SQA Statistics Year 11: Quick Reference to Formulas and Theorems | SQA 统计 Year 11：公式定理速查手册

📚 SQA Statistics Year 11: Quick Reference to Formulas and Theorems | SQA 统计 Year 11：公式定理速查手册

This handbook brings together the essential formulas, definitions and theorems required for the SQA Statistics course at Year 11. It is designed for quick revision before class tests, prelims and final examinations. Each section presents the key concept in clear English followed by the corresponding Chinese explanation, with numbered equations and worked statements to reinforce understanding.

本手册汇集了 Year 11 SQA 统计课程所必需的核心公式、定义和定理，专为单元测验、模拟考试和最终大考前的快速复习而设计。每个小节先用清晰的英文阐述关键概念，再配以对应的中文解释，并通过编号公式和应用陈述帮助巩固理解。

1. Measures of Central Tendency and Dispersion | 集中趋势和离散程度的度量

The mean of a sample is given by x̄ = Σx / n, where n is the sample size. For grouped data, use midpoints. The median is the middle value when data are ordered; for an even number of values it is the average of the two middle numbers. The mode is the most frequent value.

样本均值的公式为 x̄ = Σx / n，其中 n 为样本容量。对于分组数据，使用组中值计算。中位数是将数据排序后位于中间位置的数值；当数据个数为偶数时，取中间两个数的平均值。众数是出现频率最高的数据值。

Measures of spread include the range (max minus min), the interquartile range IQR = Q₃ – Q₁, and the standard deviation. For a population, σ = √[Σ(x – μ)² / N]; for a sample, s = √[Σ(x – x̄)² / (n – 1)].

离散程度的度量包括极差（最大值减最小值）、四分位距 IQR = Q₃ – Q₁ 和标准差。总体标准差 σ = √[Σ(x – μ)² / N]；样本标准差 s = √[Σ(x – x̄)² / (n – 1)]。

Sample variance: s² = Σ(x – x̄)² / (n – 1)
样本方差：s² = Σ(x – x̄)² / (n – 1)
Quartiles: Q₁ (25th percentile), Q₂ (median), Q₃ (75th percentile)
四分位数：Q₁（第25百分位数），Q₂（中位数），Q₃（第75百分位数）

2. Probability Rules | 概率法则

For any two events A and B, the addition rule is P(A ∪ B) = P(A) + P(B) – P(A ∩ B). If A and B are mutually exclusive, P(A ∩ B) = 0 so P(A ∪ B) = P(A) + P(B).

对于任意两个事件 A 和 B，加法法则为 P(A ∪ B) = P(A) + P(B) – P(A ∩ B)。若 A 与 B 互斥，则 P(A ∩ B) = 0，故 P(A ∪ B) = P(A) + P(B)。

Conditional probability is defined as P(A|B) = P(A ∩ B) / P(B), provided P(B) > 0. The multiplication rule then becomes P(A ∩ B) = P(A|B) × P(B).

条件概率定义为 P(A|B) = P(A ∩ B) / P(B)，要求 P(B) > 0。乘法法则可写为 P(A ∩ B) = P(A|B) × P(B)。

Two events are independent if P(A ∩ B) = P(A) × P(B), or equivalently P(A|B) = P(A).

若 P(A ∩ B) = P(A) × P(B) 或等价地 P(A|B) = P(A)，则事件 A 和 B 相互独立。

Complement rule: P(A’) = 1 – P(A)
互补法则：P(A’) = 1 – P(A)
Tree diagrams multiply probabilities along branches for ∩ and add across branches for ∪.
树状图沿分支相乘得到交集概率，分支间相加得到并集概率。

3. Binomial Distribution | 二项分布

A binomial experiment has a fixed number of trials n, each with two outcomes (success/failure), constant probability of success p, and independent trials. The random variable X ~ B(n, p) counts the number of successes.

