📚 Year 11 Eduqas Statistics: Quick Reference Formula & Theorem Handbook | Year 11 Eduqas 统计：公式定理速查手册
This comprehensive reference handbook summarises the essential formulas, notation and key theorems required for the Year 11 Eduqas GCSE Statistics course. Whether you are revising data handling, probability models or quality control, this guide provides a structured, bilingual quick check of every important statistical tool you need to master.
这本综合速查手册汇总了 Year 11 Eduqas GCSE 统计课程中必备的公式、符号和核心定理。无论你正在复习数据处理、概率模型还是质量控制，本指南都以结构化的双语形式为你提供需要掌握的重要统计工具，方便快速查阅。
1. Measures of Central Tendency | 集中趋势测量
The mean of a sample (x̄) is the sum of all the data values divided by the number of values, n. It represents the centre of mass of the data and is calculated using x̄ = Σx / n.
样本均值 (x̄) 是所有数据值之和除以数据个数 n。它是数据的平衡中心，计算公式为 x̄ = Σx / n。
x̄ = Σx / n
The median is the middle value when the data are arranged in order. For an odd number of observations, it is the (n+1)/2 th value; for an even number, it is the average of the n/2 th and (n/2)+1 th values.
中位数是将数据排序后位于中间的值。如果观测值个数为奇数，中位数是第 (n+1)/2 个值；如果为偶数，则是第 n/2 个与第 (n/2)+1 个值的平均数。
The mode is the value or category that occurs most frequently in a data set. A set of data may have one mode, more than one mode (bimodal or multimodal) or no mode at all if all values are unique.
众数是数据集中出现频率最高的值或类别。一组数据可能有一个众数、多个众数（双峰或多峰），或者当所有值唯一时没有众数。
When data are severely skewed, the median often provides a more representative measure of location than the mean, which can be pulled away by extreme values.
当数据严重偏斜时，中位数往往比均值更具代表性，因为均值会被极端值拉偏。
2. Measures of Dispersion | 离散程度测量
The range is the simplest measure of spread, defined as the difference between the maximum and minimum values: Range = Max − Min.
极差（全距）是最简单的离散程度度量，定义为最大值与最小值之差：极差 = 最大值 − 最小值。
The interquartile range (IQR) is the spread of the middle 50% of the data: IQR = Q₃ − Q₁, where Q₁ is the lower quartile and Q₃ is the upper quartile. It is unaffected by outliers.
四分位距 (IQR) 是中间 50% 数据的散布：IQR = Q₃ − Q₁，其中 Q₁ 为下四分位数，Q₃ 为上四分位数。它不受异常值影响。
The (sample) standard deviation, denoted by s, measures how much the data deviate from the mean on average. For ungrouped data:
（样本）标准差记为 s，衡量数据平均偏离均值的程度。对于未分组数据：
s = √[ Σ(x − x̄)² / (n − 1) ]
The square of the standard deviation, s², is called the variance. Both are always non‑negative; a larger value indicates greater spread.
标准差的平方 s² 称为方差。两者始终非负；数值越大表示数据越分散。
Quartiles are found using (n+1)-based positions: Q₁ is at position (n+1)/4, Q₂ (median) at (n+1)/2, and Q₃ at 3(n+1)/4 for ungrouped data.
四分位数基于 (n+1) 的位置查找：对于未分组数据，Q₁ 位于 (n+1)/4 位置，Q₂（中位数）位于 (n+1)/2，Q₃ 位于 3(n+1)/4。
3. Mean and Standard Deviation from Grouped Data | 分组数据的均值与标准差
In a frequency table with values xᵢ and frequencies fᵢ, the estimated mean is x̄ = Σ(fᵢ × xᵢ) / Σfᵢ. Use class midpoints as xᵢ when data are grouped into intervals.
在具有值 xᵢ 和频数 fᵢ 的频数表中，估计均值为 x̄ = Σ(fᵢ × xᵢ) / Σfᵢ。当数据被分入区间时，使用组中值作为 xᵢ。
x̄ = Σ(fx) / Σf
The standard deviation for grouped data uses a similar adjustment, replacing individual values with their class midpoints weighted by frequency:
分组数据的标准差采用类似调整，用组中值乘频数加权代替单个值：
s = √[ Σ f(x − x̄)² / (Σf − 1) ]
An alternative computational formula often simplifies calculations: s = √{ [Σ f x² − (Σ f x)²/Σf] / (Σf − 1) }. Always use the sample divisor (Σf − 1) unless the whole population is given.
常用简化计算式：s = √{ [Σ f x² − (Σ f x)²/Σf] / (Σf − 1) }。除非给出的是总体，否则总是使用样本除数 Σf − 1。
When estimating quartiles from grouped data, linear interpolation within the relevant class interval is required. The pth percentile formula can be applied: position = p/100 × total frequency.
