📚 Year 12 CCEA Psychology: Essential Formulas & Theorems Quick Guide | CCEA 12 年级心理学核心公式定理速查手册
Mastering the mathematical and conceptual tools of AS Psychology is essential for tackling research methods questions with confidence. This quick-reference guide brings together all the key formulas, statistical tests, and fundamental theorems that Year 12 CCEA Psychology students need at their fingertips. You'll find clear explanations, worked notation, and bilingual support to help you navigate descriptive statistics, inferential tests, and the core principles that underpin psychological research.
掌握 AS 心理学的数学与概念工具，是自信应对研究方法题目的关键。这份速查手册汇集了 CCEA Year 12 心理学学生必须掌握的所有核心公式、统计检验和基本原理。你将会找到清晰的解释、符号演示以及中英对照支持，帮助你轻松驾驭描述统计、推断检验以及支撑心理学研究的核心原理。
1. Measures of Central Tendency | 集中趋势的度量
The three main measures summarise the centre of a data set. The mean is calculated as the sum of all scores divided by the number of scores. The median is the middle value when data are ordered. The mode is the most frequently occurring value. In psychological research, the mean is used for interval/ratio data unless there are outliers, in which case the median is preferred.
三大集中趋势指标概括了数据的中心。 平均值 是所有分数的总和除以分数的个数。 中位数 是数据排序后的中间值。 众数 是出现频率最高的值。在心理学研究中，平均值用于等距/等比数据，但若存在异常值，则优先选择中位数。
| Measure | Formula / Definition | Best used for |
| Mean | x̄ = Σx / n | Symmetric, no outliers |
| Median | Middle score (n+1)/2 th value | Skewed data, ordinal data |
| Mode | Most frequent value | Nominal data |
2. Range and Interquartile Range | 全距与四分位距
The range is the simplest measure of spread: highest value minus lowest value. It is sensitive to extreme scores. The interquartile range (IQR) is the difference between the third quartile (Q3) and the first quartile (Q1), covering the middle 50% of data. IQR is more robust against outliers and is often used alongside the median.
全距 是最简单的离散度量：最大值减去最小值。它对极端分数敏感。四分位距 (IQR) 是第三四分位数 (Q3) 与第一四分位数 (Q1) 之差，覆盖了中间 50% 的数据。IQR 对异常值更具抵抗力，常与中位数结合使用。
Range = Xmax − Xmin
IQR = Q3 − Q1
3. Variance and Standard Deviation | 方差与标准差
Variance and standard deviation tell us how much scores deviate from the mean. The sample standard deviation (s) is the square root of the average squared deviation from the mean, using n−1 for degrees of freedom. The larger the standard deviation, the greater the spread. These measures are essential for parametric tests in psychology.
方差和标准差告诉我们分数相对于平均值的偏离程度。样本标准差 (s) 是各分数与平均值离差平方和的平均值的平方根，除以 n−1 以计算自由度。标准差越大，数据的离散度越大。这些度量对心理学中的参数检验至关重要。
s = √[ Σ(x − x̄)² / (n − 1) ]
The variance is simply s². In practice, the standard deviation is used because it is in the original units of measurement.
方差就是 s²。在实际应用中，使用标准差是因为它保留了原始测量单位。
4. Standard Scores (z-scores) | 标准分数（z 分数）
A z-score tells us how many standard deviations a raw score is above or below the mean. It allows researchers to compare scores from different distributions. A positive z-score indicates above average; a negative z-score indicates below average. z-scores are fundamental to probability and significance testing using normal distribution tables.
z 分数表示一个原始分数与平均值之间相差了多少个标准差。它使研究者能够比较来自不同分布的分数。正值 z 分数表示高于平均值；负值表示低于平均值。z 分数是利用正态分布表进行概率和显著性检验的基础。
z = (x − x̄) / s
Once calculated, you can look up the corresponding probability to determine how unusual a score is. In psychological testing, z-scores are commonly used to standardise psychometric instruments.
