📚 Year 11 AQA Statistics: Formula & Theorem Quick-Reference Handbook | 11年级 AQA 统计：公式定理速查手册
This concise handbook brings together all the essential formulas and theorems you will meet in Year 11 AQA Statistics. Use it for quick revision, homework checks, and last-minute exam practice. Every entry is presented first in English, followed immediately by its Chinese translation, so you can switch between languages without losing the thread.
这本简明手册汇集了 11 年级 AQA 统计课程中所有必须掌握的公式和定理。它可以用于快速复习、作业核查和考前冲刺。每个条目都先给出英文说明，紧接着是中文翻译，让你在双语切换时不中断思路。
1. Measures of Central Tendency | 集中趋势度量
The arithmetic mean of a data set is the sum of all values divided by the number of observations. It is the most commonly used average.
数据集的算术平均值是所有数值之和除以观测个数。它是使用最广泛的一种平均数。
x̄ = Σx / n
When data are grouped into a frequency table, the mean is calculated by multiplying each midpoint by its frequency, summing those products and dividing by the total frequency.
当数据以频数表分组时，平均值等于每个组中值乘以对应频数之和再除以总频数。
x̄ = Σfx / Σf
The median is the middle value when the data are arranged in order. For an odd number of observations, it is the (n+1)/2 th value; for an even number, it is the average of the n/2 th and (n/2)+1 th values.
中位数是将数据排序后位于正中间的数值。当观测数 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。
The mode is the value or category that occurs most frequently. A data set can have one mode (unimodal), two modes (bimodal), or more.
众数是出现频率最高的数值或类别。数据集可以有一个众数（单峰）、两个众数（双峰）或更多。
A weighted mean gives different importance to different observations. It is useful when combining means from groups of unequal size.
加权平均数对不同观测赋予不同的重要性，特别适合合并大小不等的组别平均数。
Weighted mean = Σwx / Σw
2. Measures of Dispersion | 离散程度度量
The range is the simplest measure of spread: it is the difference between the largest and smallest values in the data set.
极差是最简单的离散度量，等于数据中最大值与最小值之差。
Range = xₘₐₓ − xₘᵢₙ
The interquartile range (IQR) measures the spread of the middle 50% of the data. It is the difference between the upper quartile (Q₃) and the lower quartile (Q₁).
四分位距 (IQR) 衡量中间 50% 数据的离散程度，等于上四分位数 (Q₃) 与下四分位数 (Q₁) 之差。
IQR = Q₃ − Q₁
The lower quartile is the ¼(n+1) th value when data are ordered; the upper quartile is the ¾(n+1) th value. In cumulative frequency diagrams, quartiles can be read directly from the curve.
下四分位数位于排序数据的 ¼(n+1) 位置，上四分位数位于 ¾(n+1) 位置。在累积频率图中可直接从曲线上读取四分位数。
Variance and standard deviation quantify how far the data points are spread out from the mean. The standard deviation is the square root of the variance. When working with a sample, we divide by (n − 1) to obtain an unbiased estimate.
方差和标准差度量数据点偏离平均值的程度。标准差是方差的平方根。处理样本时，除以 (n − 1) 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)]
For a frequency distribution, the formula becomes s² = Σf(x − x̄)² / (Σf − 1) when using grouped data.
对于频数分布，使用分组数据时公式变为 s² = Σf(x − x̄)² / (Σf − 1)。
3. Data Representation & Frequency Density | 数据表示与频数密度
In a histogram for unequal class widths, the height of each bar is given by frequency density, not by frequency. This ensures that the area of the bar is proportional to the frequency.
在组距不相等的直方图中，每一条的高度由频数密度而不是频数决定，以保证条形的面积与频数成正比。
Frequency density = Frequency / Class width
To estimate the median from a cumulative frequency graph, find the position on the y-axis at 50% of the total frequency, draw a horizontal line to the curve, and then a vertical line down to the x-axis.
从累积频率图估计中位数时，在 y 轴上找到总频数的 50% 位置，作水平线与曲线相交，再向下作垂线到 x 轴。
For a grouped frequency table, the modal class is the class with the highest frequency density. The class containing the median is found by locating the (n/2)th observation in a cumulative frequency column.
对于分组频数表，模态组是频数密度最高的组；通过累积频数列找到第 (n/2) 个观测值所在的组即为中位数组。
4. Probability Rules | 概率法则
The probability of an event A, denoted P(A), always lies between 0 and 1 inclusive. A value of 0 means the event is impossible; a value of 1 means it is certain.
事件 A 的概率 P(A) 始终在 0 到 1 之间（含两端）。0 表示事件不可能发生，1 表示事件必然发生。
= P(A) + P(B). When events are not mutually exclusive, the general addition rule is used.
对于互斥事件 A 和 B（不能同时发生的事件），加法法则成立：P(A 或 B) = P(A) + P(B)。当事件不互斥时，使用一般加法法则。
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
The conditional probability of A given B is the probability that A occurs, knowing that B has already occurred. It is defined only when P(B) > 0.
在事件 B 已发生的条件下事件 A 发生的条件概率，记为 P(A|B)，仅在 P(B) > 0 时有定义。
P(A | B) = P(A ∩ B) / P(B)
Two events A and B are independent if the occurrence of one does not affect the probability of the other. For independent events, the multiplication rule simplifies.
如果两个事件 A 和 B 的发生互不影响，则它们独立。对于独立事件，乘法法则可以简化。
P(A ∩ B) = P(A) × P(B) (if A and B are independent)
Independence can be tested by checking whether P(A|B) = P(A) or P(B|A) = P(B).
