📚 GCSE AQA Statistics: Formula and Theorem Quick Reference Handbook | GCSE AQA 统计：公式定理速查手册
This article provides a concise, bilingual summary of the most important formulas, theorems and statistical techniques required for the AQA GCSE Statistics specification. Each section presents the core concept in English, immediately followed by its Chinese translation, making it easy to revise key definitions, notation and calculation methods. Whether you are preparing for an exam or consolidating your understanding, this handbook aims to put all the essential formulae at your fingertips.
本文为 AQA GCSE 统计课程提供了一份简洁的中英双语速查手册，涵盖了考试所需的重要公式、定理和统计方法。每个部分先用英文陈述核心概念，随后紧跟中文翻译，便于你复习关键定义、符号和计算过程。无论你是在备考冲刺还是巩固知识，这本手册都能让你快速查阅所有核心公式。
1. Measures of Central Tendency | 集中趋势度量
For an ungrouped set of data values x₁, x₂, …, xₙ, the mean is given by x̄ s and n is the number of values. The median is the middle value when the data are ordered; for an even number of observations it is the mean of the two central values. The mode is the value that occurs most frequently.
对于未分组数据 x₁, x₂, …, xₙ，均值的计算公式为 x̄ = Σx / n，其中 Σx 是所有数值的总和，n 是数据的个数。中位数是将数据排序后位于中间位置的数值；当观测个数为偶数时，中位数是中间两个数的均值。众数是出现频率最高的数值。
d data presented in a frequency table, the estimated mean is calculated using midpoints: x̄ oint and f is the frequency of each class. The modal class is the class with the highest frequency.
对于以频数表给出的分组数据，估算均值需要用到组中值：x̄ = Σ(f × x) / Σf，其中 x 是各组的组中值，f 是对应的频数。众数所在的组是频数最高的组别。
Mean: x̄ = Σx / n Grouped mean: x̄ = Σfx / Σf
均值：x̄ = Σx / n 分组均值：x̄ = Σfx / Σf
2. Quartiles and Percentiles | 四分位数与百分位数
ough. The interquartile range (IQR) = Q₃ − Q₁, and it measures the spread of the middle 50% tiles divide the data into 100 equal parts; the k-th percentile is the value below which k% of the data falls.
下四分位数 Q₁ 是排序数据四分之一处的位置值；上四分位数 Q₃ 是四分之三处的位置值。四分位距 IQR = Q₃ − Q₁，反映了中间 50% 数据的散布程度。百分位数将数据分为 100 等份；第 k 百分位数是指有 k% 的数据落在它以下的值。
For discrete data, the position of Q₁ is given by (n+1)/4 and Q₃ by 3(n+1)/4; linear interpolation may be used for grouped continuous data.
对于离散数据，Q₁ 的位置由 (n+1)/4 给出，Q₃ 的位置由 3(n+1)/4 给出；对于连续型分组数据，可采用线性插值法计算四分位数。
IQR = Q₃ − Q₁
3. Measures of Dispersion | 离散程度的度量
lue. The variance measures the average squared deviation from the mean. For a sample of data, the standard deviation s is the square root of the variance:
极差 = 最大值 − 最小值。方差衡量的是各数据与均值之差的平方的平均值。对于样本数据，标准差 s 是方差的平方根：
s = √[ Σ(x − x̄)² / (n−1) ]
When data are presented in a frequency table, the formula becomes s = √[ Σf(x − x̄ the table as a sample. In some GCSE contexts the divisor Σf may be used for a population.
当数据通过频数表给出时，若视为样本，公式变为 s = √[ Σf(x − x̄)² / (Σf − 1) ]。在部分 GCSE 场景中，若视为总体，分母可使用 Σf。
The standard deviation is expressed in the same units as the original data, making it more interpretable than variance.
标准差与原始数据的单位相同，因此比方差更容易解释。
4. Graphical Representations and Box Plots | 图形表示与箱线图
kewness. The left whisker extends from the minimum to Q₁, and the right whisker from Q₃ to the maximum; outliers may be plotted separately if the 1.5 × IQR rule is applied.
箱线图（盒子须图）显示了最小值、Q₁、中位数、Q₃ 和最大值。它有助于比较分布和识别偏态。左须从最小值延伸至 Q₁，右须从 Q₃ 延伸至最大值；若使用 1.5 × IQR 法则，离群值可单独标出。
Cumulative frequency diagrams plot the running total of frequencies against the upper class boundaries. The median and quartiles can be read from the graph by drawing horizontal lines at the appropriate cumulative frequencies.
累积频数图将频数的累加和与组距上限对应绘制。通过在相应累积频数处画水平线，可从图中读出中位数和四分位数。
5. Basic Probability | 基础概率
r of outcomes. For any event A, 0 ≤ P(A) ≤ 1. The probability that A does not occur is P(A') = 1 − P(A).
如果随机试验的所有结果等可能发生，则事件 A 的概率为 P(A) = 有利结果数 / 总结果数。对于任意事件 A，0 ≤ P(A) ≤ 1。A 不发生的概率为 P(A') = 1 − P(A)。
If two events A and B are mutually exclusive, they cannot occur at the same time and P(A ∪ B) = P(A) + P(B). If events are independent, P(A ∩ B) = P(A) × P(B).
若事件 A 和 B 互斥，则它们不能同时发生，且 P(A ∪ B) = P(A) + P(B)。若事件相互独立，则 P(A ∩ B) = P(A) × P(B)。
P(A') = 1 − P(A) P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
6. Conditional Probability | 条件概率
The probability of event A given that event B has occurred is written P(A | B) and is calculated as P(A | B) = P(A ∩ he light of new information and is often illustrated using tree diagrams or Venn diagrams.