计算出 z 后，可以查找对应的概率值，以判断该分数有多不寻常。在心理测评中，z 分数通常用于标准化心理测量工具。
5. Spearman's Rank Correlation Coefficient | 斯皮尔曼等级相关系数
Spearman's rho (ρ or rs) is a non-parametric test for correlation between two sets of ranked data. It assesses the strength and direction of a monotonic relationship. The value ranges from −1 (perfect negative) through 0 (no correlation) to +1 (perfect positive). It is widely used in AS Psychology for ordinal data or when the assumptions of Pearson's r are violated.
斯皮尔曼等级相关系数 (ρ 或 rs) 是一种用于两组等级数据之间相关性的非参数检验。它评估单调关系的强度和方向。取值范围从 −1（完全负相关）到 0（无相关）再到 +1（完全正相关）。当数据为顺序尺度或违反 Pearson r 的假设时，此方法在 AS 心理学中广泛使用。
rs = 1 − [ 6 Σd² / (n(n² − 1)) ]
Here, d is the difference between the ranks of each pair, and n is the number of pairs. After calculating rs, you compare it against a critical value table for the given sample size to determine significance.
其中 d 是每对数据的等级差，n 是数据对的数量。计算出 rs 后，将其与特定样本量下的临界值表进行比较，以判断显著性。
6. Chi-Square Test (χ²) | 卡方检验 (χ²)
The chi-square test for independence is used to examine whether there is a significant association between two categorical (nominal) variables. It compares the observed frequencies with the frequencies expected if there were no association. The test is a staple in Year 12 research methods when analysing data from questionnaires or observational categories.
独立性卡方检验用于考察两个类别（名义）变量之间是否存在显著关联。它将观察到的频数与没有关联时预期的频数进行比较。在 Year 12 研究方法中，分析问卷调查或观察类别的数据时，该检验是必备工具。
χ² = Σ [ (O − E)² / E ]
mns − 1). Calculated χ² is compared with a critical value to decide whether to reject the null hypothesis.
O 为观察频数，E 为预期频数（行合计 × 列合计 ÷ 总计）。自由度 (df) = (行数 − 1) × (列数 − 1)。计算出的 χ² 与临界值比较，以决定是否拒绝零假设。
7. Probability and Significance Levels | 概率与显著性水平
sychology, the conventional significance level is p ≤ 0.05, meaning there is less than a 5% erved (or more extreme) result if the null hypothesis is true. If p ≤ alpha, the result is 'significant' we , and we reject H₀ in favour of H₁. Setting alpha too low increases the risk of a Type II error.
在心理学中，惯用的显著性水平为 p ≤ 0.05，这意味着观察到的结果纯粹由偶然因素造成的概率低于 5%。p 值是在零假设成立时，获得当前结果（或更极端结果）的概率。如果 p ≤ α（显著性水平），结果即为“显著”，我们拒绝 H₀ 而接受 H₁。将 α 设置得过低会增加 II 型错误的风险。
- Null hypothesis (H₀): no effect or no difference / 零假设：无效果或无差异
- Alternative hypothesis (H₁): there is an effect or difference / 备择假设：有效果或有差异
8. Type I and Type II Errors | I 型错误与 II 型错误
A Type I error occurs when we reject a true null hypothesis — a 'false positive'. The probability of this is alpha, usually 0.05. A Type II error occurs when we fail to reject a false null hypothesis — a 'false negative'. The probability of a Type II error is beta. Power (1 − β) is the ability to detect a real effect. Understanding these errors is a key theorem in hypothesis testing and helps psychologists evaluate the reliability of conclusions.
I 型错误 发生在拒绝了正确的零假设时——即“假阳性”。其概率为 α，通常为 0.05。II 型错误 发生在未能拒绝错误的零假设时——即“假阴性”。II 型错误的概率为 β。统计检验力 (1 − β) 是指检测出真实效应的能力。理解这两种错误是假设检验中的核心定理，有助于心理学家评估结论的可靠性。
| H₀ is true | H₀ is false | |
| Reject H₀ | Type I error (α) | Correct decision (1 − β) |
| Do not reject H₀ | Correct decision (1 − α) | Type II error (β) |
9. Choosing the Right Statistical Test | 选择恰当的统计检验
Selecting a statistical test depends on three questions: (1) Is the research testing a difference or correlation? (2) What is the level of measurement (nominal, ordinal, interval)? (3) Is the design independent groups, repeated measures, or matched pairs? The table below summarises the common tests you need for Year 12 CCEA Psychology. Always justify your choice in the exam.