二项试验具有固定的试验次数 n、每次试验两种结果（成功/失败）、成功概率 p 恒定且各次试验相互独立。随机变量 X ~ B(n, p) 表示成功的次数。

The probability of exactly r successes is given by the binomial formula: P(X = r) = ⁿCᵣ × pʳ × (1 – p)ⁿ⁻ʳ, where ⁿCᵣ = n! / [r!(n – r)!].

恰好取得 r 次成功的概率由二项公式给出：P(X = r) = ⁿCᵣ × pʳ × (1 – p)ⁿ⁻ʳ，其中组合数 ⁿCᵣ = n! / [r!(n – r)!]。

Key parameters: Mean E(X) = np, Variance Var(X) = np(1 – p). The standard deviation is √[np(1 – p)].

关键参数：期望（均值）E(X) = np，方差 Var(X) = np(1 – p)，标准差为 √[np(1 – p)]。

Property	Formula
Probability mass	ⁿCᵣ pʳ (1 – p)ⁿ⁻ʳ
Mean	np
Variance	np(1 – p)

4. Normal Distribution | 正态分布

The normal distribution is defined by its mean μ and standard deviation σ. The probability density function is symmetric and bell-shaped. About 68% of values lie within 1σ of μ, 95% within 2σ, and 99.7% within 3σ.

正态分布由其均值 μ 和标准差 σ 定义。概率密度函数呈对称的钟形。大约 68% 的数据落在 μ ± 1σ 内，95% 落在 μ ± 2σ 内，99.7% 落在 μ ± 3σ 内。

To standardise a value x from N(μ, σ²), use the z-score: z = (x – μ) / σ. The standard normal distribution N(0,1) can then be used with tables to find probabilities.

为了将来自 N(μ, σ²) 的值 x 标准化，使用 z 分数：z = (x – μ) / σ。随后可利用标准正态分布 N(0,1) 的表查找概率。

When working with sample means, the sampling distribution of x̄ is N(μ, σ²/n). The standard error is σ/√n.

处理样本均值时，x̄ 的抽样分布为 N(μ, σ²/n)，标准误为 σ/√n。

Standard normal table gives Φ(z) = P(Z ≤ z) for Z ~ N(0,1).
标准正态分布表提供 Φ(z) = P(Z ≤ z)，其中 Z ~ N(0,1)。
To find range covering middle p%, use z-values from table for (1+p)/2.
求覆盖中间 p% 的范围时，利用表中对应 (1+p)/2 的 z 值。

5. Sampling and the Central Limit Theorem | 抽样与中心极限定理

A sampling distribution describes how a statistic (e.g., sample mean) varies over all possible samples of a given size. The Central Limit Theorem (CLT) states that for sufficiently large sample size n (typically n ≥ 30), the sampling distribution of the sample mean will be approximately normal regardless of the population’s shape, with mean μ and standard deviation σ/√n.

抽样分布描述统计量（如样本均值）在所有可能样本中的变化情况。中心极限定理指出，当样本量 n 足够大（通常 n ≥ 30）时，样本均值的抽样分布近似服从正态分布，其均值为 μ，标准差为 σ/√n，而不管总体分布的形状如何。

The standard error of the mean is SE = σ/√n when σ is known; if σ is unknown, use the estimated standard error s/√n. The precision of the estimate improves as n increases.

若 σ 已知，均值的标准误为 SE = σ/√n；若 σ 未知，则使用估计标准误 s/√n。估计的精度随 n 增大而提高。

For a sample proportion p̂, the sampling distribution is approximately normal when np ≥ 10 and n(1 – p) ≥ 10, with SE = √[p(1 – p)/n].

对于样本比例 p̂，当 np ≥ 10 且 n(1 – p) ≥ 10 时，其抽样分布近似正态，标准误为 SE = √[p(1 – p)/n]。

6. Confidence Intervals | 置信区间

A confidence interval gives a range of plausible values for a population parameter. For a population mean with known σ, the 95% CI is x̄ ± z* × (σ/√n), where z* = 1.96 for 95% confidence.

