📚 Year 10 CIE Statistics: Quick-reference Formula & Theorem Handbook | CIE Year 10 统计：公式定理速查手册
This quick-reference handbook brings together the essential formulas and theorems for Year 10 CIE Statistics. Use it for rapid revision, exam preparation, and building confidence with data handling, probability, and statistical diagrams.
本速查手册汇集了 Year 10 CIE 统计课程的核心公式与定理。可用于快速复习、备考冲刺，并帮助你更自信地掌握数据处理、概率与统计图表。
1. Measures of Central Tendency | 集中趋势度量
Measures of central tendency help us find a typical or central value in a data set. The three main measures are the mean, median and mode.
集中趋势度量帮助我们找到数据集中的典型值或中心值。三个最主要的度量是平均数、中位数和众数。
Mean for ungrouped data: Mean (x̄) = Σx / n, where Σx is the sum of all values and n is the number of values.
未分组数据的平均数： 平均数 (x̄) = Σx / n，其中 Σx 是所有数值之和，n 是数值的个数。
Mean for grouped data: : Estimated mean = Σ(f × x) / Σf, where f is the frequency of each class and x is the class mid-point.
分组数据的平均数： 估计平均数 = Σ(f × x) / Σf，其中 f 是每组的频数，x 是组中值。
Median for ungrouped data: Arrange the values in order. If n is odd, the median is the (n+1)/2 -th value. If n is even, the median is the average of the n/2 -th and (n/2 + 1)-th values.
未分组数据的中位数： 将数据按顺序排列。若 n 为奇数，中位数为第 (n+1)/2 个数值；若 n 为偶数，中位数为第 n/2 个数值与第 (n/2 +1) 个数值的平均数。
Mode: The value that occurs most frequently. A data set may have no mode, exactly one mode, or more than one mode.
众数： 出现次数最多的数值。一组数据可能没有众数、只有一个众数，或拥有多个众数。
2. Measures of Spread | 离散程度度量
Measures of spread describe how widely scattered the data are. The simplest measures are the range and the interquartile range (IQR).
离散程度度量描述数据的分散情况。最简单的度量是极差和四分位距 (IQR)。
Range: Range = maximum value – minimum value.
极差： 极差 = 最大值 – 最小值。
Interquartile Range (IQR): IQR = upper quartile (Q₃) – lower quartile (Q₁).
四分位距 (IQR)： IQR = 上四分位数 (Q₃) – 下四分位数 (Q₁)。
Variance for a population (σ²): σ² = Σ(x – ean and N is the population size. A more practical formula for calculation is σ² = (Σx²)/n – x̄², where x̄ is the sample mean when applied to a full data set.
总体方差 (σ²)： σ² = Σ(x – μ)² / N，其中 μ 为总体平均数，N 为总体大小。更常用的计算公式是 σ² = (Σx²)/n – x̄²，当处理完整数据集时 x̄ 为样本平均数。
Standard Deviation (σ): σ = √variance. It represents the average distance of each data point from the mean.
标准差 (σ)： σ = √方差。它表示各个数据点与平均数之间的平均距离。
3. Quartiles and Percentiles | 四分位数与百分位数
Quartiles divide an ordered data set into four equal parts. The lower quartile (Q₁), median (Q₂) and upper quartile (Q₃) are used to summarise the data.
四分位数将有序数据集分成四个相等的部分。下四分位数 (Q₁)、中位数 (Q₂) 和上四分位数 (Q₃) 常用于概括数据。
Position of the lower quartile: Q₁ is at the (n+1)/4 -th position for ungrouped data. If the position is not an integer, interpolation or averaging the surrounding values is used.
下四分位数位置： 对于未分组数据，Q₁ 位于第 (n+1)/4 个位置。若位置不是整数，则通过对邻近数值进行插值或取平均值求得。
Position of the upper quartile: Q₃ is at the 3(n+1)/4 -th position. The same interpolation rule applies.
上四分位数位置： Q₃ 位于第 3(n+1)/4 个位置。同样适用插值规则。
Percentiles: The k-th percentile is the value below which k% of the data fall. Its position is given by (k/100) × (n + 1).
