GCSE Edexcel Statistics: Formulae & Theorems Quick Reference | GCSE Edexcel 统计：公式定理速查手册

📚 GCSE Edexcel Statistics: Formulae & Theorems Quick Reference | GCSE Edexcel 统计：公式定理速查手册

This quick reference guide compiles all the essential formulae, definitions and theorems you need for the GCSE Edexcel Statistics examination. Keep it handy for last‑minute revision and use it to check your understanding as you work through past papers.

这份速查手册汇集了 GCSE Edexcel 统计考试所需的所有核心公式、定义与定理。适合考前快速复习，也可在做历年真题时随时查阅，巩固理解。

1. Measures of Central Tendency | 集中趋势的度量

The arithmetic mean of a set of values is the sum of the values divided by the number of values. It is the most commonly used measure of average.

算术平均值是一组数值的总和除以数值的个数。它是最常用的平均数指标。

Mean: x̄ = Σx / n

均值：x̄ = Σx / n

If the data are given as a frequency table, the mean is calculated by multiplying each value by its frequency, summing these products and dividing by the total frequency.

如果数据以频数表形式给出，均值等于每个值乘以其频数，求和后再除以总频数。

Mean from a frequency table: x̄ = Σfx / Σf

频数表均值：x̄ = Σfx / Σf

The median is the middle value when the data are arranged in order. For ungrouped data, the position of the median is (n + 1)/2. If this is a decimal, take the mean of the two surrounding values.

中位数是将数据排序后处于中间位置的数值。对于未分组数据，中位数的位置为 (n + 1)/2。若得到小数，则取相邻两个值的平均数。

The mode (or modal class for grouped data) is the value or category that occurs most frequently.

众数（或分组数据的众数组）是出现次数最多的值或类别。

2. Measures of Dispersion | 离散程度的度量

Dispersion measures the spread of the data. The simplest measure is the range, which is the difference between the largest and smallest values.

离散程度衡量数据的分散情况。最简单的量数是极差，即最大值与最小值之差。

Range = Max – Min

极差 = 最大值 – 最小值

The interquartile range (IQR) gives the spread of the middle 50% of the data and is less affected by outliers.

四分位距 (IQR) 表示中间 50% 数据的分散程度，受异常值的影响较小。

IQR = Q₃ – Q₁

IQR = 上四分位数 Q₃ – 下四分位数 Q₁

Variance and standard deviation measure how far each value deviates from the mean. For a population, the variance is the mean of the squared deviations. For a sample, we usually divide by (n – 1) to obtain an unbiased estimate.

方差和标准差度量每个数值偏离均值的程度。对于总体，方差是偏差平方的平均值；对于样本，通常除以 (n – 1) 以获得无偏估计。

Population variance: σ² = Σ(x – μ)² / N

总体方差：σ² = Σ(x – μ)² / N

Sample variance: s² = Σ(x – x̄)² / (n – 1)

样本方差：s² = Σ(x – x̄)² / (n – 1)

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

标准差就是方差的正平方根。

SD = √variance

标准差 = √方差

3. Quartiles and Box Plots | 四分位数与箱线图

Quartiles split an ordered data set into four equal parts. The lower quartile (Q₁) marks the value below which 25% of the data fall; the upper quartile (Q₃) marks the value below which 75% of the data fall.

四分位数将有序数据集分成四个等份。下四分位数 (Q₁) 是 25% 数据点落在其下的值；上四分位数 (Q₃) 是 75% 数据点落在其下的值。

For ungrouped data, find the position of Q₁ as (n + 1)/4 and Q₃ as 3(n + 1)/4. Interpolate if necessary. For grouped data, cumulative frequency graphs are used to estimate quartiles.

对于未分组数据，Q₁ 的位置为 (n + 1)/4，Q₃ 的位置为 3(n + 1)/4，必要时进行内插。对于分组数据，常利用累积频率图来估计四分位数。

A box plot (box‑and‑whisker diagram) displays the five‑number summary: minimum, Q₁, median, Q₃ and maximum, giving a clear visual of the data distribution.

