IGCSE Cambridge Statistics Formula & Theorem Quick Reference | IGCSE 剑桥统计：公式定理速查手册

📚 IGCSE Cambridge Statistics Formula & Theorem Quick Reference | IGCSE 剑桥统计：公式定理速查手册

This quick reference summarises the core formulas and definitions needed for the Cambridge IGCSE Statistics course. Use it alongside past papers and topic questions for efficient revision.

本速查手册汇总了剑桥 IGCSE 统计课程中必须掌握的核心公式与定义，适合配合历年真题与分主题练习高效复习。

1. Types of Data and Sampling | 数据类型与抽样

Data can be qualitative (labels or categories, such as eye colour) or quantitative (numerical values). Quantitative data are discrete if they can only take particular values, and continuous if they can take any value in an interval.

数据可分为定性数据（类别的标签，如眼睛颜色）和定量数据（数值型）。定量数据若只能取某些特定值则为离散数据，若可在某一区间内任意取值则为连续数据。

For stratified sampling, the sample from each group is proportional to the group’s size. The required number from a stratum is calculated as:

分层抽样中，每层抽取的样本量与该层大小成比例。某层所需样本数计算如下：

Stratum sample size = (stratum size / population size) × total sample size

Random sampling gives each member an equal chance of selection; systematic sampling selects every kth member; cluster sampling selects whole groups; quota sampling continues until set numbers in categories are met.

随机抽样使每个成员被抽中的概率相同；系统抽样每隔固定间隔 k 抽取一个；整群抽样直接抽取整个群体；配额抽样按各类别的设定人数抽取直到满足配额。

A sampling frame is a list of all members in the population. Bias can occur if some members are excluded, over-represented, or if non-responses are ignored.

抽样框是总体中所有成员的名单。如果某些成员被排除、被过度代表或未回答被忽略，就可能产生偏差。

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2. Frequency Tables and Charts | 频数表与统计图

For a histogram with unequal class widths, the height of each bar is given by the frequency density. The area of a bar represents the frequency.

在组距不等的直方图中，每个条形的高度由频数密度给出。条形的面积表示频数。

Frequency density = frequency / class width

A cumulative frequency curve is plotted using the upper class boundary against the cumulative frequency. The median is read at half the total frequency, the lower quartile at one quarter, and the upper quartile at three quarters.

累积频数曲线以每个组的上端点为横坐标、累积频数为纵坐标绘制。中位数对应总频数的一半处，下四分位数对应四分之一处，上四分位数对应四分之三处。

For a pie chart, the angle of each sector is proportional to its frequency. To find the angle:

饼图中每个扇形的角度与其频数成比例。计算角度的公式为：

Sector angle = (frequency / total frequency) × 360°

Stem-and-leaf diagrams preserve original data values and allow median and quartiles to be found directly. Box-and-whisker plots show the minimum, Q1, median, Q3 and maximum on a single scale.

茎叶图保留原始数据值，并可直接找到中位数和四分位数。箱线图在一条数轴上同时显示最小值、下四分位数、中位数、上四分位数和最大值。

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3. Measures of Central Tendency | 集中趋势量数

The arithmetic mean is the sum of all values divided by the number of values. For raw data:

算术平均数是所有数值之和除以数值的个数。对于未分组数据：

x̄ = Σx / n

For a frequency distribution, multiply each value by its frequency and divide by the total frequency:

对于频数分布，将每个数值乘以其频数再除以总频数：

x̄ = Σfx / Σf

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

中位数是将数据按顺序排列后的中间值。如果 n 为奇数，则为第 (n+1)/2 个值；如果 n 为偶数，则为第 n/2 个与第 (n/2 + 1) 个值的平均数。

The mode is the value that occurs most often. A data set can have one mode, more than one mode, or no mode.

众数是出现次数最多的数值。一组数据可以有一个众数、多个众数或没有众数。

When two or more groups are combined, the overall mean is found by weighting each group mean by its size:

当两个或多个组合并时，总平均数需要用各组大小对各组平均数加权：

x̄_combined = (n₁x̄₁ + n₂x̄₂) / (n₁ + n₂)

A weighted mean uses weights to reflect different

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