IGCSE CCEA Statistics Formula and Theorem Quick Reference | IGCSE CCEA 统计：公式定理速查手册

📚 IGCSE CCEA Statistics Formula and Theorem Quick Reference | IGCSE CCEA 统计：公式定理速查手册

This quick reference summarises the core formulae, definitions and theorems required for IGCSE CCEA Statistics. Use it alongside past-paper practice to develop speed and confidence in data handling, probability and inference.

本速查手册汇总 IGCSE CCEA 统计课程的核心公式、定义与定理。配合历年真题练习，可提升数据处理、概率与推断的速度和信心。

1. Data Types and Frequency Distributions | 数据类型与频数分布

Statistical data can be categorical, ordinal, discrete or continuous. A frequency distribution groups raw data into classes, and the class width is the difference between consecutive lower class boundaries.

统计数据可分为类别型、有序型、离散型和连续型。频数分布将原始数据分组，组距是相邻下限类边界之差。

Class width = upper class boundary − lower class boundary — 组距 = 上限类边界 − 下限类边界
Frequency density = frequency ÷ class width — 频数密度 = 频数 ÷ 组距
Cumulative frequency is the running total of frequencies — 累积频数是频数逐项累计的总和

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

The mean, median and mode summarise the centre of a data set. For ungrouped data, the arithmetic mean is the sum of all values divided by the number of values.

均值、中位数和众数用于概括数据集的中心。对于未分组数据，算术平均数是所有数值之和除以数值个数。

Mean: x̄ = Σx ÷ n

均值：x̄ = Σx ÷ n

For grouped data, use the class midpoint x as the representative value. The median is calculated by interpolation within the median class.

对于分组数据，以组中点 x 作为代表值。中位数通过在中位数所在组内插值计算。

Grouped mean: x̄ = Σfx ÷ Σf

分组均值：x̄ = Σfx ÷ Σf

Median: L + [(n/2 − F) ÷ f] × w

中位数：L + [(n/2 − F) ÷ f] × w

L = lower class boundary of the median class — L = 中位数所在组的下限边界
F = cumulative frequency before the median class — F = 中位数所在组之前的累积频数
f = frequency of the median class — f = 中位数所在组的频数
w = class width — w = 组距

3. Measures of Spread and Position | 离散程度与位置度量

The range and interquartile range measure how spread out the data are. Quartiles divide an ordered data set into four equal parts.

极差和四分位距用于衡量数据的离散程度。四分位数将有序数据集分成四个相等的部分。

Range = highest value − lowest value

极差 = 最大值 − 最小值

IQR = Q₃ − Q₁

四分位距 = Q₃ − Q₁

The position of the kth percentile is given by (k/100) × n. When this is not an integer, round up to locate the ordered value.

第 k 百分位的位置由 (k/100) × n 给出。当结果不是整数时，向上取整以确定排序后的数值。

Percentile position = (k ÷ 100) × n

百分位位置 = (k ÷ 100) × n

4. Variance and Standard Deviation | 方差与标准差

Variance and standard deviation are the most useful measures of spread because they use every data value. Standard deviation is the square root of variance and has the same units as the data.

方差和标准差是最有用的离散程度度量，因为它们使用了每一个数据值。标准差是方差的平方根，与数据具有相同的单位。

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)

For grouped data, multiply each squared deviation by the class frequency f.

对于分组数据，将每个平方偏差乘以组频数 f。

Standard deviation: s = √[Σ(x − x̄)² ÷ (n − 1)]

标准差：s = √[Σ(x − x̄)² ÷ (n − 1)]

5. Basic Probability Rules | 概率基本法则

Probability is a measure of the chance that an event occurs, always between 0 and 1 inclusive. The general addition rule adjusts for any overlap between two events.

概率是对事件发生机会的度量，始终在 0 到 1 之间。一般加法法则需要对两个事件之间的重叠进行修正。

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

If A and B are mutually exclusive, they cannot occur together, so the intersection probability is zero.

如果 A 与 B 互斥，则它们不能同时发生，因此交事件的概率为零。

Mutually exclusive: P(A ∪ B) = P(A) + P(B)

互斥事件：P(A ∪ B) = P(A) + P(B)

The complement rule says the probability that event A does not happen is one minus the probability that it does happen.

补事件法则指出，事件 A 不发生的概率等于 1 减去事件 A 发生的概率。

P(A′) = 1 − P(A)

6. Conditional Probability and Independence | 条件概率与独立性

Conditional probability calculates the chance of event A occurring given that event B has already occurred. It is found by dividing the joint probability by the probability of the condition.

条件概率计算在事件 B 已经发生的条件下事件 A 发生的概率。它等于联合概率除以条件事件的概率。

P(A | B) = P(A ∩ B) ÷ P(B)

Two events are independent if the occurrence of one does not change the probability of the other. This can be tested using either the multiplication rule or the conditional probability definition.

