📚 Year 11 AQA Statistics: Formulas and Theorems Quick Reference | Year 11 AQA 统计：公式定理速查手册
This quick reference handbook compiles every essential formula and theorem required for Year 11 AQA Statistics. Use it to review definitions, check notation, and memorise the algebraic expressions that will appear throughout your assessments.
本速查手册汇集了 Year 11 AQA 统计学所有核心公式与定理，适用于核对定义、复习符号和掌握考试中频繁出现的代数表达式。
1. Measures of Central Tendency | 集中趋势测量
The sample mean (arithmetic average) summarises the centre of a data set.
样本均值（算术平均）用来概括数据集的中心位置。
x̄ = ∑x / n
where ∑x is the sum of individual observations and n is the sample size. The median is the middle value when data are arranged in order; if n is even, take the average of the two central numbers. The mode is the most frequently occurring value.
其中 ∑x 是所有观测值的总和，n 为样本容量。中位数是有序排列后位于中间位置的数值；若 n 为偶数，则取中间两个数的平均值。众数是出现频次最高的值。
2. Measures of Dispersion | 离散程度测量
Dispersion tells us how spread out the data are. The range is the difference between the largest and smallest values.
离散程度反映数据的分布广度。极差为最大值与最小值之差。
Range = xmax − xmin
The interquartile range (IQR) measures the spread of the middle 50% of observations.
四分位距 (IQR) 衡量中间 50% 数据的分布宽度。
IQR = Q₃ − Q₁
Outliers can be identified as observations less than Q₁ − 1.5 × IQR or greater than Q₃ + 1.5 × IQR. Variance and standard deviation quantify dispersion around the mean. The population variance (when the whole population is available) is:
离群值可被识别为小于 Q₁ − 1.5×IQR 或大于 Q₃ + 1.5×IQR 的观测值。方差和标准差量化了围绕均值的离散程度。总体方差（有整个总体数据时）为：
σ² = ∑(x − μ)² / N
The sample variance uses (n − 1) as divisor to give an unbiased estimate.
样本方差以 (n − 1) 为除数从而给出无偏估计。
s² = ∑(x − x̄)² / (n − 1)
Standard deviation is the square root of the variance.
标准差为方差的正平方根。
s = √[∑(x − x̄)² / (n − 1)]
3. Probability Basics | 概率基础
The addition rule handles probabilities of events that can happen together.
加法法则计算可能同时发生的事件的概率。
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
For mutually exclusive events, P(A ∩ B) = 0. Conditional probability refines a probability given that another event has occurred.
对于互斥事件，P(A ∩ B) = 0。条件概率修正了在另一事件已发生下的概率。
P(A | B) = P(A ∩ B) / P(B), P(B) > 0
The multiplication rule follows directly from this definition.
乘法法则直接由定义导出。
P(A ∩ B) = P(A) P(B | A) = P(B) P(A | B)
Two events are independent if and only if P(A ∩ B) = P(A) P(B).
当且仅当 P(A ∩ B) = P(A) P(B) 时两个事件独立。
4. Discrete Probability Distributions | 离散概率分布
The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success p.
二项分布用于描述在一系列独立试验中成功次数的分布，每次试验成功概率相同为 p。
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) = nCr pr (1−p)n−r
Here nCr is the binomial coefficient, n! / [r! (n−r)!]. The mean and variance of the binomial distribution are:
此处 nCr 为二项系数，即 n! / [r! (n−r)!]。二项分布的期望与方差为：
E(X) = np, Var(X) = np(1−p)
5. Normal Distribution | 正态分布
A continuous random variable that follows a normal distribution is fully described by its mean μ and variance σ².
服从正态分布的连续随机变量完全由均值 μ 和方差 σ² 描述。
X ~ N(μ, σ²)
To find probabilities, we standardise the variable to follow N(0, 1).
