📚 Year 11 WJEC Statistics: Quick Reference Formula & Theorem Handbook | Year 11 WJEC 统计：公式定理速查手册
This handbook provides a concise collection of essential formulas, definitions, and theorems for the Year 11 WJEC GCSE Statistics course. Use it for quick revision, ensuring you have all the key tools at your fingertips before the exam. Each section pairs an English explanation with its Chinese counterpart, helping bilingual learners master the content with confidence.
本手册浓缩了 Year 11 WJEC GCSE 统计课程的核心公式、定义与定理，方便快速复习，确保考试前掌握全部关键工具。每一节均提供中英双语对照讲解，帮助双语学习者扎实掌握知识。
1. Measures of Central Tendency | 集中趋势度量
The arithmetic mean (average) of a set of n values is calculated by summing all the values and dividing by the number of values.
x̄ = Σx / n
where Σx is the sum of all data points and n is the sample size.
一组n个数据的算术平均数（均值）通过将所有数值相加再除以数据个数得出。
x̄ = Σx / n
其中 Σx 为数据总和，n 为数据个数。
For a frequency distribution, where each value (or class midpoint) x occurs with frequency f, the estimated mean is:
Estimated mean = Σ(f·x) / Σf
对于频数分布，用组中点 x 和对应的频数 f 计算估计平均数：
估计平均数 = Σ(f·x) / Σf
The median for ungrouped data is the middle value when the data are arranged in order. For an odd number of values, it is the central value; for an even number, it is the average of the two central values.
未分组数据的中位数是将数据排序后处于中间位置的数值。奇数个数据时取正中间的值，偶数个数据时取中间两个数的平均值。
For grouped data, the median class is identified using cumulative frequency, and the median is estimated by linear interpolation:
Median = L + ((n/2 − F) / f) × w
where L = lower boundary of the median class, n = total frequency, F = cumulative frequency before the median class, f = frequency of the median class, and w = class width.
对于分组数据，借助累积频数确定中位数组，然后用线性插值估计中位数：
中位数 = L + ((n/2 − F) / f) × w
其中 L 为中位数组下限，n 为总频数，F 为中位数组之前的累积频数，f 为中位数组的频数，w 为组距。
The mode is the value (or class) that occurs most frequently. A data set may have no mode, one mode (unimodal), or more than one mode (bimodal/multimodal).
众数是出现次数最多的数值（或人数最多的组）。一组数据可能没有众数，也可能有一个众数（单峰）或多个众数（双峰/多峰）。
2. Measures of Spread | 离散程度
The range is the simplest measure of spread:
Range = Maximum value − Minimum value
极差是最简单的离散程度指标：
极差 = 最大值 − 最小值
The interquartile range (IQR) is the difference between the upper quartile (Q₃) and the lower quartile (Q₁):
IQR = Q₃ − Q₁
四分位距 (IQR) 是上四分位数 Q₃ 与下四分位数 Q₁ 之差：
IQR = Q₃ − Q₁
Variance and standard deviation measure how spread out the data are around the mean. For a population or a full set of data, the variance σ² is:
σ² = Σ(x − x̄)² / n
The standard deviation is the square root of the variance:
σ = √[ Σ(x − x̄)² / n ]
方差和标准差衡量数据围绕均值的分散程度。对于总体或完整数据集，方差 σ² 为：
σ² = Σ(x − x̄)² / n
标准差是方差的平方根：
σ = √[ Σ(x − x̄)² / n ]
A computationally efficient alternative formula for standard deviation is:
σ = √[ (Σx² / n) − (x̄)² ]
标准差还有一个便于计算的等价公式：
σ = √[ (Σx² / n) − (x̄)² ]
If the data represent a sample and we wish to estimate the population standard deviation, we divide by (n − 1) instead of n to obtain the sample standard deviation, s.
如果数据是样本，并希望估计总体标准差，则用 (n−1) 代替 n 来计算样本标准差 s。
3. Quartiles and Box Plots | 四分位数与箱线图
Quartiles divide an ordered data set into four equal parts. 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. A common method for finding the position of Q₁ is (n+1)/4, and for Q₃ it is 3(n+1)/4; always follow the convention specified by your exam board.
四分位数将排序后的数据分成四等份。下四分位数 Q₁ 是数据下半部分的中位数，上四分位数 Q₃ 是上半部分的中位数。确定 Q₁ 位置的常用方法是 (n+1)/4，Q₃ 的位置为 3(n+1)/4；考试请以评分标准约定的方法为准。
A box plot (box-and-whisker plot) displays the five-number summary: minimum, Q₁, median, Q₃, maximum. A box is drawn from Q₁ to Q₃ with a line at the median, and whiskers extend to the smallest and largest non-outlier values.
箱线图（盒须图）展示五数概括：最小值、Q₁、中位数、Q₃、最大值。矩形箱体从 Q₁ 画到 Q₃，并在中位数处画一条线；须线延伸到非异常值的最小和最大值。
Outliers are identified using fences:
Lower fence = Q₁ − 1.5 × IQR
Upper fence = Q₃ + 1.5 × IQR
Any data point below the lower fence or above the upper fence is considered an outlier and is marked with a separate symbol, often a dot.
异常值通过“围栏”识别：
下围栏 = Q₁ − 1.5 × IQR
上围栏 = Q₃ + 1.5 × IQR
低于下围栏或高于上围栏的数据点视为异常值，在箱线图中用独立的点标出。
4. Probability Rules | 概率法则
For any two events A and B, the addition rule is:
P(A or B) = P(A) + P(B) − P(A and B)
对于任意两个事件 A 和 B，加法法则为：
P(A 或 B) = P(A) + P(B) − P(A 且 B)
If A and B are mutually exclusive (cannot occur simultaneously), then P(A and B) = 0, so:
P(A or B) = P(A) + P(B)
如果 A 与 B 互斥（不能同时发生），则 P(A 且 B) = 0，因此：
P(A 或 B) = P(A) + P(B)
For independent events, the multiplication rule is:
P(A and B) = P(A) × P(B)
对于独立事件，乘法法则为：
P(A 且 B) = P(A) × P(B)
Conditional probability is the probability of event A occurring given that event B has already occurred:
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