📚 Year 11 Eduqas Statistics: Formula and Theorem Quick Reference Guide | Year 11 Eduqas 统计：公式定理速查手册
This guide provides a rapid, at-a-glance summary of the essential formulas and theorems you need for Year 11 Eduqas Statistics. Each section presents the key concepts with brief English explanations followed immediately by Chinese translations, making it an ideal tool for revision, homework, and last-minute exam preparation.
本指南为你快速总览 Year 11 Eduqas 统计课程中的核心公式与定理。每个小节都先用英文简要解释关键概念，紧接着给出中文翻译，便于复习、完成作业以及考前冲刺使用。
1. Measures of Central Tendency | 集中趋势度量
The arithmetic mean of a sample is calculated by summing all data values and dividing by the number of observations.
样本的算术平均数通过将所有数据值相加后除以观察值的个数计算得出。
x̄ = Σx / n
The median is the middle value when the data are arranged in order. For an even number of observations, it is the average of the two central numbers.
中位数是将数据排序后位于中间位置的数值。当观察值个数为偶数时，中位数是中间两个数的平均值。
The mode is the value that occurs most frequently in a data set. A set can have one mode, more than one mode, or no mode at all.
众数是数据集中出现频率最高的值。一个集合可能有一个众数、多个众数或者没有众数。
For grouped data, the modal class is the class interval with the highest frequency, and the median is estimated using linear interpolation.
对于分组数据，众数所在的组是频率最高的组别，中位数则通过线性插值法进行估计。
Estimated Median = L + [(n/2 − F) / f] × w
2. Measures of Dispersion | 离散度量
The range is the difference between the largest and smallest observations. It is the simplest measure of spread.
极差是最大值与最小值之差，是最简单的离散程度度量。
Range = Max − Min
The interquartile range (IQR) is the difference between the upper quartile (Q₃) and the lower quartile (Q₁), representing the spread of the middle 50% of the data.
四分位距（IQR）是上四分位数（Q₃）与下四分位数（Q₁）的差值，代表中间50%数据的离散程度。
IQR = Q₃ − Q₁
Variance measures the average squared deviation from the mean. For a sample, we divide by (n − 1) to obtain an unbiased estimate.
方差衡量各数据与均值之差的平方的平均水平。样本方差除以 (n − 1) 以得到无偏估计。
s² = Σ(x − x̄)² / (n − 1)
The standard deviation is the square root of the variance and is expressed in the same units as the original data.
标准差是方差的平方根，其单位与原始数据相同。
s = √[Σ(x − x̄)² / (n − 1)]
3. Probability Rules | 概率规则
The probability of an event A lies between 0 and 1 inclusive, where 0 indicates impossibility and 1 indicates certainty.
事件 A 的概率介于 0 和 1 之间（含端点），0 表示不可能发生，1 表示必然发生。
0 ≤ P(A) ≤ 1
The addition rule for mutually exclusive events states that the probability of A or B occurring is the sum of their individual probabilities.
互斥事件的加法法则：事件 A 或 B 发生的概率等于各自概率之和。
P(A ∪ B) = P(A) + P(B)
When events are not mutually exclusive, we must subtract the probability of their intersection to avoid double counting.
若事件并非互斥，则需减去它们交集的概率以避免重复计算。
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
For independent events, the multiplication rule says the probability of both A and B occurring is the product of their probabilities.
对于独立事件，乘法法则指出 A 和 B 同时发生的概率等于各自概率的乘积。
P(A ∩ B) = P(A) × P(B)
4. Conditional Probability | 条件概率
Conditional probability P(A|B) is the probability of event A occurring given that event B has already occurred.
条件概率 P(A|B) 表示在事件 B 已经发生的条件下事件 A 发生的概率。
P(A|B) = P(A ∩ B) / P(B)
This formula is the foundation for solving problems involving two-way tables, tree diagrams, and Venn diagrams. It applies as long as P(B) > 0.
