📚 Year 10 Cambridge Statistics: Core Knowledge Review | Year 10 剑桥统计：核心知识点梳理
Mastering statistics at the Year 10 Cambridge level means building a solid foundation in handling data — from collecting and presenting it to drawing meaningful conclusions. This article walks you through every essential topic, explaining concepts clearly so you can tackle IGCSE questions with confidence. Let's dive into the core knowledge you must have.
掌握 Year 10 剑桥统计意味着在数据处理方面打下坚实基础——从收集、呈现数据到得出有意义的结论。本文将带你梳理每一个核心主题，清晰讲解概念，让你自信应对 IGCSE 题目。让我们一起梳理这些必备知识。
1. Types of Data | 数据类型
Data can be classified as qualitative or quantitative. Qualitative data describes categories or qualities, such as eye colour or type of vehicle. Quantitative data involves numbers and can be either discrete or continuous. Discrete data can only take specific values, often counted in whole numbers — for example, the number of students in a class. Continuous data can take any value within a range and is usually measured, such as height or time.
数据可以分为定性数据和定量数据。定性数据描述类别或性质，例如眼睛的颜色或车辆类型。定量数据涉及数字，可以是离散数据或连续数据。离散数据只能取特定的值，通常以整数计数——例如一个班级的学生人数。连续数据可以在一个范围内取任意值，通常需要通过测量得到，比如身高或时间。
2. Collecting Data and Sampling | 数据收集与抽样
Reliable conclusions start with good data collection. You need to know the difference between primary and secondary data. Primary data is collected firsthand by the investigator (e.g., a survey), while secondary data uses existing sources (e.g., government statistics). When you can't study an entire population, you take a sample. A random sample gives every member an equal chance of being chosen, avoiding bias. Stratified sampling divides the population into distinct groups and takes a proportional random sample from each, ensuring representation.
可靠的结论始于良好的数据收集。你需要了解原始数据和二手数据的区别。原始数据由研究者亲自收集（如问卷调查），而二手数据使用现有来源（如政府统计数据）。当无法研究整个总体时，就需要进行抽样。随机抽样使每个成员被选中的概率相等，从而避免偏差。分层抽样将总体分成不同的组，并从每组中按比例随机抽取样本，确保代表性。
3. Frequency Tables and Charts | 频率表与图表
A frequency table is the simplest way to organise data, listing each data value alongside its frequency. For grouped data, we organise values into class intervals. From a frequency table you can quickly identify the mode — the value with the highest frequency. Tally charts help you count raw data accurately before filling in frequencies.
频率表是整理数据最简单的方法，它列出每个数据值及其对应的频数。对于分组数据，我们将数值划分到组距中。通过频率表你可以快速锁定众数——即出现频数最高的值。计数图表可以帮助你在填写频数之前准确地统计原始数据。
4. Bar Charts and Pie Charts | 条形图与饼图
Bar charts are used to display categorical or discrete data with bars of equal width; the height of each bar represents the frequency. In a bar chart, bars do not touch each other, emphasising that categories are separate. Pie charts show proportions of a whole; each sector angle is calculated using the formula:
条形图用于展示类别数据或离散数据，柱子宽度相等，每根柱子的高度代表频数。在条形图中，柱子之间不接触，突出类别是分离的。饼图则展示各部分占整体的比例；每个扇形的圆心角可以通过公式计算：
Sector angle = (Frequency / Total frequency) × 360°
扇形角度 = (频数 ÷ 总频数) × 360°
Always check that the sum of all sector angles equals 360°.
务必检查所有扇形角度之和等于360°。
5. Histograms and Frequency Density | 直方图与频率密度
A histogram looks similar to a bar chart but is used for continuous grouped data. Crucially, the bars touch each other, and the area of each bar is proportional to the frequency. When class intervals are unequal, you must use frequency density to plot the vertical axis:
直方图看起来与条形图相似，但用于连续分组数据。关键是，柱子彼此接触，且每个柱子的面积与频数成正比。当组距不相等时，必须使用频率密度来绘制纵轴：
Frequency density = Frequency ÷ Class width
频率密度 = 频数 ÷ 组距
Remember: in a histogram, it's the area of the bar that tells you the frequency, not just the height.
记住：在直方图中，柱子的面积表示频数，而不仅仅是高度。
6. Cumulative Frequency Graphs | 累积频率图
A cumulative frequency graph shows the running total of frequencies up to each upper class boundary. To construct it, add a cumulative frequency column to your grouped table and plot points at the upper boundary of each interval. Join the points with a smooth curve. From this graph you can find the median, lower quartile (Q₁) and upper quartile (Q₃) by reading across from the 50th, 25th and 75th percentiles of the total cumulative frequency.
