📚 IGCSE WJEC Statistics: Formula and Theorem Quick-Reference Handbook | IGCSE WJEC 统计：公式定理速查手册
This quick-reference handbook summarises the key formulae and theorems required for the IGCSE WJEC Statistics specification. It covers data collection, averages, spread, probability, discrete and normal distributions, correlation, regression, time series and index numbers. Use it alongside past-paper practice to check definitions, apply formulas accurately, and develop calculator-free estimation skills where relevant.
本速查手册汇总了 IGCSE WJEC 统计课程的核心公式与定理，涵盖数据收集、集中趋势、离散程度、概率、离散分布与正态分布、相关与回归、时间序列以及指数。建议结合历年真题练习使用，以便核对定义、准确套用公式，并在适用场景下提升无计算器估算能力。
1. Data Types and Sampling | 数据类型与抽样
Data can be qualitative, meaning categorical, or quantitative, meaning numerical. Quantitative data may be discrete, taking distinct values, or continuous, taking any value within an interval. A population is the whole set of individuals or items, while a sample is a subset used to estimate population features.
数据可以是定性数据，即分类数据；也可以是定量数据，即数值数据。定量数据可进一步分为离散数据，取互不相同的数值，或连续数据，在某一区间内可取任意值。总体是所有个体或项目的集合，样本则是用于估计总体特征的子集。
For a stratified sample, the number selected from a stratum is proportional to its population share:
n_stratum = (N_stratum / N_population) × n_sample
分层抽样中，每层抽取的样本数量与该层总体数量占总体的比例成正比。
Random sampling removes selection bias because every member has an equal chance of selection. Stratified sampling keeps proportional representation of groups, which is useful when subgroups differ in size or characteristics.
随机抽样可减少选择偏差，因为每个成员被选中的机会相等。分层抽样保持各组的比例代表性，在子组大小或特征不同时尤为适用。
2. Charts and Diagrams | 统计图与图表
A histogram uses area to represent frequency. With equal class widths, frequency is proportional to bar height, but with unequal class widths the frequency density must be calculated:
Frequency density = frequency / class width
直方图用面积表示频数。组距相等时，频数与柱高成正比；组距不相等时，必须计算频率密度，即频数除以组距。
A cumulative frequency curve is used to estimate medians, quartiles and percentiles. On a box plot, the box runs from Q₁ to Q₃ with the median marked inside, and whiskers extend to the highest and lowest values that are not outliers.
累积频率曲线用于估计中位数、四分位数和百分位数。在箱线图中，箱体从下四分位数 Q₁ 延伸到上四分位数 Q₃，中位数标在箱内，触须延伸到非异常值的最低值和最高值。
Outliers are often defined as values below Q₁ – 1.5 × IQR or above Q₃ + 1.5 × IQR, where IQR is the interquartile range.
异常值通常定义为低于 Q₁ – 1.5 × IQR 或高于 Q₃ + 1.5 × IQR 的数值，其中 IQR 为四分位距。
3. Measures of Central Tendency | 集中趋势
The mean of raw data is the sum of all values divided by the number of values:
x̄ = Σx / n
原始数据的平均数等于所有数值之和除以数据个数。
For a frequency distribution, the mean is calculated using frequencies as weights:
x̄ = Σfx / Σf
频数分布中，平均数等于各组数值乘以对应频数之和除以总频数。
The median position in ordered data is (n + 1) / 2. For grouped data, the median is estimated by interpolation:
Median = L + [ (n/2 – F_below) / f_median ] × class width
有序数据的中位数位置为 (n + 1) / 2。对于分组数据，中位数可用插值公式估计：中位数等于中位数组下界 L 加上 (n/2 减去前一累计频数) 除以中位数组频数，再乘以组距。
The mode is the most frequent value or class. A distribution is positively skewed when mean > median > mode, and negatively skewed when mean < median < mode.
众数是最常出现的数值或组。当平均数大于中位数且中位数大于众数时，分布为正偏态；当平均数小于中位数且中位数小于众数时，分布为负偏态。
4. Measures of Dispersion | 离散程度
The range is the difference between the largest and smallest values. The interquartile range is the difference between the upper and lower quartiles:
IQR = Q₃ – Q₁
极差是最大值与最小值之差。四分位距是上四分位数 Q₃ 与下四分位数 Q₁ 之差。
Variance for ungrouped data measures the average squared deviation from the mean:
σ² = Σ(x – μ)² / n = Σx² / n – μ²
未分组数据的方差衡量各数据与总体均值离差平方的平均数，也等于数据平方的平均数减去均值平方。
Standard deviation is the square root of variance and has the same units as the original data:
σ = √[ Σ(x – μ)² / n ]
标准差是方差的平方根，与原数据具有相同单位，表示数据围绕均值的平均离散程度。
For grouped data, use class midpoints x as representative values:
σ² = Σf(x – μ)² / Σf
分组数据使用组中值 x 作为代表值计算方差，公式为频数乘以组中值与均值之差的平方和除以总频数。
5. Probability Rules | 概率法则
The addition rule gives the probability that either event A or event B occurs:
P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
加法法则给出事件 A 或事件 B 发生的概率：等于各自概率之和减去同时发生的概率。
If A and B are mutually exclusive, they cannot occur together, so P(A ∩ B) = 0, and:
P(A ∪ B) = P(A) + P(B)
若 A 与 B 互斥，则它们不能同时发生，因此 P(A ∩ B) = 0，或事件的概率等于两个概率直接相加。
Conditional probability is the probability of A given that B has occurred:
P(A | B) = P(A ∩ B) / P(B)
条件概率表示在 B 已发生的条件下 A 发生的概率，等于联合概率除以 B 的概率。
For independent events, P(A ∩ B) = P(A) × P(B). Relative frequency estimates probability as frequency / total trials.
独立事件同时发生的概率等于各自概率相乘。相对频率用发生次数除以总试验次数来估计概率。
6. Expectation and Variance of Discrete Variables | 离散变量的期望与方差
For a discrete random variable X, the expectation is the probability-weighted average of its values:
E(X) = Σ x P(X = x)
离散随机变量 X 的期望值等于每个取值 x 乘以其对应概率之和。
The variance of X is the expectation of the squared deviation from the mean:
Var(X) = Σ x² P(X = x) – [E(X)]²
X 的方差等于每个取值平方乘以其概率之和减去期望值的平方。
Standard deviation of X
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