百分位数： 第 k 百分位数是指有 k% 的数据落在该值以下。其位置由 (k/100) × (n + 1) 给出。
4. Representing Data (Box Plots and Histograms) | 数据展示（箱线图与直方图）
A box plot uses the five-number summary: minimum, Q₁, median, Q₃ and maximum. It shows the spread and skewness but not detailed shape.
箱线图使用五数汇总：最小值、Q₁、中位数、Q₃ 和最大值。它能显示数据的散布与偏态，但无法展示详细分布形态。
Drawing a box plot: rs to the minimum and maximum values (or to 1.5 × IQR boundaries if outliers are defined).
绘制箱线图： 从 Q₁ 到 Q₃ 绘一个矩形，并在中位数处画一条线。须线延伸至最小值和最大值（若定义异常值则至 1.5 × IQR 的界限）。
Histogram and frequency density: rea of each bar is proportional to frequency. Frequency density = frequency / class width.
直方图与频率密度： 在直方图中，每个长方形的面积与频数成正比。频率密度 = 频数 / 组距。
Constructing a histogram: Use frequency density on the vertical axis and class boundaries on the horizontal axis. The height of each bar represents the frequency density.
构建直方图： 纵轴为频率密度，横轴为组限。每个长方形的高表示频率密度。
5. Cumulative Frequency | 累积频数
A cumulative frequency curve (ogive) is used to estimate the median, quartiles and percentiles from grouped data.
累积频数曲线（肩形图）用于从分组数据中估计中位数、四分位数和百分位数。
Cumulative frequency table: Add up the frequencies successively. Plot the cumulative frequency against the upper class boundary for each interval.
累积频数表： 将频数依次累加。以各组的上限为横坐标，累积频数为纵坐标描点。
Finding the median from an ogive: Locate the cumulative frequency value equal to half the total frequency (Σf / 2) on the vertical axis, draw a horizontal line to the curve, then read down to the horizontal axis.
从累积频数曲线找中位数： 在纵轴上找出等于总频数一半 (Σf / 2) 的累积频数值，画水平线与曲线相交，再向下读横轴值。
Finding quartiles: Use Σf / 4 for Q₁ and (3Σf)/4 for Q₃, following the same method.
寻找四分位数： 用 Σf / 4 找 Q₁，用 (3Σf)/4 找 Q₃，方法相同。
6. Probability Basics | 概率基础
Probability measures how likely an event is to occur. It is always a number between 0 and 1 inclusive.
概率衡量某个事件发生的可能性大小。概率值总是在 0 到 1 之间（含 0 和 1）。
Basic probability formula: le outcomes / total number of possible outcomes, provided all outcomes are equally likely.
基本概率公式： P(A) = 有利结果的数量 / 所有可能结果的总数，前提是所有结果等可能发生。
Complementary events: The probability that event A does not happen is P(A') = 1 – P(A).
互补事件： 事件 A 不发生的概率为 P(A') = 1 – P(A)。
Expected frequency: Expected number of occurrences = P(event) × number of trials.
期望频数： 期望发生次数 = P(事件) × 试验次数。
7. Combined Events and Tree Diagrams | 组合事件与树状图
Combined events involve two or more simple events. Addition and multiplication rules are used to calculate probabilities, while tree diagrams help map out sequential experiments.
组合事件涉及两个或两个以上简单事件。加法法则和乘法法则用于计算概率，树状图则有助于梳理顺序试验。
Addition rule for mutually exclusive events: P(A or B) = P(A) + P(B) when A and B cannot happen at the same time.
互斥事件的加法法则： 当 A 与 B 不能同时发生时，P(A 或 B) = P(A) + P(B)。
Addition rule for non-mutually exclusive events: P(A or B) = P(A) + P(B) – P(A and B).
非互斥事件的加法法则： P(A 或 B) = P(A) + P(B) – P(A 且 B)。
Multiplication rule for independent events: P(A and B) = P(A) × P(B) when events A and B are independent.