累积频率图显示截至每个组上边界的累计频数。构建时，在分组表中添加累积频率列，并在每个区间的上边界处描点，然后用平滑曲线连接。通过该图你可以找到中位数、下四分位数 (Q₁) 和上四分位数 (Q₃)，只需从总累积频数的第50、第25和第75百分位数位置水平读取即可。
7. Averages: Mean, Median, Mode | 平均数：平均值、中位数、众数
The three measures of central tendency each summarise data differently. The mean is calculated by adding all values and dividing by how many there are. For grouped data, use the midpoint of each interval multiplied by its frequency:
三种集中趋势度量以不同方式概括数据。平均值是将所有数值相加后除以数据个数。对于分组数据，使用每个区间的中点值乘以其频数：
Mean ≈ Σ(midpoint × frequency) / Σ frequency
平均值 ≈ Σ(中点 × 频数) / Σ 频数
The median is the middle value when data are ordered; if the number of values n is even, the median is the mean of the two middle values. The mode is the most frequent value. Each average has strengths — the mean uses all data, the median is unaffected by outliers, and the mode shows the most typical case.
中位数是将数据排序后的中间值；若数据个数 n 为偶数，则中位数为中间两数的平均值。众数是出现次数最多的值。每种平均数都有各自的特点——平均值使用了全部数据，中位数不受异常值影响，众数则反映最常见的情况。
8. Range and Interquartile Range | 极差与四分位距
Spread is just as important as an average. The range is a simple measure of dispersion:
离散程度与平均数同等重要。极差是一种简单的离散度量：
Range = Highest value – Lowest value
极差 = 最大值 – 最小值
However, the range can be distorted by outliers. A more robust measure is the interquartile range (IQR):
然而，极差可能被异常值扭曲。更稳健的度量是四分位距 (IQR)：
IQR = Upper quartile (Q₃) – Lower quartile (Q₁)
IQR = 上四分位数 (Q₃) – 下四分位数 (Q₁)
The IQR captures the middle 50% of the data, making it less sensitive to extreme values.
IQR 涵盖了中间 50% 的数据，因此对极端值不那么敏感。
9. Box Plots | 箱线图
epresents the interquartile range, with the median line shown inside. Whiskers extend from the box to the minimum and maximum, provided there are no outliers. Outliers may be plotted as separate points if they lie more than 1.5 × IQR beyond the quartiles. Box plots are excellent for comparing distributions between two or more data sets at a glance.
箱线图（或箱须图）展示了五数概括：最小值、下四分位数、中位数、上四分位数和最大值。矩形盒子代表四分位距，内部标有中位数线。若无异常值，须线从盒子两端延伸至最小值和最大值。如果数据点超过四分位数1.5 × IQR的范围，则作为异常值单独标出。箱线图非常适合快速比较两组或多组数据的分布。
10. Scatter Graphs and Correlation | 散点图与相关性
Scatter graphs help visualise the relationship between two variables. When points follow an upward trend, correlation is positive; a downward trend indicates negative correlation. If no pattern is visible, there is no correlation. Correlation does not imply causation. You should be able to draw a line of best fit, which can be used to make predictions. A strong correlation means points cluster closely around the line of best fit, whereas a weak correlation shows greater scatter.
散点图有助于可视化两个变量之间的关系。点呈现上升趋势时，是正相关；下降趋势表明负相关。如果没有可见的模式，则是不相关。相关关系不代表因果关系。你需要会画最佳拟合线，并用它进行预测。强相关意味着点紧密聚集在最佳拟合线附近，而弱相关则表现出较大的离散性。
11. Basic Probability | 基础概率
Probability measures how likely an event is, expressed as a number between 0 and 1 or as a fraction, decimal or percentage. The probability of an event A is:
概率衡量事件发生的可能性，用0到1之间的数字表示，也可以是分数、小数或百分比。事件 A 的概率为：
P(A) = Number of favourable outcomes / Total number of possible outcomes
P(A) = 有利结果数 / 所有可能结果总数
exclusive events, P(A or B) = P(A) + P(B). The probability of something not happening is 1 – learn to calculate expected frequency by multiplying probability by the number of trials.
对于互斥事件，P(A 或 B) = P(A) + P(B)。事件不发生的概率为 1 – P(事件)。在 Year 10 你还会使用树状图处理组合事件，并学习用概率乘以试验次数来计算期望频数。
12. Common Mistakes and Exam Tips | 常见错误与考试技巧
When plotting cumulative frequency graphs, remember to use the upper class boundary, not the midpoint. In histograms, check if class widths vary — if they do, frequency density must be used, otherwise the graph is misleading. For averages from grouped data, state that you are calculating an estimate because the exact values are unknown. In probability questions, ensure fractions are simplified and tree diagram branches are correctly labelled with probabilities summing to 1. Always show your working clearly, especially when reading values from graphs; draw guide lines on the diagram.
绘制累积频率图时，记住使用上组边界，而不是中点。在直方图中，检查组距是否不同——若不同，必须使用频率密度，否则图形会产生误导。对于分组数据的平均数，需说明你计算的是估算值，因为确切的数值未知。在概率题中，确保分数已化简，树状图的分支标上概率且总和为1。答题时始终清晰展示解题步骤，特别是从图中读取数值时，要在图上画出辅助线。
Published by TutorHao | Statistics Revision Series | aleveler.com
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