两个事件相互独立是指一个事件的发生不会改变另一个事件发生的概率。这可以通过乘法法则或条件概率定义来检验。

Independent: P(A ∩ B) = P(A) × P(B)

相互独立：P(A ∩ B) = P(A) × P(B)

Equivalently, A and B are independent if P(A | B) = P(A) provided P(B) > 0.

等价地，如果 P(B) > 0 且 P(A | B) = P(A)，则 A 与 B 相互独立。

7. Discrete Random Variables and Expectation | 离散随机变量与期望

A discrete random variable X takes countable values. Its expected value E(X) is the long-run average of X, and variance measures the spread of the distribution.

离散随机变量 X 取可数个值。其期望值 E(X) 是 X 的长期平均值，方差衡量分布的离散程度。

E(X) = Σ[x × P(X = x)]

A faster formula for variance is the mean of the square minus the square of the mean.

方差的一个更快捷公式是平方的均值减去均值的平方。

Var(X) = E(X²) − [E(X)]²

The total probability for all possible values of X must equal 1.

随机变量 X 所有可能取值的概率总和必须等于 1。

ΣP(X = x) = 1

8. Binomial Distribution | 二项分布

The binomial distribution models the number of successes in a fixed number of independent trials. Each trial has only two outcomes and a constant probability of success p.

二项分布用于描述固定次数的独立试验中成功次数的分布。每次试验只有两种结果，且成功概率 p 保持不变。

X ~ B(n, p)

The probability of exactly r successes is given by the binomial probability formula.

恰好有 r 次成功的概率由二项概率公式给出。

P(X = r) = C(n, r) × pʳ × (1 − p)ⁿ⁻ʳ

The mean, variance and standard deviation of a binomial random variable are simple multiples of n and p.

二项随机变量的均值、方差和标准差是 n 与 p 的简单乘积关系。

Mean = np Var(X) = np(1 − p) SD = √[np(1 − p)]

均值 = np 方差 = np(1 − p) 标准差 = √[np(1 − p)]

9. Normal Distribution and Standardised Scores | 正态分布与标准化分数

The normal distribution is a continuous probability distribution that is symmetric about the mean. To compare values from different normal distributions, convert raw scores to z-scores.

正态分布是一种关于均值对称的连续概率分布。为了比较不同正态分布中的数据，需要将原始分数转换为 z 分数。

z = (x − μ) ÷ σ

The standard normal distribution has mean 0 and standard deviation 1. The empirical rule gives useful approximations for symmetric data.

标准正态分布的均值为 0，标准差为 1。经验法则为对称数据提供了有用的近似比例。

About 68% of values lie within 1σ of the mean — 约 68% 的数据落在均值 ± 1σ 内
About 95% within 2σ — 约 95% 落在均值 ± 2σ 内
About 99.7% within 3σ — 约 99.7% 落在均值 ± 3σ 内

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 r always lies between −1 and 1.

相关用于衡量两个变量之间线性关系的强度和方向。积矩相关系数 r 始终在 −1 到 1 之间。

Sxy = Σxy − (Σx × Σy) ÷ n

Sxx = Σx² − (Σx)² ÷ n Syy = Σy² − (Σy)² ÷ n

r = Sxy ÷ √(Sxx × Syy)

The least squares regression line predicts y from x and passes through the mean point (x̄, ȳ).

最小二乘回归线用于由 x 预测 y，并通过均值点 (x̄, ȳ)。

y = a + bx

b = Sxy ÷ Sxx a = ȳ − b × x̄

11. Sampling and Confidence Intervals | 抽样与置信区间

When a sample is drawn from a population, the sample mean is used to estimate the population mean. The standard error measures the variability of the sample mean.

从总体中抽样时，样本均值用于估计总体均值。标准误衡量样本均值的变异性。

Standard error = σ ÷ √n

标准误 = σ ÷ √n

For a population with known standard deviation σ, a 95% confidence interval for the population mean is constructed using the 1.96 z-value.

当总体标准差 σ 已知时，总体均值的 95% 置信区间使用 1.96 这一 z 值来构建。

95% CI = x̄ ± 1.96 × (σ ÷ √n)

95% 置信区间 = x̄ ± 1.96 × (σ ÷ √n)

The margin of error is the half-width of the confidence interval. Larger samples produce smaller standard errors and narrower intervals.

误差边际是置信区间的一半宽度。样本量越大，标准误越小，置信区间越窄。

12. Index Numbers and Moving Averages | 指数与移动平均

Index numbers compare a quantity or price at a given time with a base value. A moving average smooths out short-term fluctuations in time series data.

指数用于将某一时期的数据或价格与基准值进行比较。移动平均用于平滑时间序列数据中的短期波动。

Index number = (value ÷ base value) × 100

指数 = (数值 ÷ 基准值) × 100

For a moving average, take the mean of a fixed number of consecutive values and move one period at a time. Seasonal variation can then be estimated by comparing raw values with moving averages.

计算移动平均时，取固定个连续数值的均值，然后每次向后移动一个时期。随后可通过比较原始值与移动平均值估计季节变动。

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IGCSE CCEA Statistics Formula and Theorem Quick Reference | IGCSE CCEA 统计：公式定理速查手册