Year 12 AQA Statistics: Formula & Theorem Quick Reference | AQA Year 12 统计公式定理速查手册

📚 Year 12 AQA Statistics: Formula & Theorem Quick Reference | AQA Year 12 统计公式定理速查手册

This quick-reference handbook summarises the essential formulas and theorems required for the Year 12 AQA Statistics module. Use it to reinforce your memory and clarify key concepts before assessments.

本速查手册总结了 Year 12 AQA 统计学模块所需的核心公式和定理，可用于考前强化记忆、理清关键概念。

1. Measures of Central Tendency | 集中趋势度量

The sample mean for raw data x₁, x₂, …, xₙ is the sum of all observations divided by the number of observations. For grouped frequency data, multiply each class’s midpoint by its frequency.

对于原始数据 x₁, x₂, …, xₙ，样本均值为所有观测值之和除以观测值个数。对于分组频率数据，将每一组的组中值乘以频数再求和。

x̄ = (∑xᵢ)/n for grouped data x̄ = (∑fx)/∑f

The median is the middle value when data are ordered. If n is odd, the median is the (n+1)/2 th value. For grouped data, linear interpolation within the median class is used.

中位数是排序后居中的数值。若 n 为奇数，中位数为第 (n+1)/2 个数据。对于分组数据，需在中位数所在组内使用线性插值。

The mode is the most frequently occurring value. A data set may have no mode, one mode, or several modes.

众数是出现次数最多的数值。数据可能没有众数、有一个众数或多个众数。

2. Measures of Dispersion | 离散程度度量

The range is the difference between the largest and smallest values. The interquartile range (IQR) measures the spread of the middle 50% of data.

极差是最大值与最小值之差。四分位距 (IQR) 衡量中间 50% 数据的分散程度。

Range = max – min IQR = Q₃ – Q₁

Variance and standard deviation quantify how much the data deviate from the mean. For a population we use σ² and σ; for a sample we often use s² and s, dividing by n−1.

方差和标准差量化了数据偏离均值的程度。总体使用 σ² 和 σ，样本则常用 s² 和 s，分母为 n−1。

σ² = (∑(xᵢ – μ)²)/N s² = (∑(xᵢ – x̄)²)/(n – 1)

A useful computational form for the sum of squares Sxx avoids repeated subtraction of the mean.

一个便于计算的平方和 Sxx 公式可以避免重复减去均值。

Sxx = ∑(xᵢ – x̄)² = ∑xᵢ² – (∑xᵢ)²/n

The standard deviation is simply the positive square root of the variance.

标准差就是方差的正平方根。

s = √s² σ = √σ²

3. Probability Basics | 概率基础

For an experiment with equally likely outcomes, the probability of an event A is the number of favourable outcomes divided by the total number of outcomes. Probabilities are always between 0 and 1.

对于等可能结果的随机试验，事件 A 的概率等于有利结果数除以所有可能结果总数。概率值总在 0 和 1 之间。

P(A) = n(A)/n(S)

The addition rule handles the union of events. If events are mutually exclusive, their intersection is empty.

加法法则处理事件的并集。若事件互斥，则它们的交集为空。

P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

If mutually exclusive: P(A ∪ B) = P(A) + P(B)

The complement of A contains all outcomes not in A. Its probability provides a handy shortcut.

A 的补集包含所有不在 A 中的结果，利用补集概率可以简化计算。

P(A’) = 1 – P(A)

4. Conditional Probability & Independence | 条件概率与独立性

Conditional probability is the probability of event B occurring given that A has already occurred. It is fundamental for updating probabilities with new information.

条件概率是在事件 A 已发生的条件下事件 B 发生的概率，它是根据新信息更新概率的基础。

P(B|A) = P(A ∩ B) / P(A), P(A) > 0

The multiplication rule rearranges the conditional formula and is especially useful for multi-stage experiments and tree diagrams.

乘法法则由条件概率公式变形得到，在处理多阶段试验和概率树时特别有用。

P(A ∩ B) = P(A) × P(B|A) = P(B) × P(A|B)

Two events are independent if the occurrence of one does not affect the probability of the other. This can be tested with either the intersection condition or the conditional condition.

若一个事件的发生不影响另一个事件发生的概率，则两事件独立。可以用交集条件或条件概率条件来检验。

A and B independent ⟺ P(A ∩ B) = P(A) × P(B) or P(B|A) = P(B)

5. Discrete Random Variables | 离散随机变量

A discrete random variable takes a countable number of values, each with an assigned probability. The probabilities must sum to 1.

离散随机变量取值为可数个，每个取值对应一个概率，所有概率之和必为 1。

∑P(X = x) = 1

The expected value E(X) is the probability-weighted average of the possible values, often seen as the long-run average.

期望值 E(X) 是以概率为权的加权平均值，可理解为长期平均。

E(X) = ∑x P(X = x)

The expected value of the square, E(X²), is needed to compute variance efficiently.

需要先计算平方的期望 E(X²)，才能高效求得方差。

E(X²) = ∑x² P(X = x)

Variance measures the spread of the distribution. It can be found from the mean of the squares minus the square of the mean.