该公式是解决双向表、树状图和韦恩图问题的基础，只要 P(B) > 0 即可应用。
Rearranging the definition gives the general multiplication rule for any two events, not just independent ones.
将该定义变形后，可得到适用于任意两个事件的一般乘法公式，而不仅限于独立事件。
P(A ∩ B) = P(B) × P(A|B)
Three common representations help calculate conditional probabilities: frequency tables (contingency tables), probability tree diagrams, and Venn diagrams with overlapping sets.
有三种常用工具可以帮助计算条件概率：频率表（列联表）、概率树形图以及带有交集区域的韦恩图。
5. Binomial Distribution | 二项分布
A binomial distribution arises when there are a fixed number n of independent trials, each with the same probability p of success. The random variable X counts the number of successes.
当进行固定次数 n 次独立试验，且每次试验的成功概率 p 相同时，就服从二项分布。随机变量 X 表示成功的次数。
X ~ B(n, p)
The probability of obtaining exactly r successes is given by the binomial probability formula, which uses the binomial coefficient.
恰好获得 r 次成功的概率由二项概率公式求得，其中用到了二项式系数。
P(X = r) = ⁿCᵣ × pʳ × (1 − p)ⁿ⁻ʳ
Here the binomial coefficient ⁿCᵣ is also written as n!/(r!(n−r)!) and can be read from Pascal’s triangle or calculated using a calculator.
这里二项式系数 ⁿCᵣ 也写作 n!/(r!(n−r)!)，可以从帕斯卡三角形读取或使用计算器求得。
The mean (expected value) and variance of a binomial distribution are straightforward to compute without listing all outcomes.
二项分布的均值（期望）和方差很容易计算，无需列出所有可能的结果。
Mean = np
Variance = np(1 − p)
6. Normal Distribution | 正态分布
The normal distribution is a continuous symmetric bell-shaped curve defined by its mean μ and standard deviation σ. Many real-world variables follow this pattern.
正态分布是一种连续、对称的钟形曲线，由其均值 μ 和标准差 σ 定义。许多现实世界中的变量均服从这一模式。
X ~ N(μ, σ²)
To find probabilities, we convert any normal variable to the standard normal distribution Z with mean 0 and standard deviation 1.
为求概率，我们将任意正态变量转化为均值为 0、标准差为 1 的标准正态分布 Z。
z = (x − μ) / σ
The empirical rule (68-95-99.7 rule) states that approximately 68% of values lie within μ ± σ, 95% within μ ± 2σ, and 99.7% within μ ± 3σ.
经验法则（68-95-99.7 规则）指出，约有 68% 的数值落在 μ ± σ 范围内，95% 落在 μ ± 2σ 内，99.7% 落在 μ ± 3σ 内。
Probability calculations for ranges use the standard normal table or calculator functions. For example, P(a < X < b) is found by standardising both boundaries.
区间概率的计算需使用标准正态分布表或计算器功能。例如，求 P(a < X < b) 时需对区间两端进行标准化。
7. Confidence Intervals | 置信区间
A confidence interval provides a range of plausible values for an unknown population parameter, such as the population mean, based on a sample.
置信区间给出总体参数（例如总体均值）的一个合理取值范围，基于样本数据得出。
When the population standard deviation σ is known, the confidence interval for the population mean μ uses the Z-distribution critical value.
当总体标准差 σ 已知时，总体均值 μ 的置信区间使用 Z 分布的临界值。
CI = x̄ ± z* × (σ / √n)
More commonly, σ is unknown and is estimated by the sample standard deviation s. Then the interval relies on the t-distribution with n−1 degrees of freedom.
更常见的情况是 σ 未知，用样本标准差 s 来估计。此时区间基于自由度为 n−1 的 t 分布。
CI = x̄ ± t*ₙ₋₁ × (s / √n)
The margin of error is half the width of the confidence interval and decreases as sample size increases. A 95% confidence level is most frequently used.
