📚 CIE Year 10 Statistics: Core Concepts Summary | CIE Year 10 统计：核心知识点梳理
Year 10 CIE Statistics builds a foundation in handling data, summarising information, and understanding chance. This article walks through the essential topics you will meet in the syllabus, from types of data to basic regression, with clear explanations and examples to support your revision.
Year 10 CIE 统计课程为数据处理、信息概括和概率理解打下基础。本文将梳理教学大纲中的核心主题，从数据类型到基础回归，配合清晰的解释和示例，帮助你高效复习。
1. Types of Data | 数据类型
Data can be classified as categorical or numerical. Categorical data describe qualities (e.g. eye colour, type of transport) and may be nominal or ordinal. Numerical data record quantities and are either discrete (countable, like number of siblings) or continuous (measurable, like height).
数据可分为类别数据和数值数据。类别数据描述特征（如眼睛颜色、交通方式），可分为名义型或顺序型。数值数据记录数量，可以是离散型（可计数，如兄弟姐妹人数）或连续型（可测量，如身高）。
Understanding data type is crucial because it determines which charts and summary statistics are appropriate. For example, you cannot calculate a meaningful mean for eye colour, but you can for height.
理解数据类型至关重要，因为它决定了适合的图表和汇总统计量。例如，你不能为眼睛颜色计算有意义的平均数，但可以为身高计算。
2. Sampling Methods | 抽样方法
A sample is a subset of a population. Good sampling reduces bias. Random sampling gives every member an equal chance of being chosen, often using random number tables or generators.
样本是总体的一个子集。好的抽样方法能减少偏差。随机抽样使每个成员被选中的机会均等，通常使用随机数表或生成器。
Stratified sampling divides the population into distinct groups (strata) and randomly selects from each in proportion to its size. This ensures key subgroups are represented.
分层抽样将总体划分为不同的组（层），并按各层大小比例随机选取。这保证了关键子群体都能被代表。
Systematic sampling selects every kᵗʰ item from a list after a random start. Quota sampling picks individuals based on pre‑set characteristics until quotas are filled, but it is non‑random and may introduce bias.
系统抽样在随机起点后，从名单中每隔k个选取一个。定额抽样根据预设特征选取个体直到满额，但非随机，可能引入偏差。
3. Charts and Diagrams | 图表与图形
Bar charts display categorical or discrete data using bars of uniform width, with gaps between them. The height of each bar represents frequency.
条形图使用等宽且有间隙的条形展示类别或离散数据。每个条形的高度代表频数。
portions of a whole. The angle of each sector is calculated as (frequency / total) × 360°.
饼图展示各部分在整体中的占比。每个扇形的角度由 (频数 / 总数) × 360° 计算得出。
y. For unequal class widths, frequency density = frequency ÷ class width must be plotted on the vertical axis.
直方图用于分成区间的连续数据。条形之间无间隙，条形的面积与频数成正比。当组距不等时，纵轴须使用频数密度 = 频数 ÷ 组距。
Stem‑and‑leaf diagrams preserve original data while showing distribution. The stem represents the leading digit(s) and the leaf the trailing digit. An ordered stem‑and‑leaf diagram makes finding medians and quartiles easy.
茎叶图保留了原始数据的同时展示分布。茎表示前导数字，叶表示末位数字。有序茎叶图便于求中位数和四分位数。
4. Measures of Central Tendency | 集中趋势度量
The mean (average) for ungrouped data is sum of all values divided by the number of values.
未分组数据的平均数（均值）是所有数值之和除以数值个数。
x̄ = Σx / n
The median is the middle value when data are ordered. For an even number of observations, it is the average of the two middle values. The median is resistant to extreme values.
中位数是数据排序后位于中间的值。如果数据个数为偶数，则取中间两个值的平均数。中位数不受极端值影响。
The mode is the most frequently occurring value. For grouped data, the modal class is the class interval with the highest frequency.
众数是出现次数最多的值。对于分组数据，众数所在组是频数最高的组距。
Choosing the appropriate average depends on the data distribution and the presence of outliers. The mean uses all data but is sensitive to skewness.
选择合适的平均数取决于数据分布和异常值的存在。均值利用了所有数据，但对偏斜敏感。
5. Measures of Dispersion | 离散程度度量
e = maximum value − minimum value. It is simple but ignores the spread of the middle data.
极差 = 最大值 − 最小值。它简单，但忽略了中间数据的离散状况。
₁. Q₁ is the lower quartile (25th percentile) and Q₃ is the upper quartile (75th percentile). IQR describes the spread of the middle 50% of data and is less affected by outliers.
四分位距 (IQR) = Q₃ − Q₁。Q₁ 是下四分位数（第25百分位数），Q₃ 是上四分位数（第75百分位数）。IQR 描述中间50%数据的分散程度，较少受异常值影响。
Standard deviation measures average distance of data points from the mean. For a set of n values,
标准差衡量数据点与均值的平均距离。对于一组n个数值,
s = √( Σ(x − x̄)² / n )
The variance is s². A larger standard deviation indicates greater variability. When working with grouped data, replace x with the midpoint of each interval and multiply squared deviations by frequency.
