📚 Year 10 CCEA Sociology: Quick Reference Handbook of Formulas and Theorems | Year 10 CCEA 社会学：公式定理速查手册
In CCEA Year 10 Sociology, you need to master both quantitative research methods and key theoretical frameworks. This compact guide brings together essential statistical formulas used in sociological investigations and classical sociological theorems that underpin your understanding of society. Use it to revise central tendency, dispersion, rates, and the foundational propositions of Marx, Durkheim, and Thomas.
在 CCEA 10年级社会学课程中，你需要掌握定量研究方法与关键理论框架。这本紧凑的手册汇集了社会学调查中使用的核心统计公式以及支撑你对社会理解的经典社会学定理。用它来复习集中趋势、离散度、比率以及马克思、涂尔干和托马斯的奠基性命题。
1. Formula: Arithmetic Mean | 公式：算术平均数
The arithmetic mean (average) is a measure of central tendency. It is found by adding up all the values in a dataset and dividing by the total number of values. Sociologists use the mean to report typical income, screen time, or test scores in a sample.
算术平均数（均值）是一种集中趋势度量。它通过将数据集中所有数值相加，然后除以数值总数得出。社会学家使用均值报告样本中的典型收入、屏幕时间或测试分数。
Mean (x̄) = Σx / n
Here x represents each individual data value, Σ means ‘sum of’, and n is the number of observations. The mean is sensitive to extreme values, so it works best for symmetrical distributions.
在此 x 代表每个单独数据值，Σ 表示“总和”，n 是观测个数。均值对极端值敏感，因此最适合对称分布。
2. Formula: Median | 公式：中位数
The median is the middle value when data are placed in order from smallest to largest. It splits the distribution into two equal halves and is not influenced by outliers, making it a better measure of central tendency for skewed data like house prices or crime counts.
中位数是将数据按从小到大排序后位于中间的值。它将分布平分为两半，且不受异常值影响，因此对于像房价或犯罪数量这样的偏态数据，它是更好的集中趋势度量。
For an odd number of observations, the median is the value at position (n+1)/2. For an even number, it is the average of the two middle values.
对于奇数个观测值，中位数位于 (n+1)/2 位置的数值；对于偶数个观测值，则是中间两个数值的平均数。
Median position = (n + 1) ÷ 2
3. Formula: Mode | 公式：众数
The mode is the value that appears most frequently in a dataset. A distribution can have one mode (unimodal), two modes (bimodal), or more. The mode is particularly useful for categorical variables, such as the most common marital status or favourite leisure activity.
众数是数据集中出现频率最高的值。一个分布可以有一个众数（单峰）、两个众数（双峰）或更多。众数对于类别变量特别有用，例如最常见的婚姻状况或最喜爱的休闲活动。
If no value repeats, there is no mode. If multiple values tie for the highest frequency, all can be reported.
如果没有重复值，就没有众数。如果有多个值并列最高频率，都可以报告。
4. Formula: Range | 公式：全距
Range is the simplest measure of dispersion. It tells you how spread out the data are by subtracting the smallest value from the largest value. While easy to compute, it can be heavily affected by a single outlier.
全距是最简单的离散度度量。它通过用最大值减去最小值告诉你数据的分散程度。虽然计算简单，但它极易受单个异常值的影响。
Range = Maximum value - Minimum value
For example, if ages in a youth group range from 11 to 19, the range is 8 years.
例如，如果一个青年群体的年龄从11岁到19岁，全距为8岁。
5. Formula: Interquartile Range | 公式：四分位距
The interquartile range (IQR) measures the spread of the middle 50% of data, reducing the impact of outliers. It is the difference between the upper quartile (Q₃) and the lower quartile (Q₁). Q₁ is the median of the lower half of data, Q₃ the median of the upper half.
四分位距（IQR）衡量中间50%数据的散布程度，减少了异常值的影响。它是上四分位数（Q₃）与下四分位数（Q₁）之差。Q₁为数据下半部分的中位数，Q₃为上半部分的中位数。
IQR = Q₃ - Q₁
A smaller IQR indicates that the central values are tightly clustered; a larger IQR shows greater variability.
较小的IQR表明中心值紧密聚集；较大的IQR则显示出更大的变异性。
6. Formula: Percentage Change | 公式：百分比变化
Percentage change quantifies how much a variable has grown or shrunk over time. Sociologists use it to analyse trends, such as the rise in divorce rates or the fall in religious attendance.
百分比变化量化变量随时间增长或缩减的幅度。社会学家用它来分析趋势，例如离婚率的上升或宗教参与率的下降。
Percentage Change = (New Value - Original Value) / Original Value × 100%
A positive result means an increase; a negative result means a decrease. Always use the original value as the denominator.
正数结果表示增加；负数结果表示减少。始终以原始值作为分母。
7. Formula: Ratio and Proportion | 公式：比率与比例
Ratios compare the relative size of two groups, often expressed as ‘part to part’. Proportions express a part relative to the whole, typically as a decimal or fraction. In sociology, you encounter gender ratios, dependency ratios, and ethnic proportions.
比率比较两个群体的相对大小，常表示为“部分比部分”。比例表示整体中的一部分，通常以小数或分数形式呈现。在社会学中，你会遇到性别比、抚养比和族裔比例。
| Concept | Formula | Example |
|---|---|---|
| Ratio | Group A : Group B | Males per 100 females |
| Proportion | Part / Whole | 0.15 of households |
Dependency ratio, a common sociological measure, compares dependants to the working‑age population.
抚养比，一种常见的社会学衡量指标，将非劳动人口与劳动年龄人口进行比较。
Dependency Ratio = (Population aged 0–14 + 65+) / Working-age population × 100
8. Formula: Rate per 1,000 | 公式：每千人比率
Sociological rates, such as birth rate, death rate, and crime rate, are standardised per 1,000 people to allow fair comparisons across regions or years regardless of population size.
社会学比率，如出生率、死亡率和犯罪率，被标准化为每千人，以便在地区或年份之间进行公平比较，不受人口规模影响。
Rate = (Number of events / Total population) × 1,000
cords 200 burglaries in a year, the burglary rate is (200 ÷ 50,000) × 1,000 = 4 per 1,000.
例如，如果一个人口5万的城镇一年记录了200起入室盗窃，则入室盗窃率为 (200 ÷ 50,000) × 1,000 = 每千人4起。
9. Theorem: Marx’s Surplus Value Rate | 定理：马克思的剩余价值率
Karl Marx’s theory of exploitation centres on the extraction of surplus value. Workers are paid only a fraction of the value they produce; the rest is appropriated by capitalists as profit. The rate of surplus value quantifies this exploitation.
卡尔·马克思的剥削理论核心在于榨取剩余价值。工人只获得他们所创造价值的一部分；其余部分被资本家以利润形式占有。剩余价值率量化了这种剥削。
Surplus Value Rate (s′) = s / v
Here s stands for surplus value (unpaid labour) and v for variable capital (wages paid to labour). The higher the ratio, the greater the exploitation of workers according to Marxian sociology.
在此 s 代表剩余价值（无酬劳动），v 代表可变资本（支付给劳动力的工资）。根据马克思主义社会学，该比率越高，对工人的剥削越严重。
10. Theorem: Durkheim’s Social Integration and Suicide |
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