📚 Year 12 SQA Statistics: Formula and Theorem Quick Reference | SQA 12年级统计：公式定理速查手册
This concise handbook brings together the essential formulas and theorems you need for Year 12 SQA Statistics. Use it to reinforce your understanding of data description, probability models, inferential methods, and statistical relationships.
这本简明手册汇集了12年级SQA统计课程必备的公式与定理，可用于巩固你对数据描述、概率模型、推断方法以及统计关系的理解。
1. Descriptive Statistics | 描述性统计
The sample mean measures the centre of a data set.
样本均值衡量数据集的中心位置。
x̄ = Σx / n
The unbiased sample variance and standard deviation are:
无偏样本方差及标准差为：
s² = Σ(x − x̄)² / (n − 1)
s = √s²
The interquartile range (IQR) describes spread resistant to outliers:
四分位数距（IQR）描述对异常值具有抵抗性的离散程度：
IQR = Q₃ − Q₁
| Minimum | Q₁ | Median (Q₂) | Q₃ | Maximum |
2. Probability Theory | 概率论基础
For any two events A and B, the addition rule is:
对于两个事件A和B，加法法则为：
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
If A and B are independent, P(A ∩ B) simplifies to:
若A与B独立，则P(A ∩ B)简化为：
P(A ∩ B) = P(A) × P(B)
Conditional probability and the multiplication rule are linked by:
条件概率与乘法法则通过下式关联：
P(A|B) = P(A ∩ B) / P(B), P(B) > 0
A partition formula often used with Bayes’ theorem is:
常用于贝叶斯定理的全概率公式为：
P(B) = Σ P(B|Aᵢ) P(Aᵢ)
3. Discrete Random Variables | 离散随机变量
The expected value (mean) of a discrete random variable X is:
离散随机变量X的期望值（均值）为：
E(X) = μ = Σ x P(X = x)
Variance and the useful computing formula are:
方差及其便捷计算公式为：
Var(X) = Σ (x − μ)² P(X = x) = E(X²) − [E(X)]²
Linear transformations affect expectation and variance as follows:
线性变换对期望与方差的影响如下：
E(aX + b) = a E(X) + b
Var(aX + b) = a² Var(X)
4. Binomial Distribution | 二项分布
When a fixed number n of independent trials each have success probability p, X ~ B(n, p) and:
当固定次数n的独立试验每次成功概率为p时，X ~ B(n, p)，且：
P(X = r) = ⁿCᵣ pʳ (1 − p)ⁿ⁻ʳ, r = 0,1,…,n
The mean and variance of a binomial count are quick to obtain:
二项计数的均值和方差计算简便：
E(X) = np, Var(X) = np(1 − p)
Conditions for a binomial model: fixed n, independent trials, constant p, two outcomes per trial.
二项模型的条件：固定n、独立试验、恒定的p、每次试验两种结果。
5. Poisson Distribution | 泊松分布
For events occurring independently at a constant average rate λ, X ~ Po(λ) with:
对于以恒定平均率λ独立发生的事件，X ~ Po(λ)，且有：
P(X = x) = e⁻λ λˣ / x!, x = 0,1,2,…
Both the mean and the variance equal the rate parameter:
均值与方差相等，都等于速率参数：
E(X) = λ, Var(X) = λ
The Poisson can approximate a binomial when n is large and p is small:
当n很大而p很小时，泊松分布可近似二项分布：
X ~ B(n, p) ≈ Po(np) (with λ = np, n ≥ 50 and np < 10 often recommended)
6. Normal Distribution | 正态分布
If X ~ N(μ, σ²), the standardised Z-score transforms any normal value:
若X ~ N(μ, σ²)，标准化的Z分数可将任意正态值转换：
Z = (X − μ) / σ, Z ~ N(0, 1²)
Probabilities are found using standard normal tables; for a continuity correction when approximating a discrete distribution:
概率通过标准正态表查找；当逼近离散分布时使用连续性校正：
P(X ≤ k) ≈ P(Z ≤ (k + 0.5 − μ)/σ) (for discrete to normal)
Binomial normal approximation rule of thumb: np ≥ 5 and n(1 − p) ≥ 5.
二项正态近似的经验规则：np ≥ 5 且 n(1 − p) ≥ 5。
7. Sampling & Central Limit Theorem | 抽样与中心极限定理
The distribution of the sample mean X̄ has mean μ and standard error σ/√n:
样本均值X̄的分布具有均值μ和标准误σ/√n：
E(X̄) = μ, SD(X̄) = σ / √n
If the population is normal, X̄ is exactly normal. Otherwise, the Central Limit Theorem states that for large n (usually n ≥ 30), X̄ is approximately normal:
若总体为正态，X̄精确正态；否则，中心极限定理指出，对于大样本（通常n ≥ 30），X̄近似正态：
X̄ ~ N(μ, σ²/n) approximately
For a sample proportion p̂ from a binomial population, its distribution under repeated sampling satisfies:
对于来自二项总体的样本比例p̂，其重复抽样分布满足：
E(p̂) = p, Var(p̂) = p(1 − p)/n, p̂ ≈ N
8. Confidence Intervals | 置信区间
A 100(1 − α)% confidence interval for the population mean μ (σ known) is:
总体均值μ（σ已知）的100(1 − α)%置信区间为：
x̄ ± z* × (σ / √n)
When σ is unknown and the sample size is small (or population is normal), use the t-distribution with degrees of freedom n − 1:
当σ未知且样本量较小（或总体正态）时，使用自由度为n − 1的t分布：
x̄ ± t*ₙ₋₁ × (s / √n)
Confidence interval for a population proportion p (large sample) is given by:
总体比例p的大样本置信区间为：
p̂ ± z* √[p̂(1 − p̂) / n]
| Confidence level | 90% | 95% | 99% |
| z* | 1.645 | 1.960 | 2.576 |
9. Hypothesis Testing | 假设检验
The null hypothesis H₀ and alternative hypothesis H₁ are statements about a parameter. The test statistic for a mean (σ known) is:
原假设H₀与备择假设H₁是关于参数的陈述。均值（σ已知）的检验统计量为：
z = (x̄ − μ₀) / (σ / √n)
When σ is unknown, use the one-sample t-test:
当σ未知时，使用单样本t检验：
t = (x̄ − μ₀) / (s / √n), df = n − 1
For a binomial test, the p-value is calculated using the exact binomial probability. The decision rule: if p-value ≤ α, reject H₀.
对于二项检验，p值通过精确二项概率计算。决策规则：若p值 ≤ α，则拒绝H₀。
10. Correlation & Regression | 相关与回归
Pearson’s product-moment correlation coefficient r assesses linear association:
皮尔逊积矩相关系数r衡量线性关联程度：
r = Σ (xᵢ − x̄)(yᵢ − ȳ) / √[ Σ (xᵢ − x̄)² × Σ (yᵢ − ȳ)² ]
The least-squares regression line of y on x is y = a + bx, where:
y对x的最小二乘回归线为y = a + bx，其中：
b = Σ (xᵢ − x̄)(yᵢ − ȳ) / Σ (xᵢ − x̄)²
a = ȳ − b x̄
The coefficient of determination R² = r² tells us the proportion of variance in y explained by x.
决定系数R² = r²告诉我们y中被x解释的方差比例。
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