📚 SQA Statistics Formula & Theorem Quick Reference Handbook | SQA 统计公式定理速查手册
This handbook provides a concise summary of the essential formulas and theorems you need for Year 11 SQA Statistics. Each entry is presented with clear notation, a brief explanation, and a Chinese translation to support bilingual learners. Use it for quick revision, homework help, and exam preparation.
本手册为 Year 11 SQA 统计学课程总结了必备公式与定理。每一条均配有清晰的符号说明、简要解析及中文翻译，方便双语学习者快速查阅，用于复习、作业和考前冲刺。
1. Measures of Central Tendency | 集中趋势的度量
The three principal measures of central tendency summarise a data set with a single representative value.
三种主要的集中趋势量数用一个代表性数值概括整个数据集。
Mean (x̄): The arithmetic average.
平均数 (x̄)：算术平均值。
x̄ = Σx / n
Where Σx is the sum of all data values and n is the sample size.
其中 Σx 是所有数据值的总和，n 是样本容量。
Median: The middle value when data are ordered.
中位数：将数据排序后位于中间的值。
For odd n: the (n+1)/2-th value. For even n: the average of the n/2-th and (n/2 + 1)-th values.
当 n 为奇数时，取第 (n+1)/2 个值；当 n 为偶数时，取第 n/2 与第 (n/2 +1) 个值的平均数。
Mode: The most frequently occurring value(s).
众数：出现次数最多的数值。
2. Measures of Dispersion | 离散程度的度量
Dispersion describes how spread out the data are. Key measures include the range, interquartile range, and standard deviation.
离散程度描述数据分布的广窄。主要度量包括极差、四分位距和标准差。
Range: maximum − minimum
极差：最大值 − 最小值
Interquartile Range (IQR): IQR = Q₃ − Q₁
四分位距 (IQR)：IQR = Q₃ − Q₁
Where Q₁ is the lower quartile (25th percentile) and Q₃ is the upper quartile (75th percentile).
其中 Q₁ 为下四分位数（第25百分位数），Q₃ 为上四分位数（第75百分位数）。
Semi-interquartile range: (Q₃ − Q₁)/2
四分位差：(Q₃ − Q₁)/2
3. Variance and Standard Deviation | 方差与标准差
Variance and standard deviation measure the average squared deviation from the mean. For a sample, we use n−1 in the denominator to obtain an unbiased estimate.
方差和标准差测量数据与均值之差的平方平均值。样本计算时分母用 n−1 以得到无偏估计。
Sample variance: s² = Σ(x − x̄)² / (n − 1)
Sample standard deviation: s = √[ Σ(x − x̄)² / (n − 1) ]
Alternative computational formula:
另一个便于手工计算的公式：
s = √[ (Σx² − (Σx)²/n) / (n − 1) ]
The standard deviation has the same units as the original data and is always non‑negative.
标准差的单位与原始数据相同，且永远是非负数。
4. Basic Probability Rules | 基本概率法则
Probability measures how likely an event is to occur, with a value between 0 and 1. The sample space S contains all possible outcomes.
概率衡量事件发生的可能性，取值在0到1之间。样本空间 S 包含所有可能的结果。
P(A) = number of outcomes in A / total number of outcomes in S
Addition Rule for mutually exclusive events:
互斥事件的加法法则：
P(A ∪ B) = P(A) + P(B)
If A and B are NOT mutually exclusive:
若 A 与 B 不是互斥事件：
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Multiplication Rule for independent events:
独立事件的乘法法则：
P(A ∩ B) = P(A) × P(B)
Complement Rule: P(not A) = 1 − P(A)
互补事件法则：P(非 A) = 1 − P(A)
5. Conditional Probability | 条件概率
Conditional probability is the probability of an event A occurring given that event B has already occurred. It is fundamental for understanding 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。它是理解相依事件的基础。
P(A | B) = P(A ∩ B) / P(B), provided P(B) > 0
Rearranging gives the general multiplication rule for any two events:
变形后得到适用于任意两个事件的一般乘法法则：
P(A ∩ B) = P(A | B) × P(B) = P(B | A) × P(A)
Two events are independent if and only if P(A | B) = P(A), or equivalently P(A ∩ B) = P(A) × P(B).
两个事件相互独立当且仅当 P(A | B) = P(A)，或等价地 P(A ∩ B) = P(A) × P(B)。
6. Binomial Distribution | 二项分布
A binomial experiment consists of a fixed number of independent trials, each with exactly two possible outcomes (success/failure) and a constant probability of success, p.
二项试验包含固定次数的独立试验，每次试验只有成功与失败两种可能结果，且成功概率 p 保持不变。
Let X be the number of successes in n trials. Then X ~ B(n, p).
设 X 为 n 次试验中成功的次数，则 X ~ B(n, p)。
P(X = r) = ⁿCᵣ × pʳ × (1 − p)ⁿ⁻ʳ
Where ⁿCᵣ = n! / [r!(n − r)!] is the binomial coefficient.
其中 ⁿCᵣ = n! / [r!(n − r)!] 为二项式系数。
The binomial distribution is appropriate when the trials are independent and the probability p is constant.
当试验相互独立且概率 p 恒定时，使用二项分布是合适的。
7. Mean and Variance of Binomial Distribution | 二项分布的期望与方差
For a binomial random variable X ~ B(n, p), the mean (expected value) and variance are given by simple formulas.
对于二项随机变量 X ~ B(n, p)，其均值（期望）与方差有简洁的公式。
Mean: μ = E(X) = np
Variance: σ² = Var(X) = np(1 − p)
Standard deviation: σ = √[np(1 − p)]
These results assume np and n(1−p) are both sufficiently large when using normal approximations (typically both > 5).