独立事件的乘法法则： 当事件 A 和 B 独立时，P(A 且 B) = P(A) × P(B)。
Tree diagrams: Multiply probabilities along branches to find the probability of an outcome. Add probabilities of relevant branches for combined events.
树状图： 沿分支相乘概率来求得某一结果的概率；将相关分支的概率相加来求得组合事件的概率。
8. Conditional Probability | 条件概率
Conditional probability is the probability of an event A occurring given that event B has already occurred. It helps analyse dependent events.
条件概率是指在事件 B 已经发生的条件下，事件 A 发生的概率。它用于分析相依事件。
Conditional probability formula: P(A|B) = P(A ∩ B) / P(B), provided P(B) > 0.
条件概率公式： P(A|B) = P(A ∩ B) / P(B)，其中 P(B) > 0。
Rearranged for intersection: P(A ∩ B) = P(A|B) × P(B). This is especially useful in tree diagrams with dependent stages.
乘法形式的变形： P(A ∩ B) = P(A|B) × P(B)。在带有相依阶段的树状图中，这一公式非常有用。
9. Scatter Plots and Correlation | 散点图与相关性
A scatter plot displays the relationship between two quantitative variables. Correlation describes the direction and strength of a linear relationship.
散点图展示两个定量变量之间的关系。相关性描述线性关系的方向与强度。
Types of correlation: Positive correlation (as one variable increases, the other tends to increase), negative correlation (one increases, the other decreases), and no correlation (no clear pattern).
相关性的种类： 正相关（一个变量增加，另一个也倾向增加）、负相关（一个增加，另一个减少）和无相关（无明显模式）。
Strength of correlation: The closer the points lie to a straight line, the stronger the correlation. Perfect correlation implies all points lie exactly on a line.
相关性强度： 点越接近一条直线，相关性越强。完全相关意味着所有点都精确地落在一条直线上。
10. Equation of a Line of Best Fit | 最佳拟合直线方程
The line of best fit (regression line) models the linear relationship seen in a scatter plot. It can be drawn by eye or calculated using the least squares method.
最佳拟合直线（回归线）对散点图中的线性关系进行建模。它可以通过直观画线或使用最小二乘法计算得出。
Equation form: y = a + bx or y = mx + c, where b (or m) is the gradient and a (or c) is the y-intercept.
方程形式： y = a + bx 或 y = mx + c，其中 b（或 m）是斜率，a（或 c）是 y 轴截距。
Least squares gradient (b): b = Σ(x – x̄)(y – ȳ) / Σ(x – x̄)². In computational form: b = (nΣxy – Σx Σy) / (nΣx² – (Σx)²).
最小二乘斜率 (b)： b = Σ(x – x̄)(y – ȳ) / Σ(x – x̄)²。计算式为：b = (nΣxy – Σx Σy) / (nΣx² – (Σx)²)。
Intercept (a): a = ȳ – b x̄, where x̄ is the mean of x-values and ȳ is the mean of y-values.
截距 (a)： a = ȳ – b x̄，其中 x̄ 是 x 值的平均数，ȳ 是 y 值的平均数。
11. Sampling Techniques | 抽样方法
Sampling is the process of selecting a subset of individuals from a population to estimate characteristics of the whole population. Understanding bias is crucial.
抽样是指从总体中选出一部分个体以估计整体特征的过程。理解偏差与否至关重要。
Random sampling: Every member of the population has an equal chance of being chosen. This reduces bias but may not always be practical.
随机抽样： 总体中每一个体都有相等的机会被选中。这可以减少偏差，但并非始终可行。
Stratified sampling: The population is divided into distinct groups (strata), and a random sample is taken from each group in proportion to its size.
分层抽样： 将总体划分为不同的组（层），并按各层大小比例从每一层中随机抽取样本。
Systematic sampling: Choose subjects at regular intervals from a list, e.g. every 10th person. The starting point must be random.
系统抽样： 从名单中按固定间隔选取样本，例如每隔 10 人抽取一人。起点必须随机选择。
Bias considerations: Convenience sampling (selecting those easiest to reach) often leads to bias and should be avoided.
偏差注意事项： 便利抽样（选择最容易接触到的样本）常导致偏差，应当避免。
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