📚 Sex and Gender as Categorical Variables in Statistics | 统计中的生理性别与社会性别分类变量
In Edexcel A-Level Mathematics, topics such as ‘sex and gender’ do not appear as biological or sociological theory. Instead, they appear as categorical variables in statistical problems. Understanding how to classify, sample, display, and test proportions for such variables is an important part of the statistics content.
在 Edexcel A-Level 数学中，’生理性别与社会性别’ 这一类主题并不是作为生物学或社会学理论出现的。它们作为统计问题中的分类变量出现。理解如何对这些变量进行分类、抽样、展示以及检验比例，是统计学内容的重要组成部分。
1. Variables in Statistics | 统计中的变量
In statistics, a variable is any characteristic that can take different values across individuals. We first classify variables as qualitative or quantitative. Qualitative variables describe categories or labels, while quantitative variables describe numerical measurements.
在统计学中，变量是指可以在不同个体间取不同值的特征。我们首先将变量分为定性变量和定量变量。定性变量描述类别或标签，定量变量描述数值测量。
Sex and gender are used in surveys to sort individuals into groups. In many exam questions they appear as two categories such as ‘male’ and ‘female’. This makes them qualitative, not numerical.
生理性别和社会性别在调查中用于将个体分组。在许多考试题中，它们以 ‘男’ 和 ‘女’ 两个类别出现。因此它们属于定性变量，而不是数值变量。
2. Sex and Gender as Categorical Data | 作为分类数据的生理性别与社会性别
A categorical variable places each observation into exactly one category. Sex recorded as male/female is categorical. Gender, when recorded as a self-reported label, is also categorical, even if more than two categories are offered.
分类变量将每个观测值归入恰好一个类别。以男/女记录的生理性别是分类变量。社会性别若按自我报告的标签记录，即使类别多于两个，也属于分类变量。
In A-Level exam contexts, data is usually simplified to a binary classification. However, good statistical practice recognises that gender categories may be non-binary, and the data collection method affects conclusions.
在 A-Level 考试情境中，数据通常简化为二元分类。但良好的统计实践认识到社会性别类别可能是非二元的，数据收集方式会影响结论。
3. Nominal Data and Coding | 名义数据与编码
Sex and gender usually produce nominal data: categories have no natural order. We can code them as 0 and 1 for analysis, for example 0 = male and 1 = female, but the numbers are labels only.
生理性别和社会性别通常产生名义数据：类别没有自然顺序。我们可以将其编码为 0 和 1 以便分析，例如 0 = 男，1 = 女，但这些数字只是标签。
Coding categorical variables is not the same as making them quantitative. You cannot calculate a meaningful mean of 0/1 codes if the categories are merely labels; you can only summarise counts and proportions.
对分类变量编码并不等于将其变为定量变量。如果类别只是标签，你就不能对 0/1 编码计算有意义的平均值；你只能汇总频数和比例。
| Variable / 变量 | Data type / 数据类型 | Example / 示例 |
|---|---|---|
| Sex | Nominal categorical | male, female |
| Gender | Nominal categorical | man, woman, non-binary |
| Coded sex | Coded labels, not quantitative | 0 = male, 1 = female |
4. Sampling and Gender Strata | 抽样与性别分层
In stratified sampling, the population is divided into groups called strata. Sex or gender is often used as a stratum because it is easy to define and may be related to the variable being measured.
在分层抽样中，总体被分成称为层的组。生理性别或社会性别常被用作分层变量，因为它易于定义，并且可能与所测量变量有关。
If a college has 600 male and 400 female students, a stratified sample of size 50 should include 30 males and 20 females. This is calculated by:
如果一所学院有 600 名男生和 400 名女生，容量为 50 的分层样本应包含 30 名男生和 20 名女生。计算如下：
n_male = (600 ÷ 1000) × 50 = 30
n_female = (400 ÷ 1000) × 50 = 20
Stratified sampling by sex or gender can improve representativeness, but it requires an up-to-date sampling frame and clear definitions of categories.
按生理性别或社会性别进行分层抽样可以提高代表性，但这需要最新的抽样框架和清晰的类别定义。
5. Displaying Gender Data | 展示性别数据
Categorical data such as sex or gender can be displayed using bar charts, pie charts, and two-way tables. A bar chart shows frequencies, and the bars should be separated because the categories are distinct and unordered.
如生理性别或社会性别这样的分类数据可以用条形图、饼图和双向表展示。条形图显示频数；由于类别互异且无序，条形之间应留有空隙。
A two-way table is useful for comparing gender with another variable, for example preferred subject or handedness. From the table, we can calculate row or column proportions and comment on association.
双向表可用于比较性别与另一个变量，例如偏好科目或利手性。根据该表，我们可以计算行比例或列比例，并评论是否存在关联。
6. Probability and the Binomial Model | 概率与二项模型
When sex or gender is recorded as two categories and we count the number of individuals in one category, the binomial distribution often applies. If X is the number of female students in a random sample of size n, and the population proportion of females is p, then X ~ B(n, p).
当生理性别或社会性别被记录为两个类别，并且我们计算某一类别中的个体数时，二项分布通常适用。如果 X 是随机样本容量 n 中女生人数，总体女生比例为 p，则 X ~ B(n, p)。
The binomial model requires independent trials, a fixed sample size, two outcomes, and a constant probability p. In reality, if sampling without replacement from a small population, independence is only approximate. Edexcel questions usually assume the sample is small relative to the population.
二项模型要求独立试验、固定样本容量、两种结果以及恒定概率 p。实际上，如果从小总体中不放回抽样，独立性只是近似。Edexcel 题目通常假设样本相对于总体足够小。
7. Hypothesis Testing with a Gender Proportion | 性别比例的假设检验
A common exam task is to test whether the proportion of females in a school differs from a national proportion. The null hypothesis might be H₀: p = 0.5, and the alternative H₁: p ≠ 0.5 for a two-tailed test.
常见考试任务是检验某校女生比例是否与全国比例不同。原假设可能为 H₀：p = 0.5，备择假设为 H₁：p ≠ 0.5，用于双侧检验。
An observation such as ‘6 females in a sample of 20’ can be tested using X ~ B(20, 0.5). We compute P(X ≤ 6) ≈ 0.0577. In a two-tailed test at the 5% significance level, each tail has
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