置信区间给出了总体参数的合理取值范围。当 σ 已知时，均值的 95% 置信区间为 x̄ ± z* × (σ/√n)，其中 95% 置信水平对应的 z* = 1.96。

When σ is unknown (which is common), use the t-distribution: x̄ ± t* × (s/√n). The critical value t* comes from the t-table with n – 1 degrees of freedom.

当 σ 未知时（这是常见情况），使用 t 分布：x̄ ± t* × (s/√n)。临界值 t* 根据自由度 n – 1 从 t 分布表查得。

For a population proportion, the 95% CI is p̂ ± z* × √[p̂(1 – p̂)/n]. The margin of error is the half-width of the interval.

对于总体比例，95% 置信区间为 p̂ ± z* × √[p̂(1 – p̂)/n]。误差界限即区间的半宽。

90% CI uses z* = 1.645; 99% CI uses z* = 2.576.
90% 置信区间使用 z* = 1.645；99% 置信区间使用 z* = 2.576。
Increasing the confidence level widens the interval.
提高置信水平会加宽区间。

7. Hypothesis Testing (Single Proportion or Mean) | 假设检验（单一比例或均值）

A hypothesis test begins with the null hypothesis H₀ (e.g., μ = μ₀ or p = p₀) and the alternative H₁ (two-tailed, or one-tailed with < or >). Select a significance level α, typically 0.05.

假设检验从零假设 H₀（如 μ = μ₀ 或 p = p₀）和备择假设 H₁（双尾，或单尾使用 < 或 >）开始。选择一个显著性水平 α，通常取 0.05。

Calculate the test statistic. For a mean with known σ: z = (x̄ – μ₀) / (σ/√n). For a proportion: z = (p̂ – p₀) / √[p₀(1 – p₀)/n]. Compare this value with the critical value(s) from the standard normal table, or find the p-value.

计算检验统计量。在 σ 已知时，均值的检验统计量为 z = (x̄ – μ₀) / (σ/√n)；比例的检验统计量为 z = (p̂ – p₀) / √[p₀(1 – p₀)/n]。将计算值与标准正态表查得的临界值比较，或求出 p 值。

When σ is unknown, use the t-test: t = (x̄ – μ₀) / (s/√n) with df = n – 1. Reject H₀ if the test statistic falls in the rejection region or if p-value < α.

当 σ 未知时，使用 t 检验：t = (x̄ – μ₀) / (s/√n)，自由度为 n – 1。若检验统计量落在拒绝域内，或 p 值 < α，则拒绝 H₀。

Type I error: rejecting a true H₀ (probability = α).
第一类错误：拒绝为真的 H₀（概率为 α）。
Type II error: failing to reject a false H₀ (β). Power = 1 – β.
第二类错误：未拒绝错误的 H₀（β）。检验功效 = 1 – β。

8. Correlation and Regression | 相关与回归

The Pearson product-moment correlation coefficient r measures the strength and direction of a linear relationship between two variables. It is calculated as r = Sₛ / √(S₠ Sₜ), where S₠ = Σ(x – x̄)², Sₜ = Σ(y – ȳ)², Sₛ = Σ(x – x̄)(y – ȳ). The value of r lies between –1 and 1.

皮尔逊积差相关系数 r 衡量两个变量之间线性关系的强度和方向。计算公式为 r = Sₛ / √(S₠ Sₜ)，其中 S₠ = Σ(x – x̄)²，Sₜ = Σ(y – ȳ)²，Sₛ = Σ(x – x̄)(y – ȳ)。r 的取值在 –1 到 1 之间。

The least-squares regression line of y on x has equation y = a + bx, where the slope b = Sₛ / S₠ and the intercept a = ȳ – b x̄. This line minimises the sum of squared vertical residuals.

y 对 x 的最小二乘回归直线方程为 y = a + bx，其中斜率 b = Sₛ / S₠，截距 a = ȳ – b x̄。该直线使垂直残差的平方和最小。

For a hypothesis test on the correlation, the test statistic is t = r × √(n – 2) / √(1 – r²) with df = n – 2, used to test H₀: ρ = 0.