方差为 s²。标准差越大，变异性越强。处理分组数据时，用每组的组中值代替 x，并将平方偏差乘以频数。
6. Cumulative Frequency and Box Plots | 累积频率与箱线图
A cumulative frequency table adds frequencies step by step. The cumulative frequency curve (ogive) is plotted using upper class boundaries on the x‑axis and cumulative frequency on the y‑axis.
累积频率表逐步累加频数。累积频率曲线（ogive）以组上限为横轴、累积频率为纵轴绘制。
The median, Q₁, Q₃ can be estimated from the curve by locating the positions (n/2, n/4, 3n/4) on the vertical axis and reading off the corresponding values. The inter‑percentile range can also be found for any percentile.
通过从纵轴上 (n/2, n/4, 3n/4) 的位置水平读到曲线再垂直读至横轴，即可估算中位数、Q₁、Q₃。任何百分位距也可类似求得。
A box plot (box‑and‑ l outliers. Outliers are often defined as values more than 1.5 × IQR below Q₁ or above Q₃.
箱线图展示最小值、Q₁、中位数、Q₃ 和最大值。它能反映对称性、偏态和潜在的异常值。异常值通常定义为小于 Q₁ − 1.5×IQR 或大于 Q₃ + 1.5×IQR 的数据点。
7. Basic Probability | 基础概率
Probability measures the likelihood of an event, ranging from 0 (impossible) to 1 (certain). It can be expressed as a fraction, decimal, or percentage.
概率度量事件发生的可能性，范围从0（不可能）到1（确定）。可用分数、小数或百分数表示。
tcomes / total number of outcomes. The sum of probabilities of all possible outcomes is 1.
对于等可能结果，P(事件) = 有利结果数 / 总结果数。所有可能结果的概率之和为 1。
exclusive events, P(A or B) = P(A) + P(B). The complement rule states P(not A) = 1 − P(A).
若两个事件不能同时发生，则它们互斥。对于互斥事件，P(A 或 B) = P(A) + P(B)。互补规则为 P(非 A) = 1 − P(A)。
y exclusive, the addition rule accounts for overlap: P(A or B) = P(A) + P(B) − P(A and B).
当事件不互斥时，加法法则需扣除重叠部分：P(A 或 B) = P(A) + P(B) − P(A 且 B)。
8. Tree Diagrams and Combined Events | 树形图与组合事件
Tree diagrams help visualise sequences of events. Each branch shows an outcome with its probability, and probabilities along a path multiply.
树形图有助于可视化事件序列。每个分支显示一个结果及其概率，沿一条路径的概率相乘。
ependent events, probabilities on the second branch are conditional, such as P(B given A).
若事件独立，P(A 且 B) = P(A) × P(B)。对于相依事件，第二个分支上的概率为条件概率，如 P(B|A)。
Tree diagrams are especially useful when experiments are repeated, like tossing a coin then drawing a card without replacement.
当实验重复进行时（如抛硬币然后不放回抽牌），树形图特别有用。
9. Scatter Diagrams and Correlation | 散点图与相关性
A scatter diagram plots bivariate data with one variable on each axis. It reveals the type and strength of relationship between the two variables.
散点图将双变量数据绘制在坐标轴上，每个轴代表一个变量。它能显示两个变量之间关系的类型和强度。
Positive correlation means as one variable increases, the other tends to increase. Negative correlation means as one increases, the other tends to decrease. No correlation appears as a random cloud of points.
正相关意味着一个变量增加时另一个也往往增加。负相关意味着一个增加时另一个往往减少。无相关表现为散乱无规律的云状点。
The strength of linear correlation is measured by the product‑ ate strong linear association, while values near 0 suggest weak or no linear relationship.
线性相关强度由积矩相关系数 r 度量，其值在 −1 到 +1 之间。接近 ±1 表示强线性关联，接近0 表示弱或无线性的关系。
10. Simple Linear Regression | 简单线性回归
onship. The equation of a straight line is y = a + bx, where b is the slope and a is the y‑intercept.
如果散点图显示出线性相关，可以绘制最佳拟合线来建立关系模型。直线方程为 y = a + bx，其中 b 为斜率，a 为 y轴截距。
The line is usually drawn 'by eye' passing through the mean point (x̄, ȳ). In CIE exams, you may be asked to draw the line, find its equation, and use it for interpolation (predicting within the data range) but avoid extrapolation beyond the range as it may be unreliable.
这条线通常‘目测’绘制，并经过均值点 (x̄, ȳ)。在 CIE 考试中，你可能需要画出该直线、求其方程，并用于内插（在数据范围内预测），但应避免范围外的外推，因为它可能不可靠。
The slope b can be estimated by rise over run from the line, and the intercept a read from the graph. The line can then predict y for a given x or vice versa.
斜率 b 可通过图上竖直增量与水平增量之比估算，截距 a 可直接从图上读取。然后可利用该直线由给定 x 预测 y，或反之。
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