箱线图（盒须图）通过展示五数概括——最小值、Q₁、中位数、Q₃ 和最大值——直观呈现数据分布。

4. Estimating Mean and Median for Grouped Data | 分组数据的均值与中位数估算

When data are grouped into intervals, exact values are unknown, so we use midpoints to estimate the mean.

当数据被分成组区间后，无法得知精确值，因此使用组中值来估算均值。

Estimated mean = Σ(midpoint × frequency) / Σ frequency

估算均值 = Σ(组中值 × 频数) / 总频数

The median for grouped data is found by identifying the median class using cumulative frequency, then applying linear interpolation.

对于分组数据的中位数，先通过累积频率确定中位数组，然后使用线性插值法计算。

Median = L + [ ( (n/2) – Fₚ ) / fₘ ] × w

中位数 = L + [ ( (n/2) – Fₚ ) / fₘ ] × w

where L is the lower boundary of the median class, n is the total frequency, Fₚ is the cumulative frequency before the median class, fₘ is the frequency of the median class, and w is the class width.

其中 L 是中位数组的下界，n 是总频数，Fₚ 是中位数组之前的累积频数，fₘ 是中位数组的频数，w 是组距。

5. Basic Probability | 概率基础

The probability of an event A, written P(A), is the proportion of times A would occur in a large number of trials, provided outcomes are equally likely.

事件 A 的概率，记为 P(A)，是指在等可能结果假设下，大量试验中 A 发生的次数比例。

P(A) = Number of favourable outcomes / Total number of outcomes

P(A) = 有利结果数目 / 所有可能结果的总数

Probabilities always lie between 0 and 1 inclusive. The sum of probabilities of all possible outcomes is 1.

概率值总是在 0 到 1 之间（含）。所有可能结果的概率之和为 1。

For mutually exclusive events A and B (they cannot happen at the same time), the addition rule applies.

对于互斥事件 A 和 B（不能同时发生），适用加法法则。

P(A ∪ B) = P(A) + P(B)

P(A 或 B) = P(A) + P(B)

The relative frequency of an event obtained from an experiment can be used as an estimate for its probability.

由实验得到的事件相对频率可用作其概率的估计值。

6. Conditional Probability and Tree Diagrams | 条件概率与树图

Conditional probability, P(A|B), is the probability that A occurs given that B has already occurred.

条件概率 P(A|B) 是在已知 B 已发生的前提下，A 发生的概率。

P(A|B) = P(A ∩ B) / P(B)

P(A|B) = P(A 且 B) / P(B)

Two events are independent if the occurrence of one does not affect the probability of the other. For independent events A and B, P(A|B) = P(A) and P(A ∩ B) = P(A) × P(B).

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。对于独立事件 A 和 B，有 P(A|B) = P(A) 且 P(A ∩ B) = P(A) × P(B)。

Probability tree diagrams are very helpful for visualizing multi‑stage experiments. Multiply probabilities along the branches to find the probability of a combination of outcomes; add up the probabilities of relevant end‑points for combined events.

概率树图非常有助于直观展示多步试验。沿分支相乘得到各结果组合的概率；将相关终点的概率相加得到复合事件的概率。

7. Expectation and Variance of Discrete Random Variables | 离散型随机变量的期望与方差

The expected value (or mean) of a discrete random variable X is a weighted average of its possible values, with the weights being their probabilities.

离散型随机变量 X 的期望值（或均值）是其可能取值以对应概率为权的加权平均值。

E(X) = Σ x · P(X = x)

期望 E(X) = Σ x · P(X = x)

The variance of X, Var(X), measures the spread of the distribution. The standard deviation is √Var(X).

X 的方差 Var(X) 衡量分布的离散程度。标准差为 √Var(X)。

Var(X) = Σ (x – μ)² P(X = x) = Σ x² P(X = x) – μ²

方差 Var(X) = Σ (x – μ)² P(X = x) = Σ x² P(X = x) – μ²

These formulae are essential for analysing probability distributions and for tackling expected profit or loss problems.

这些公式是分析概率分布以及解决期望利润或损失问题的基础。

8. Binomial Distribution | 二项分布

A binomial distribution arises when a trial has two possible outcomes (success and failure), is repeated a fixed number of times, and each trial is independent with constant probability of success p.