从分组数据估计四分位数时，需要在相应组距内进行线性插值。可采用第 p 百分位公式：位置 = p/100 × 总频数。
4. Probability Rules and Notation | 概率规则与符号
The probability of an event A, P(A), satisfies 0 ≤ P(A) ≤ 1, where 0 indicates impossibility and 1 indicates certainty. For all possible outcomes in the sample space S, P(S) = 1.
事件 A 的概率 P(A) 满足 0 ≤ P(A) ≤ 1，0 表示不可能，1 表示必然。对于样本空间 S 中所有可能结果，有 P(S) = 1。
The complement rule states that the probability an event does not occur is P(not A) = 1 − P(A).
互补规则：事件不发生的概率为 P(非 A) = 1 − P(A)。
For mutually exclusive events A and B (they cannot happen together), the addition rule is P(A or B) = P(A) + P(B). If events are not mutually exclusive, use P(A or B) = P(A) + P(B) − P(A and B).
对于互斥事件 A 和 B（不能同时发生），加法规则为 P(A 或 B) = P(A) + P(B)。若事件不互斥，则使用 P(A 或 B) = P(A) + P(B) − P(A 且 B)。
For independent events, P(A and B) = P(A) × P(B). Conditional probability is defined as P(A|B) = P(A and B) / P(B), provided P(B) > 0.
对于独立事件，P(A 且 B) = P(A) × P(B)。条件概率定义为 P(A|B) = P(A 且 B) / P(B)，当 P(B) > 0。
Tree diagrams are used to map multi‑stage trials; probabilities along branches multiply, and probabilities of all branches from a single node sum to 1.
树状图用于展示多阶段试验；沿分支的概率相乘，且从同一节点发出的所有分支概率之和为 1。
5. Binomial Distribution | 二项分布
A binomial distribution models the number of successes, X, in n independent trials each with constant probability of success p. The distribution is denoted as X ~ B(n, p).
二项分布用于描述在 n 次独立试验中成功的次数 X，每次试验成功概率恒为 p。分布记为 X ~ B(n, p)。
The probability of obtaining exactly r successes is given by the binomial formula:
恰好获得 r 次成功的概率由二项公式给出：
P(X = r) = C(n, r) × pʳ × qⁿ⁻ʳ
Here q = 1 − p, and C(n, r) = n! / [r! (n − r)!] is the binomial coefficient counting combinations of r items from n.
其中 q = 1 − p，C(n, r) = n! / [r! (n − r)!] 是二项式系数，计算从 n 个中选取 r 个的组合数。
The mean (expected value) of a binomial distribution is μ = np, and the variance is σ² = npq.
二项分布的均值（期望）为 μ = np，方差为 σ² = npq。
Cumulative probabilities P(X ≤ k) can be obtained from statistical tables or by summing individual terms. The distribution is symmetrical when p = 0.5 and becomes skewed when p is near 0 or 1.
累积概率 P(X ≤ k) 可从统计表中查得或通过累加各项求出。当 p = 0.5 时分布对称，p 接近 0 或 1 时呈现偏态。
6. Standardised Scores (z-scores) | 标准分数（z 分数）
A standardised score, or z‑score, measures how many standard deviations a raw value X lies away from the mean of its distribution. It is defined as:
标准分数（z 分数）衡量原始值 X 与其分布均值之间相距多少个标准差。定义为：
z = (X − μ) / σ
A positive z‑score indicates the value is above the mean; a negative z‑score indicates it is below the mean. z‑scores allow comparison across different distributions.
正 z 分数表示数值高于均值；负 z 分数表示低于均值。z 分数可用于跨分布比较。
When the population parameters μ and σ are unknown, the sample mean x̄ and sample standard deviation s are used to compute an estimated z‑score: z ≈ (X − x̄) / s.
当总体参数 μ 和 σ 未知时，可用样本均值 x̄ 和样本标准差 s 计算近似 z 分数：z ≈ (X − x̄) / s。
z‑scores are essential for using standard normal distribution tables to find proportions and percentiles.
z 分数对于使用标准正态分布表查找比例和百分位数至关重要。
7. Normal Distribution | 正态分布
Many continuous variables follow a normal distribution, often denoted as N(μ, σ²). The graph is a symmetric, bell‑shaped curve centred at μ, with points of inflection at μ ± σ.
许多连续变量遵循正态分布，通常记为 N(μ, σ²)。其图形是对称的钟形曲线，以 μ 为中心，拐点位于 μ ± σ 处。
The standard normal distribution has mean 0 and standard deviation 1: Z ~ N(0, 1). Any normal variable can be converted to a standard normal variable using z = (X − μ) / σ.
标准正态分布的均值为 0，标准差为 1：Z ~ N(0, 1)。任何正态变量都可通过 z = (X − μ) / σ 转化为标准正态变量。
Approximately 68% of data fall within 1 standard deviation of the mean (μ ± σ), 95% within 2 standard deviations (μ ± 2σ), and 99.8% within 3 standard deviations (μ ± 3σ) — known as the empirical rule.