可以通过检验 P(A|B)=P(A) 或 P(B|A)=P(B) 来判断独立性。
5. Discrete Probability Distributions | 离散概率分布
The expected value (or mean) of a discrete random variable X is the long-run average of its outcomes. It is a weighted average of all possible values, each weighted by its probability.
离散随机变量 X 的期望值（或均值）是其长期平均值，等于所有可能取值与其概率乘积之和。
E(X) = Σx·P(X = x)
ar(X) = E[(X − E(X))²], which is equivalent to E(X²) − [E(X)]².
随机变量 X 的方差衡量分布的离散程度，定义为 Var(X) = E[(X − E(X))²]，也等价于 E(X²) − [E(X)]²。
Var(X) = Σ(x − E(X))²·P(X = x) = E(X²) − [E(X)]²
The binomial distribution B(n, p) models the number of successes in n independent trials, each with the same probability of success p. The probability of obtaining exactly k successes is given by the binomial formula.
二项分布 B(n, p) 描述在 n 次独立试验中成功次数的分布，每次试验成功概率为 p。恰好得到 k 次成功的概率由二项式公式给出。
P(X = k) = ⁿCₖ pᵏ (1 − p)ⁿ⁻ᵏ
For a binomial random variable, the mean and variance have simple closed forms.
对于二项随机变量，其均值和方差有简洁的表达式。
E(X) = np
Var(X) = np(1 − p)
6. The Normal Distribution | 正态分布
The normal distribution N(μ, σ²) is a continuous, bell-shaped distribution that is symmetric about its mean μ. Its shape is determined entirely by μ and the standard deviation σ.
正态分布 N(μ, σ²) 是一个连续、钟形且关于均值 μ 对称的分布，其形状完全由 μ 和标准差 σ 决定。
To find probabilities for any normal distribution, we first convert the value x into a z-score, which tells us how many standard deviations x is away from the mean.
要计算任意正态分布的概率，首先将 x 转化为 z 分数，它表示 x 距离均值多少个标准差。
z = (x − μ) / σ
can be looked up in the standard normal table, which has mean 0 and standard deviation 1.
标准化后，P(X ≤ x) = P(Z ≤ z) 可通过均值为 0、标准差为 1 的标准正态分布表查询。
Z ~ N(0, 1²)
To find an unknown mean or standard deviation given a probability, rearrange the z-score formula and use the inverse normal table.
若已知概率求未知均值和标准差，可重新排列 z 分数公式并利用反查标准正态分布表。
7. Sampling & Central Limit Theorem | 抽样与中心极限定理
A sample is a subset of a population. The way the sample is selected influences the reliability of any conclusions drawn. Random sampling gives each member of the population an equal chance of being chosen.
样本是总体的一个子集。样本的选取方式直接影响结论的可靠性。随机抽样使每个成员有相同机会被选入样本。
The sample mean x̄ is a random variable; its standard deviation is called the standard error. If the population standard deviation σ is known, the standard error of the mean is σ/√n.
样本均值 x̄ 是一个随机变量，其标准差称为标准误。若总体标准差 σ 已知，均值的标准误为 σ/√n。
SE(x̄) = σ / √n
The Central Limit Theorem (CLT) states that, for a sufficiently large sample size (typically n ≥ 30), the sampling distribution of the mean is approximately normal, regardless of the shape of the population distribution.
中心极限定理指出，当样本量足够大（通常 n ≥ 30）时，无论总体分布形状如何，样本均值的抽样分布近似正态。
x̄ ~ N(μ, σ²/n) (approximately for large n)
8. Confidence Intervals | 置信区间
A confidence interval gives a range of plausible values for an unknown population parameter. The level of confidence, usually 95%, describes how sure we are that the interval contains the true parameter.
置信区间提供了未知总体参数可能取值的范围。置信水平（通常为 95%）表示该区间包含真实参数的把握程度。
When the population standard deviation σ is known, a confidence interval for the population mean μ is constructed using the z-distribution.
当总体标准差 σ 已知时，总体均值 μ 的置信区间使用 z 分布构造。
x̄ ± z* × (σ/√n)
In most real situations σ is unknown and we must use the sample standard deviation s. Then the t-distribution with n − 1 degrees of freedom provides the correct multiplier t*.
在大多数实际情况中 σ 未知，必须使用样本标准差 s，此时应使用具有 n − 1 自由度的 t 分布给出乘数 t*。
x̄ ± t* × (s/√n)
For a population proportion p, the confidence interval is based on the sample proportion p̂ and the normal approximation.
对于总体比例 p，其置信区间基于样本比例 p̂ 和正态近似。
p̂ ± z* × √[ p̂(1 − p̂) / n ]
9. Hypothesis Testing | 假设检验
A hypothesis test is a formal procedure for deciding between two competing claims about a population parameter. The null hypothesis H₀ represents the status quo, while the alternative hypothesis H₁ represents the challenge we wish to test.
假设检验是在关于总体参数的两种对立主张之间做决定的正式程序。零假设 H₀ 代表现状，备择假设 H₁ 代表我们想要检验的挑战。
The test statistic is calculated from the sample data and measures how far the observed outcome is from what would be expected if H₀ were true. A p-value is the probability of obtaining a test statistic at least as extreme as the one observed, assuming H₀ is true.
检验统计量由样本数据计算得出，衡量观测结果与 H₀ 成立时期望值之间的差距。p 值是在 H₀ 为真时得到至少与观测结果一样极端的检验统计量的概率。
A decision is made by comparing the p-value with the significance level α (often 5% or 0.05). If p-value ≤ α, we reject H₀ in favour of H₁; otherwise, we fail to reject H₀.
将 p 值与显著性水平 α（通常为 5% 或 0.05）比较作出决定：若
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