对相关系数进行假设检验时，检验统计量为 t = r × √(n – 2) / √(1 – r²)，自由度为 n – 2，用于检验 H₀: ρ = 0。

9. Chi-Squared Tests | 卡方检验

The chi-squared test statistic is χ² = Σ (O – E)² / E, where O are the observed frequencies and E are the expected frequencies under the null hypothesis. The test is applicable to categorical data.

卡方检验统计量为 χ² = Σ (O – E)² / E，其中 O 为观测频数，E 为原假设下的期望频数。该检验适用于分类数据。

For a goodness-of-fit test, E = n × pᵤ for each category i, and the degrees of freedom are (number of categories – 1). For a test of independence in a contingency table, E = (row total × column total) / grand total, and df = (r – 1)(c – 1).

适合度检验中，每个类别的期望频数 E = n × pᵤ，自由度为（类别数 – 1）。在列联表的独立性检验中，E = （行合计 × 列合计）/ 总和，自由度 df = (r – 1)(c – 1)。

The test is reliable when all expected frequencies are at least 5. If the calculated χ² exceeds the critical value from the chi-squared table (with given df and α), reject H₀.

当所有期望频数均至少为 5 时，检验可靠。若计算得到的 χ² 大于卡方表中的临界值（对应给定的 df 和 α），则拒绝 H₀。

χ² distribution is skewed right; its shape depends on df.
χ² 分布呈右偏，形状取决于自由度 df。
Use table or calculator to find p-value corresponding to the test statistic.
利用表或计算器找出与检验统计量对应的 p 值。

10. Non-Parametric Tests (Mann-Whitney U) | 非参数检验（曼-惠特尼U检验）

The Mann-Whitney U test compares the medians of two independent samples when the normality assumption is not met. It uses the ranks of all observations combined.

曼-惠特尼 U 检验用于比较两个独立样本的中位数，适用于不满足正态性假设的情形。检验基于所有观测值的合并秩次。

Steps: combine both samples, rank all values from 1 to n₁ + n₂ (assign average ranks for ties). Calculate the sum of ranks for each group, R₁ and R₂. Then compute U₁ = n₁ n₂ + n₁(n₁ + 1)/2 – R₁, and similarly U₂ = n₁ n₂ + n₂(n₂ + 1)/2 – R₂. The test statistic U is the smaller of U₁ and U₂.

步骤：合并两个样本，对所有数值从 1 到 n₁ + n₂ 排序（结时取平均秩次）。分别计算两组秩和 R₁ 与 R₂。然后计算 U₁ = n₁ n₂ + n₁(n₁ + 1)/2 – R₁，类似地 U₂ = n₁ n₂ + n₂(n₂ + 1)/2 – R₂。检验统计量 U 取 U₁ 和 U₂ 中的较小者。

For small sample sizes, compare U with critical values from the Mann-Whitney table. For large samples (n₁, n₂ > 10), the normal approximation can be used: z = (U – μu) / σu, where μu = n₁ n₂ / 2 and σu = √[n₁ n₂ (n₁ + n₂ + 1) / 12].

小样本时，将 U 与曼-惠特尼表查得的临界值进行比较。大样本（n₁, n₂ > 10）时，可使用正态近似：z = (U – μu) / σu，其中 μu = n₁ n₂ / 2，σu = √[n₁ n₂ (n₁ + n₂ + 1) / 12]。

The test is robust to outliers and does not assume equal variances.
该检验对异常值稳健，且不要求方差齐性。
Always check assumptions and choose the test that matches the data’s structure.
务必检验假设条件，选择与数据结构匹配的检验方法。

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SQA Statistics Year 11: Quick Reference to Formulas and Theorems | SQA 统计 Year 11：公式定理速查手册