选择统计检验取决于三个问题：(1) 研究检验的是差异还是相关？ (2) 测量尺度是什么（名义、顺序、等距）？ (3) 实验设计是独立组、重复测量还是配对组？下表总结了 CCEA Year 12 心理学常见的检验。考试时务必说明你的选择理由。
| Test name | Used for | Data type | Design |
| Spearman's rho | Correlation | Ordinal | N/A |
| Chi-square (χ²) | Difference/association | Nominal | Independent |
| Mann-Whitney U | Difference | Ordinal | Independent groups |
| Wilcoxon signed ranks | Difference | Ordinal | Repeated measures / matched pairs |
10. Ethical Principles as Research Theorems | 伦理原则作为研究定理
While not mathematical theorems, the British Psychological Society (BPS) ethical principles function as immutable guidelines that protect participants. The main principles are: respect for autonomy (informed consent), beneficence (maximising benefits and minimising harm), justice (fair treatment), and scientific integrity. Deception must be justified and followed by debriefing. In Year 12, you must be able to apply these 'theorems' to evaluate psychological studies.
尽管不是数学定理，但英国心理学会（BPS）的伦理原则就像保护被试不可动摇的指南。主要原则包括：尊重自主权（知情同意）、慈善（最大化益处并最小化伤害）、公正（公平对待）和科学诚信。欺骗必须具有合理性，并在事后进行情况说明。在 Year 12 阶段，你必须能够运用这些“定理”来评估心理学研究。
- Informed consent / 知情同意
- Deception and debriefing / 欺骗与事后说明
- Right to withdraw / 退出权
- Protection from harm / 免受伤害
- Confidentiality / 保密性
11. Validity and Reliability as Core Concepts | 效度与信度作为核心概念
Validity refers to whether a test measures what it claims to measure, while reliability refers to the consistency of measurement. Internal validity asks whether the observed effects are due to the manipulation of the IV; external validity asks if findings can be generalised. Ecological validity, population validity, and temporal validity are all aspects you must evaluate. Reliability can be assessed through test-retest or inter-rater methods. These ideas are foundational 'theorems' for critiquing research.
效度指一个测试是否测量了它所宣称要测量的内容，而信度指测量的一致性。内部效度探究观察到的效应是否由自变量的操纵引起；外部效度则关注研究结果是否可以推广。生态效度、人口效度和时间效度都是需要评估的方面。信度可以通过再测信度或评分者间信度来评估。这些概念是评论研究的基础“定理”。
12. Quick-Access Formula Summary | 公式速查总结
Here is a compact overview of the key formulas for your revision. Keep this page handy when working through past papers.
以下为核心公式的简明汇总，复习时请随手翻阅，尤其适合刷历年真题时使用。
| Concept | Formula |
| Mean | x̄ = Σx / n |
| Range | Xmax − Xmin |
| Standard deviation (sample) | s = √[ Σ(x − x̄)² / (n − 1) ] |
| z-score | z = (x − x̄) / s |
| Spearman's rho | rs = 1 − [ 6 Σd² / (n(n² − 1)) ] |
| Chi-square | χ² = Σ[ (O − E)² / E ] |
| Degrees of freedom (chi-square) | df = (rows − 1)(columns − 1) |
| Mann-Whitney U (smaller of UA, UB) | U = n₁n₂ + [ n₁(n₁+1)/2 ] − R₁ |
Remember that statistical tables are provided in the exam; you are not expected to memorise critical values. Focus on understanding when to apply each formula and how to interpret the results in context.
请记住，考试中会提供统计临界值表，你无需记忆具体临界值。重点在于理解何时应用每个公式，以及如何在语境中解释结果。
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