为求概率，我们标准化变量使其服从 N(0, 1)。
Z = (X − μ) / σ
The cumulative probability P(Z < z) = Φ(z) is read from standard normal tables. Useful relations are:
累积概率 P(Z < z) = Φ(z) 可从标准正态表查得。常用关系式有：
P(Z > z) = 1 − Φ(z)
P(a < Z < b) = Φ(b) − Φ(a)
To obtain an upper bound given a probability, use inverse normal: z = Φ⁻¹(p).
已知概率求上限值使用逆正态：z = Φ⁻¹(p)。
6. Correlation and Regression | 相关与回归
Pearson’s product-moment correlation coefficient measures the strength and direction of a linear relationship.
皮尔逊积矩相关系数量化线性关系的强度与方向。
r = Sxy / √(Sxx Syy)
where Sxx = ∑(x − x̄)², Syy = ∑(y − ȳ)², and Sxy = ∑(x − x̄)(y − ȳ).
其中 Sxx = ∑(x − x̄)², Syy = ∑(y − ȳ)², Sxy = ∑(x − x̄)(y − ȳ)。
Once a linear association is confirmed, the least squares regression line of y on x is used for prediction.
确认线性关联后，可使用 y 对 x 的最小二乘回归线进行预测。
y = a + b x
The slope and intercept are:
斜率和截距为：
b = Sxy / Sxx, a = ȳ − b x̄
Residuals e = y − ŷ indicate the difference between observed and predicted values. The coefficient of determination R² = r² explains the proportion of variance accounted for by the model.
残差 e = y − ŷ 表示观测值与预测值的差异。决定系数 R² = r² 解释了模型所说明的方差比例。
7. Confidence Intervals | 置信区间
A confidence interval provides a range of plausible values for a population parameter. When the population standard deviation σ is known, the interval for the mean is:
置信区间给出了总体参数的合理取值范围。当总体标准差 σ 已知时，均值的置信区间为：
x̄ ± z* × (σ / √n)
If σ is unknown, the Student’s t-distribution is used with n−1 degrees of freedom.
若 σ 未知，则使用自由度为 n−1 的学生 t 分布。
x̄ ± t* × (s / √n)
For a population proportion p, the approximate confidence interval is:
对于总体比例 p，近似置信区间为：
p̂ ± z* √[p̂(1−p̂) / n]
Common critical z* values: 1.645 for 90%, 1.96 for 95%, and 2.576 for 99% confidence.
常用临界 z* 值：90% 置信水平为 1.645，95% 为 1.96，99% 为 2.576。
8. Hypothesis Testing | 假设检验
A statistical hypothesis test starts by stating a null hypothesis H₀ and an alternative H₁. For a test about the mean, the test statistic depends on whether σ is known.
统计假设检验从建立零假设 H₀ 和备择假设 H₁ 开始。对于均值的检验，检验统计量取决于 σ 是否已知。
z = (x̄ − μ₀) / (σ / √n) or t = (x̄ − μ₀) / (s / √n)
The p‑value is the probability of obtaining a result at least as extreme as the one observed, assuming H₀ is true. If p ≤ α (commonly α = 0.05), we reject H₀ in favour of H₁.
p 值是假定 H₀ 成立时获得至少与观测值同等极端结果的概率。若 p ≤ α（通常 α = 0.05），则拒绝 H₀ 而支持 H₁。
A critical region approach compares the test statistic with critical values corresponding to α. For a two‑tailed test, the rejection region lies in both tails.
临界值方法将检验统计量与 α 对应的临界值比较。双侧检验的拒绝域位于分布的两端。
9. Chi‑Squared Tests | 卡方检验
The chi‑squared statistic is used for goodness‑of‑fit and independence tests. It compares observed frequencies (O) with expected frequencies (E).
卡方统计量用于拟合优度检验和独立性检验，比较观测频数 (O) 与期望频数 (E)。
χ² = ∑ (O − E)² / E
For goodness‑of‑fit, degrees of freedom = k − 1 (or k − p − 1 if p parameters are estimated). For a test of independence in an r × c contingency table:
拟合优度检验的自由度为 k − 1（若估计了 p 个参数，则为 k − p − 1）。r × c 列联表独立性检验：
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