Year 10 CAIE Statistics: Formula & Theorem Quick Reference Handbook | 公式定理速查手册

📚 Year 10 CAIE Statistics: Formula & Theorem Quick Reference Handbook | 公式定理速查手册

This quick-reference handbook summarises the essential formulas and theorems for the Year 10 CAIE Statistics syllabus. It is designed for rapid revision, helping you recall the key concepts required for examinations.

本速查手册总结了 Year 10 CAIE 统计课程的核心公式与定理，旨在帮助快速复习，牢记考试所需的关键概念。

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

Measures of central tendency identify the centre of a data set. The three main measures are the mean, median and mode.

集中趋势度量用于确定数据集的中心。三种主要度量是均值、中位数和众数。

The arithmetic mean (x̄) for a list of n values x₁, x₂, …, xₙ is the sum of the values divided by n.

对于 n 个值 x₁, x₂, …, xₙ，算术均值 (x̄) 是这些值之和除以 n。

x̄ = Σx / n

For grouped data, the mean is estimated using the class midpoints x and frequencies f.

对于分组数据，均值使用组中值 x 和频数 f 进行估算。

x̄ = Σfx / Σf

The median is the middle value when the data are arranged in ascending order. If n is odd, median = the (n+1)/2 th value. If n is even, median = the mean of the n/2 th and (n/2+1)th values.

中位数是将数据按升序排列后的中间值。若 n 为奇数，中位数 = 第 (n+1)/2 个值；若 n 为偶数，中位数 = 第 n/2 个与第 (n/2+1) 个值的平均数。

The mode is the value that occurs most frequently. A data set can have one mode (unimodal), more than one mode (bimodal/multimodal) or no mode if all values occur equally often.

众数是出现频率最高的值。数据集可能有一个众数（单峰），不止一个众数（双峰/多峰），或者如果所有值出现次数一样多则无众数。

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2. Measures of Dispersion | 离散程度的度量

Dispersion measures describe the spread or variability of the data. Common measures include range, interquartile range (IQR), variance and standard deviation.

离散度量描述数据的分散或变异程度。常用度量包括极差、四分位距、方差和标准差。

The range is the difference between the maximum and minimum values.

极差是最大值与最小值之差。

Range = xₘₐₓ − xₘᵢₙ

The interquartile range is the difference between the upper quartile Q₃ and the lower quartile Q₁.

四分位距是上四分位数 Q₃ 与下四分位数 Q₁ 的差。

IQR = Q₃ − Q₁

For a population of N values with mean μ, the population variance σ² is the mean of the squared deviations.

对于均值为 μ、容量为 N 的总体，总体方差 σ² 是离差平方的平均。

σ² = Σ(x − μ)² / N

For a sample of n values with mean x̄, the sample variance s² uses (n−1) as the denominator to give an unbiased estimate.

对于均值为 x̄、容量为 n 的样本，样本方差 s² 使用 (n−1) 作为分母以给出无偏估计。

s² = Σ(x − x̄)² / (n − 1)

The standard deviation is the positive square root of the variance. It is expressed in the original units of the data.

标准差是方差的正平方根，用数据的原始单位表示。

σ = √σ²    or    s = √s²

For grouped data, the variance formulas use midpoints x and frequencies f: σ² = Σf(x − μ)² / Σf and s² = Σf(x − x̄)² / (Σf − 1).

对于分组数据，方差公式使用组中值 x 和频数 f：σ² = Σf(x − μ)² / Σf 以及 s² = Σf(x − x̄)² / (Σf − 1)。

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3. Frequency Distributions and Histograms | 频数分布与直方图

When data are grouped into classes, the frequency density is used to construct histograms so that the area of each bar is proportional to the frequency.

当数据被分组到区段中时，使用频数密度构建直方图，使得每个直方的面积与频数成正比。

Frequency density = Frequency / Class width

The class width is the difference between the upper and lower class boundaries. For continuous data, the boundaries eliminate gaps between classes.

组距是上、下组边界之差。对于连续数据，边界消除了各组之间的空隙。

In a histogram, the vertical axis represents frequency density, and the total area of all rectangles equals the total frequency.

在直方图中，纵轴表示频数密度，所有矩形的总面积等于总频数。

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4. Basic Probability Rules | 基本概率法则

Probability measures the likelihood of an event on a scale from 0 (impossible) to 1 (certain). The sum of probabilities of all possible outcomes is 1.

概率衡量事件发生的可能性，范围从 0（不可能）到 1（必然）。所有可能结果的概率之和为 1。

The complement rule states that the probability of an event A not occurring is 1 minus P(A).

互补法则指出事件 A 不发生的概率等于 1 减去 P(A)。

P(not A) = 1 − P(A)

For any two events A and B, the addition rule is:

对于任意两个事件 A 和 B，加法法则为：

P(A or B) = P(A) + P(B) − P(A and B)

If A and B are mutually exclusive (they cannot happen together), P(A and B) =

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