约 68% 的数据落在均值 ± 1 个标准差范围内 (μ ± σ)，95% 在 ± 2σ 内，99.8% 在 ± 3σ 内——这就是经验法则。
Tables of the standard normal distribution give the probability Φ(z) = P(Z ≤ z) for positive z; symmetry and complement rules are used to handle negative z‑values and upper‑tail probabilities.
标准正态分布表给出正 z 时的概率 Φ(z) = P(Z ≤ z)；利用对称性和互补规则可处理负 z 值和右侧尾概率。
8. Index Numbers | 指数
An index number measures the change in a variable over time relative to a base period, which is given the value 100. The simple price relative for an item is:
指数用于衡量变量相对于基期的变化，基期赋值为 100。单一商品的简单价比为：
Price relative = (Current price / Base‑period price) × 100
A weighted aggregate index combines several items. The Laspeyres price index uses base‑period quantities as weights, while the Paasche index uses current‑period quantities.
加权综合指数结合了多个项目。拉斯拜尔价格指数使用基期数量作为权重，派许指数则使用当期数量。
Laspeyres index = Σ(pₙ × q₀) / Σ(p₀ × q₀) × 100
Paasche index = Σ(pₙ × qₙ) / Σ(p₀ × qₙ) × 100
where p₀ is the base‑period price, pₙ is the current price, q₀ is the base‑period quantity, and qₙ is the current‑period quantity.
其中 p₀ 为基期价格，pₙ 为当期价格，q₀ 为基期数量，qₙ 为当期数量。
The Retail Price Index (RPI) and Consumer Price Index (CPI) are real‑world weighted indices that track changes in the cost of a fixed basket of goods and services.
零售价格指数 (RPI) 和消费者价格指数 (CPI) 是现实中的加权指数，追踪固定一篮子商品和服务成本的变化。
9. Quality Assurance & Control Charts | 质量保证与控制图
Quality assurance uses statistical methods to monitor and control a process. Control charts (Shewhart charts) plot sample statistics over time to detect signals of variation.
质量保证运用统计方法监控和控制流程。控制图（休哈特图）将样本统计量随时间绘制，以检出变异信号。
The central line on an x̄‑chart is the process mean μ̂, often estimated from a large number of samples. The upper control limit (UCL) and lower control limit (LCL) are typically set at ±3 standard errors from the mean:
x̄ 控制图的中心线是过程均值 μ̂，通常由大量样本估计得出。上控制限 (UCL) 和下控制限 (LCL) 通常设定在均值 ±3 个标准误处：
UCL = μ̂ + 3 σ/√n, LCL = μ̂ − 3 σ/√n
Where σ is the process standard deviation and n is the subgroup size. The factor 3 corresponds to a low probability of a false alarm when the process is in control.
其中 σ 是过程标准差，n 是子组大小。因子 3 对应过程受控时较低的误报概率。
Range charts (R‑charts) monitor the variation within each subgroup; target lines and control limits for the range are determined using statistical constants d₂, D₃, D₄.
极差图 (R 图) 监控每个子组内的变异；极差的目标线和控制限使用统计常数 d₂、D₃、D₄ 确定。
Action signals occur when a plotted point falls outside control limits or when non‑random patterns (e.g. 7 consecutive points on one side of the centre line) appear.
当绘制点落在控制限外，或出现非随机模式（如连续 7 点位于中心线同侧）时，即触发行动信号。
10. Correlation and Linear Regression | 相关与线性回归
Correlation measures the strength and direction of a linear relationship between two variables. The product‑moment correlation coefficient r is calculated as:
相关性衡量两个变量之间线性关系的强度和方向。积差相关系数 r 计算公式为：
r = Sxy / √(Sxx × Syy)
where Sxx = Σ(x − x̄)², Syy = Σ(y − ȳ)², and Sxy = Σ(x − x̄)(y − ȳ). The value of r lies between −1 and +1.
其中 Sxx = Σ(x − x̄)²，Syy = Σ(y − ȳ)²，Sxy = Σ(x − x̄)(y − ȳ)。r 取值在 −1 至 +1 之间。
An r close to +1 indicates strong positive linear correlation; close to −1 indicates strong negative correlation; near 0 suggests very weak or no linear correlation.
r 接近 +1 表示强正线性相关；接近 −1 表示强负相关；接近 0 则表明线性相关很弱或无。
The least‑squares regression line of y on x has equation ŷ = a + bx, where the slope b = Sxy / Sxx and intercept a = ȳ − bx̄. This line can be used to make predictions within the data range.
y 对 x 的最小二乘回归线方程为 ŷ = a + bx，斜率 b = Sxy / Sxx，截距 a = ȳ − bx̄。该直线可用于数据范围内的预测。
Do not extrapolate predictions far beyond the range of the collected data, as the linear relationship may not hold outside the observed interval.
切忌远超出收集数据范围进行外推预测，因为在观测区间之外线性关系可能不再成立。
11. Moving Averages & Time Series | 移动平均与时间序列
A time series is
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