当单次试验只有两种可能结果（成功与失败），重复固定次数，且每次试验独立、成功概率 p 不变时，就产生二项分布。

If X ~ B(n, p), the probability of obtaining exactly r successes in n trials is given by the binomial probability formula.

若 X ~ B(n, p)，则在 n 次试验中恰好获得 r 次成功的概率由二项概率公式给出。

P(X = r) = nCᵣ × pʳ (1 – p)ⁿ⁻ʳ

Here nCᵣ (also written as C(n, r)) is the binomial coefficient, equal to n! / [r! (n – r)!].

其中 nCᵣ（也可写作 C(n, r)）是二项式系数，等于 n! / [r! (n – r)!]。

The mean and variance of a binomial distribution are straightforward to calculate; they are often needed to answer questions about expected number of successes.

二项分布的均值和方差计算简便，回答有关期望成功次数的问题时常用。

E(X) = np

期望 E(X) = np

Var(X) = np (1 – p)

方差 Var(X) = np(1 – p)

9. Standardisation and the Normal Distribution | 标准化与正态分布

The normal distribution is a continuous, bell‑shaped curve that is fully described by its mean μ and standard deviation σ. Many real‑world measurements, such as heights and weights, are approximately normally distributed.

正态分布是一种连续、钟形的曲线，由其均值 μ 和标准差 σ 完全描述。许多现实世界测量值，如身高和体重，都近似服从正态分布。

To compare values from different normal distributions or to find probabilities using standard normal tables, we calculate the z‑score.

为了比较不同正态分布的值或使用标准正态表求概率，需要计算 z 分数。

z = (x – μ) / σ

A z‑score of 0 corresponds to the mean; positive z‑scores are above the mean, negative ones below. In the standard normal distribution (μ = 0, σ = 1), approximately 68% of data lie within 1σ of the mean.

z 分数为 0 对应均值；正值在均值之上，负值在均值之下。在标准正态分布（μ = 0, σ = 1）中，约 68% 的数据落在均值 ±1σ 范围内。

You may be required to use the standard normal distribution table to find P(Z < z) and interpret the results in context.

考试可能要求使用标准正态分布表求 P(Z < z)，并解释其在具体情境中的含义。

10. Correlation and Regression | 相关与回归

Correlation measures the strength and direction of a linear relationship between two variables. The product moment correlation coefficient (PMCC), often denoted by r, is a value between –1 and +1.

相关衡量两个变量之间线性关系的强度和方向。积矩相关系数 (PMCC) 常记为 r，取值在 –1 到 +1 之间。

For GCSE Statistics, you are usually given r or you can calculate it using a calculator. A value close to +1 indicates strong positive correlation; close to –1 indicates strong negative correlation; close to 0 indicates no linear correlation.

GCSE 统计中通常给出 r 值或可用计算器求出。值接近 +1 表示强正相关，接近 –1 表示强负相关，接近 0 表示无线性相关。

The equation of the regression line of y on x enables prediction of y from a given x.

y 对 x 的回归线方程可用于由已知 x 值预测 y 值。

y = a + b x

The gradient b and intercept a are found using the least‑squares method. You are expected to know the formulae and apply them.

斜率 b 和截距 a 通过最小二乘法求得。需要掌握公式并能应用。

b = Sxy / Sxx, where Sxy = Σ(x – x̄)(y – ȳ) and Sxx = Σ(x – x̄)²

b = Sxy / Sxx，其中 Sxy = Σ(x – x̄)(y – ȳ)，Sxx = Σ(x – x̄)²

a = ȳ – b x̄

Spearman’s rank correlation coefficient (rₛ) is used when data are ranked or when the relationship is monotonic but not necessarily linear.

斯皮尔曼等级相关系数 (rₛ) 用于排序数据或存在单调但不一定是线性的关系时。

rₛ = 1 – (6 Σd²) / (n(n² – 1))

Here d is the difference between the ranks of each pair of corresponding values. The coefficient is interpreted similarly to PMCC.

其中 d 是每对对应值的秩之差。该系数的解读方式与积矩相关系数类似。

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GCSE Edexcel Statistics: Formulae & Theorems Quick Reference | GCSE Edexcel 统计：公式定理速查手册