方差度量分布的离散程度，可由平方的期望减去期望的平方得到。

Var(X) = E(X²) – [E(X)]²

The standard deviation of X is the square root of the variance.

X 的标准差是方差的平方根。

σ = √Var(X)

6. Binomial Distribution | 二项分布

A binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success, p. The conditions are:

Fixed number of trials, n
Each trial is independent
Only two outcomes (success/failure)
Constant probability of success, p

二项分布描述了在固定次数的独立试验中成功的次数，每次试验的成功概率 p 保持不变。适用条件为：

试验次数 n 固定
各次试验相互独立
每次试验只有两种结果（成功/失败）
每次成功的概率 p 恒定

If X ~ B(n, p), the probability of obtaining exactly r successes is given by the binomial probability formula. The binomial coefficient ⁿCᵣ counts the combinations of n items taken r at a time.

若 X ~ B(n, p)，则恰好获得 r 次成功的概率由二项概率公式给出。二项系数 ⁿCᵣ 表示从 n 个元素中选取 r 个的组合数。

P(X = r) = ⁿCᵣ pʳ qⁿ⁻ʳ, where q = 1 – p

ⁿCᵣ = n! / [r!(n – r)!]

The expected value and variance for a binomial random variable are simple multiples of n, p, and q.

二项随机变量的期望值和方差是 n、p 和 q 的简单乘积形式。

E(X) = np Var(X) = npq σ = √(npq)

7. Normal Distribution | 正态分布

The normal distribution is a continuous probability distribution with a symmetric bell-shaped curve defined by its mean μ and standard deviation σ. The total area under the curve is 1.

正态分布是一种连续型概率分布，其对称的钟形曲线由均值 μ 和标准差 σ 确定。曲线下总面积为 1。

X ~ N(μ, σ²)

To find probabilities, any normal variable is transformed to the standard normal Z with mean 0 and standard deviation 1. The Z-score tells how many standard deviations a value is from the mean.

为求概率，可将任意正态变量转化为均值为 0、标准差为 1 的标准正态变量 Z。Z 值表示某个数值距离均值多少个标准差。

Z = (X – μ) / σ

Probabilities are then read from the standard normal table. Common intervals include roughly 68% within 1σ, 95% within 2σ, and 99.7% within 3σ of the mean.

随后通过标准正态表读取概率。常见的区间为：均值 ± 1σ 内约占 68%，± 2σ 内约占 95%，± 3σ 内约占 99.7%。

When summing or differencing independent normal variables, the resulting variable is also normally distributed. Means add or subtract, while variances always add.

相互独立的正态变量之和差仍服从正态分布。均值直接相加减，方差总是相加。

If X~N(μₓ, σₓ²) and Y~N(μᵧ, σᵧ²) independent, then

X ± Y ~ N(μₓ ± μᵧ, σₓ² + σᵧ²)

8. Correlation & Regression | 相关与回归

The product moment correlation coefficient r measures the strength and direction of a linear relationship between two variables. It always lies between –1 and 1.

积矩相关系数 r 衡量两个变量之间线性关系的强度和方向，其值总是在 –1 到 1 之间。

r = Sxy / √(Sxx Syy)

Before computing r, three sums of squares and products are required. They can be calculated efficiently using raw sums.

在计算 r 之前，需要先求出三个平方和与乘积和。可使用原始和高效计算。

Sxx = ∑xᵢ² – (∑xᵢ)²/n

Syy = ∑yᵢ² – (∑yᵢ)²/n

Sxy = ∑xᵢyᵢ – (∑xᵢ)(∑yᵢ)/n

The least squares regression line of y on x minimises the sum of squared vertical distances from the points to the line. It is used to predict y from a given x.

y 对 x 的最小二乘回归线使各点到直线的纵向距离平方和最小，用于由 x 预测 y。

y = a + bx where b = Sxy / Sxx, a = ȳ – b x̄

Be cautious with extrapolation: predictions are reliable only within or close to the range of observed data.

注意外推风险：仅在观测数据范围内或附近进行预测才是可靠的。

9. Linear Functions of a Random Variable | 随机变量的线性变换

Adding a constant shifts the expected value by that constant but leaves the variance unchanged. Multiplying by a constant multiplies the expected value and multiplies the variance by the square of that constant.

加常数会使期望值偏移该常数，但方差不改变。乘以常数会使期望值乘以该常数，方差则乘以该常数的平方。

E(aX + b) = aE(X) + b

Var(aX + b) = a²Var(X)

These rules are vital when transforming data into standardised scores or when applying linear models. Remember that adding b shifts the centre but does not affect dispersion.

这些法则是将数据转换为标准得分或应用线性模型时的基础。务必记住加 b 仅改变中心位置，不影响离散程度。

If we have two independent random variables X and Y, the expectation of their sum is the sum of expectations, and the variance of their sum is the sum of variances. Independence is crucial for the variance addition.

若有两个独立的随机变量 X 与 Y，它们之和的期望为期望之和，和的方差

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Year 12 AQA Statistics: Formula & Theorem Quick Reference | AQA Year 12 统计公式定理速查手册