误差界限是置信区间宽度的一半，随样本容量增大而减小。最常见的是 95% 置信水平。
8. Hypothesis Testing | 假设检验
Hypothesis testing is a formal procedure used to decide whether there is enough evidence in a sample to reject a null hypothesis H₀ in favour of an alternative H₁.
假设检验是一种规范流程，用于判断样本中是否有足够证据拒绝原假设 H₀ 而支持备择假设 H₁。
The p-value is the probability of obtaining a test statistic at least as extreme as the one observed, assuming H₀ is true. A small p-value (typically < 0.05) leads us to reject H₀.
p 值是指在 H₀ 为真的条件下，获得至少与观测值一样极端的检验统计量的概率。若 p 值很小（通常 < 0.05），则拒绝 H₀。
For a Z-test on a population mean, the test statistic is computed by standardising the sample mean. This is used when σ is known or for large samples.
对总体均值进行 Z 检验时，检验统计量通过对样本均值进行标准化求得，适用于 σ 已知或大样本情况。
z = (x̄ − μ₀) / (σ / √n)
For a t-test (σ unknown), the test statistic follows a t-distribution with (n−1) degrees of freedom. The critical region is determined by the significance level α.
进行 t 检验（σ 未知）时，检验统计量服从自由度为 (n−1) 的 t 分布。拒绝域由显著性水平 α 确定。
t = (x̄ − μ₀) / (s / √n)
9. Correlation and Regression | 相关与回归
Pearson’s product-moment correlation coefficient r measures the strength and direction of a linear relationship between two variables. It takes values between −1 and +1.
皮尔逊积矩相关系数 r 衡量两个变量之间线性关系的强度和方向，取值范围在 −1 到 +1 之间。
| r value | Interpretation |
|---|---|
| r = +1 | Perfect positive linear correlation |
| r = 0 | No linear correlation |
| r = -1 | Perfect negative linear correlation |
A shortcut formula for r eliminates the need for computing separate standard deviations if summary statistics are available.
如果已知各项汇总统计量，可以使用 r 的简洁公式，无需单独计算标准差。
r = [nΣxy − (Σx)(Σy)] / √[ (nΣx² − (Σx)²)(nΣy² − (Σy)²) ]
The least squares regression line of y on x minimises the sum of squared vertical deviations. It is used for prediction.
y 对 x 的最小二乘回归线使得铅直偏差的平方和最小，常用于预测。
y = a + bx
b = r × (sᵧ / sₓ)
a = ȳ − bx̄
10. Data Collection and Sampling | 数据收集与抽样
Random sampling methods, such as simple random sampling, give every member of the population an equal chance of being selected, reducing bias.
随机抽样方法（例如简单随机抽样）使总体中每个成员都有均等的机会被选中，从而减少偏差。
Stratified sampling divides the population into distinct subgroups (strata) and then selects a random sample from each in proportion to its size. This ensures representation.
分层抽样将总体划分为不同的子群（层），然后按各层规模的比例从中随机抽取样本，以保证代表性。
Sample size from stratum = (Stratum size / Population size) × Total sample size
Systematic sampling selects every kᵗʰ individual from a list after a random start. It is easy to implement but can introduce periodicity bias.
系统抽样在随机起点后，从名单上每隔 k 个个体抽取一个。它易于执行，但可能引入周期性偏差。
A sampling frame is a list of all members of the target population. Any discrepancy between the frame and the population leads to sampling error.
抽样框是目标总体中所有成员的名单。抽样框与总体之间的任何差异都会导致抽样误差。
Non-probability methods like quota or convenience sampling are quicker but do not allow valid statistical inference. Understanding their limitations is essential for interpreting data.
配额抽样或便利抽样等非概率方法速度更快，但无法进行有效的统计推断。理解它们的局限性对于解读数据至关重要。
Published by TutorHao | Statistics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导Cancel reply