在事件 B 已发生的条件下事件 A 发生的概率写作 P(A | B)，计算公式为 P(A | B) = P(A ∩ B) / P(B)，前提是 P(B) > 0。条件概率帮助我们根据新信息修正预测，常用树状图或韦恩图来展示。
For dependent events, the general multiplication rule is P(A ∩ B) = P(A) × P(B | . This is essential when performing calculations involving selections without replacement.
对于不独立的事件，一般乘法规则为 P(A ∩ B) = P(A) × P(B | A)。当涉及不放回抽取时，这一规则至关重要。
7. 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. If X ~ B(n, p), then the probability of exactly r successes is:
二项分布用于描述在固定次数的独立试验中成功的次数，每次试验成功的概率均为 p。若 X ~ B(n, p)，则恰好获得 r 次成功的概率为：
P(X = r) = nCr pr qn−r where q = 1 − p
The notation nCr represents the binomial coefficient, which can be calculated using a calculator or by nCr = n! / [r!(n−r)!]. The expected number of successes is E(X) = np, and the variance is Var(X) = npq.
符号 nCr 表示二项式系数，可用计算器或公式 nCr = n! / [r!(n−r)!] 求出。成功的期望次数为 E(X) = np，方差为 Var(X) = npq。
8. Normal Distribution | 正态分布
The normal distribution is a continuous probability distribution that is symmetric and bell‑shaped. It is fully described by its mean μ and standard deviation σ. Approximate proportions of observations within 1, 2 and 3 standard deviations of the mean are 68%, 95% and 99.7% respectively.
正态分布是一种对称的钟形连续概率分布，完全由其均值 μ 和标准差 σ 决定。大约有 68% 的观测值落在均值 ±1 个标准差范围内，95% 落在 ±2 个标准差内，99.7% 落在 ±3 个标准差内。
To find probabilities for a normal variable X ~ N(μ, σ²), the standardised value z is calculated, measuring how many standard deviations the observation is from the mean. z-tables are then used to obtain probabilities.
要求正态变量 X ~ N(μ, σ²) 的概率，需要计算标准值 z，它衡量了观测值与均值之间相差几个标准差。然后使用 z 表查找概率。
z = (X − μ) / σ
9. Scatter Diagrams and Correlation | 散点图与相关性
A scatter diagram shows the relationship between two variables. Positive correlation means that as one variable increases, the other tends to increase; negative correlation means that as one increases, the other tends to decrease. The strength of linear association can be measured by Spearman's rank correlation coefficient rs when a monotonic relationship is suspected.
散点图展示两个变量之间的关系。正相关意味着一个变量增加时，另一个也倾向于增加；负相关则意味着一个增加时另一个减少。当怀疑存在单调关系时，可用斯皮尔曼等级相关系数 rs 衡量线性关联的强度。
rs = 1 − (6 Σd2) / [n(n2 − 1)]
Here d is the difference between the ranks of each pair of observations, and n is the number of data pairs. The value of rs s close to ±1 indicate strong correlation, while values near 0 suggest little correlation.
式中 d 是每对观测值排序后的等级差，n 是数据对的个数。rs 的取值始终在 −1 到 +1 之间；接近 ±1 表示强相关，接近 0 表示几乎没有相关。
10. Probability Distributions and Expected Value | 概率分布与期望值
robability. The sum of all probabilities must equal 1. The expected value E(X) is the long‑run average outcome and is calculated as E(X) = Σ[x · P(X = x)].
离散概率分布列出了随机变量的每一个可能取值及其对应的概率。所有概率之和必须等于 1。期望值 E(X) 是长期的平均结果，计算公式为 E(X) = Σ[x · P(X = x)]。
(X)) = Σ[g(x) · P(X = x)]. The variance can also be expressed as Var(X) = E(X²) − [E(X)]².
对于 X 的函数，期望值为 E(g(X)) = Σ[g(x) · P(X = x)]。方差也可表示为 Var(X) = E(X²) − [E(X)]²。
11. Index Numbers | 指数
s given by (current price / base price) × 100. Weighted index numbers, such as the retail price index, combine several items, giving more importance to those with greater weights.
指数用于衡量某个量随时间发生的相对变化。基期指数通常设定为 100。单一商品的价格指数为（现价 / 基价）× 100。加权指数（例如零售物价指数）综合多种商品，给予权重更大的商品更高的比重。
d directly from the indices: percentage change = (new index − old index) / old index × 100%.
两个时期之间的百分比变化可直接由指数求出：百分比变化 =（新指数 − 旧指数）/ 旧指数 × 100%。
12. Quality Assurance and Control | 质量保证与控制
limits (often at ±2σ) and action limits (often at ±3σ). Data points outside the action limits, or unusual patterns within the limits, suggest the process may be out of control.
质量管理图用于工业中监控过程随时间的变化。它们通常包含中心线（目标值）、上/下警戒限（常设在 ±2σ）和行动限（常设在 ±3σ）。超出行动限的数据点，或者在界限内出现异常模式，都表明过程可能失控。
When interpreting control charts, students should be able to distinguish between 'common cause' variation (natural, expected) and 'special cause' variation (unusual, worthy of investigation).
在解读控制图时，学生应能区分“普通原因”变异（自然的、预期的）和“特殊原因”变异（异常的、值得调查的）。
Published by TutorHao | Statistics Revision Series | aleveler.com
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