隔 k 个成员选取）。必须避免偏差以确保样本具有代表性。问卷应使用清晰、无偏见的问题并配有合适的回答选项。

Stratified sample formula: Number from stratum = (stratum size / population size) × sample size

分层抽样公式：各层抽取数 = (层大小 / 总体大小) × 样本大小

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IGCSE CIE Maths: Statistics Key Points Review | IGCSE CIE 数学：统计考点精讲