这些结果适用于正态近似时要求 np 与 n(1−p) 都足够大（通常都大于5）。
8. Normal Distribution | 正态分布
The normal distribution is a continuous, bell‑shaped curve that is symmetric about its mean μ. It is defined by two parameters: μ (mean) and σ (standard deviation).
正态分布是一种连续的钟形曲线，关于均值 μ 对称。它由两个参数定义：μ（均值）和 σ（标准差）。
Notation: X ~ N(μ, σ²). The total area under the curve equals 1. About 68% of data lie within 1σ of μ, 95% within 2σ, and 99.7% within 3σ (Empirical Rule).
记法：X ~ N(μ, σ²)。曲线下总面积为1。约68%的数据落在 μ±1σ 内，95%在 μ±2σ 内，99.7%在 μ±3σ 内（经验法则）。
The probability that X lies between a and b is given by the area under the curve:
X 落在 a 与 b 之间的概率由曲线下面积给出：
P(a < X < b) = area under N(μ, σ²) from a to b
9. Standard Normal Distribution and Z-Scores | 标准正态分布与 Z 分数
The standard normal distribution has mean 0 and standard deviation 1: Z ~ N(0,1). Any normal variable can be standardised.
标准正态分布的均值为0，标准差为1：Z ~ N(0,1)。任意正态变量都可以标准化。
Z = (X − μ) / σ
This Z-score tells you how many standard deviations X is above or below the mean. It allows you to use standard normal tables to find probabilities.
Z 分数表示 X 距离均值有多少个标准差，正值表示高于均值，负值表示低于均值。利用 Z 分数可查阅标准正态表求概率。
For finding probabilities:
求概率时：
- P(X < a) = P(Z < (a−μ)/σ) | P(X < a) = P(Z < (a−μ)/σ)
- P(X > a) = 1 − P(Z < (a−μ)/σ) | P(X > a) = 1 − P(Z < (a−μ)/σ)
- P(a < X < b) = P(Z < (b−μ)/σ) − P(Z < (a−μ)/σ) | P(a < X < b) = P(Z < (b−μ)/σ) − P(Z < (a−μ)/σ)
10. Confidence Intervals for the Mean | 均值的置信区间
A confidence interval gives a range of plausible values for the population mean μ based on sample data. The formula depends on whether the population standard deviation σ is known.
置信区间给出基于样本数据对总体均值 μ 的合理取值范围。公式取决于总体标准差 σ 是否已知。
When σ is known or the sample size is large (n ≥ 30), use the z‑interval:
当 σ 已知或样本量较大 (n ≥ 30) 时，使用 z 区间：
x̄ ± z* × (σ / √n)
When σ is unknown and the population is approximately normal, use the t‑interval with sample standard deviation s:
当 σ 未知且总体近似正态时，使用 t 区间，以样本标准差 s 代替σ：
x̄ ± t* × (s / √n)
Here z* or t* is the critical value corresponding to the desired confidence level (e.g., 1.96 for 95% confidence using z, or from t‑table with n−1 degrees of freedom).
此处 z* 或 t* 是与置信水平对应的临界值（例如95%置信水平下 z* = 1.96；t* 需查表，自由度为 n−1）。
margin of error is the term after the ± sign. Increasing sample size narrows the interval.
误差界限就是±号后面的项。增大样本量可使区间变窄。
11. Hypothesis Testing for a Mean | 均值的假设检验
Hypothesis testing is a formal procedure to decide whether a sample provides enough evidence to reject a statement about the population mean.
假设检验是一种正式程序，用于判断样本是否提供了足够的证据以拒绝有关总体均值的陈述。
Null hypothesis H₀: μ = μ₀ (the claimed value)
原假设 H₀：μ = μ₀（声称的值）
Alternative hypothesis H₁: μ ≠ μ₀ (two‑tailed), μ > μ₀ (one‑tailed), or μ < μ₀ (one‑tailed).
备择假设 H₁：μ ≠ μ₀（双尾）, μ > μ₀（右侧单尾）或 μ < μ₀（左侧单尾）。
Test statistic when σ is known or n is large:
σ 已知或大样本时的检验统计量：
z = (x̄ − μ₀) / (σ / √n)
When σ is unknown, use the t‑statistic:
σ 未知时使用 t 统计量：
t = (x̄ − μ₀) / (s / √n)
Compare the calculated test statistic with a critical value or use a p‑value. If p‑value < significance level (α, often 0.05), reject H₀.
将计算出的检验统计量与临界值比较，或使用 p 值。若 p 值 < 显著性水平（α，常用0.05），则拒绝 H₀。
12. 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) ranges from −1 to 1.
相关衡量两个变量之间线性关系的强度和方向。积矩相关系数 (r) 的取值范围是 −1 到 1。
r = Σ(xy) − n x̄ ȳ / √[(Σx² − n x̄²)(Σy² − n ȳ²)]
Linear regression finds the equation of the straight line that best fits the data in the sense of least squares:
线性回归寻找一条在最小二乘意义下最佳拟合数据的直线方程：
y = a + bx
Where:
b = Σ(x − x̄)(y − ȳ) / Σ(x − x̄)² = (Σxy − n x̄ ȳ) / (Σx² − n x̄²)
a = ȳ − b x̄
The slope b represents the change in y for a one‑unit increase in x. The coefficient of determination r² denotes the proportion of variation in y explained by x.
斜率 b 表示 x 每增加一个单位时 y 的变化量。判定系数 r² 表示 y 的变异中有多大比例可由 x 解释。
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