📚 Year 13 SQA Psychology: Essential Formulas & Theorems Quick-Reference Handbook | SQA 心理学：公式定理速查手册
This comprehensive quick-reference guide is designed for Year 13 students preparing for SQA Psychology examinations, covering the critical statistical formulas, research methods principles, and key theoretical equations you need to master. From descriptive statistics to inferential testing and foundational laws in psychophysics, this handbook consolidates the must-know content into clear, bilingual explanations. Use it to reinforce your revision and ensure you can confidently apply every formula in exam contexts.
这本全面的速查手册专为备考 SQA 心理学的 Year 13 学生设计，涵盖了你必须掌握的关键统计公式、研究方法原则和重要理论方程。从描述统计到推断性检验，再到心理物理学的基本定律，本手册将必考内容整合为清晰的双语解释。用它来巩固复习，确保在考试情境中能够自信地应用每一个公式。
1. Measures of Central Tendency | 集中趋势量数
The mean is the arithmetic average and the most commonly used measure of central tendency, sensitive to extreme scores. It is calculated by summing all values and dividing by the number of values.
均值是算术平均数，是最常用的集中趋势量数，对极端分数敏感。计算方法为将所有数值相加后除以数值的个数。
Mean (x̄) = Σx / n
The median is the middle score when data are arranged in order, unaffected by outliers and preferred for skewed distributions.
中位数是将数据按顺序排列后位于中间的数值，不受异常值影响，适用于偏态分布。
The mode is the most frequently occurring score, useful for categorical data but may not represent the centre well in small datasets.
众数是出现频率最高的分数，对分类数据有用，但在小数据集中可能无法很好地代表中心。
2. Measures of Dispersion | 离散趋势量数
The range provides a quick sense of spread but is heavily influenced by outliers; it is simply the difference between the highest and lowest scores.
极差提供了离散程度的快速感知，但受异常值影响很大；它只是最高分与最低分之差。
Range = Xmax – Xmin
Standard deviation measures how much scores deviate from the mean on average. A larger SD indicates greater variability within the dataset.
标准差衡量分数平均偏离均值的程度。标准差越大，表示数据集内的变异性越大。
SD = √[ Σ(x – x̄)² / (n – 1) ]
Variance is the square of the standard deviation and is fundamental to many inferential statistics; the formula above uses n–1 for the sample variance to provide an unbiased estimate.
方差是标准差的平方，是许多推断统计的基础；上述公式中使用 n–1 计算样本方差，以得出无偏估计。
3. Normal Distribution and Z-Scores | 正态分布与 Z 分数
The normal distribution is a symmetrical, bell-shaped curve where the mean, median and mode coincide. In SQA Psychology, understanding its properties is crucial for hypothesis testing.
正态分布是对称的钟形曲线，均值、中位数和众数重合。在 SQA 心理学中，理解其性质对假设检验至关重要。
A Z-score tells you how many standard deviations a raw score lies from the mean, allowing comparison across different scales.
Z 分数告诉你一个原始分数距离均值有多少个标准差，允许跨不同量表进行比较。
z = (x – μ) / σ
In the standard normal distribution, approximately 68% of values fall within ±1 SD, 95% within ±1.96 SD, and 99.7% within ±3 SD. This underpins confidence intervals and significance levels.
在标准正态分布中，大约 68% 的值落在 ±1 标准差内，95% 落在 ±1.96 标准差内，99.7% 落在 ±3 标准差内。这是置信区间和显著性水平的基础。
4. Pearson’s Product-Moment Correlation Coefficient | 皮尔逊积差相关系数
Pearson’s r quantifies the strength and direction of a linear relationship between two continuous variables, ranging from –1 to +1.
皮尔逊 r 量化两个连续变量之间线性关系的强度和方向，取值范围从 –1 到 +1。
r = Σ[(x – x̄)(y – ȳ)] / √[Σ(x – x̄)² Σ(y – ȳ)²]
A positive r indicates that as one variable increases, the other tends to increase; a negative r indicates an inverse relationship. Correlation does not imply causation.
正 r 表示一个变量增加时，另一个也趋于增加；负 r 表示反向关系。相关并不意味着因果。
The coefficient of determination, r², indicates the proportion of shared variance. For instance, r=0.8 yields r²=0.64, meaning 64% of the variance is shared.
决定系数 r² 表示共享方差的比例。例如，r=0.8 得出 r²=0.64，意味着 64% 的方差是共享的。
5. Spearman’s Rank-Order Correlation | 斯皮尔曼等级相关系数
Spearman’s rho (ρ) is a non-parametric alternative to Pearson’s r, used when data are ordinal or when the relationship is monotonic but not linear.
斯皮尔曼 rho (ρ) 是皮尔逊 r 的非参数替代方法，用于顺序数据或关系为单调但非线性的情况。
ρ = 1 – [6 Σd² / n(n² – 1)]
Here, d is the difference between the ranks of each pair, and n is the number of pairs. The formula assumes no tied ranks, though corrections exist for ties in SQA exams.
其中 d 是每对数据的等级差，n 是配对数。该公式假设无相同等级，不过在 SQA 考试中，存在相同等级时有校正方法。
Spearman’s rho is robust to outliers and is frequently used with Likert-scale data in psychological research.
斯皮尔曼 rho 对异常值稳健，经常用于心理学研究中的李克特量表数据。
6. Chi-Square Test for Independence | 卡方独立性检验
The chi-square test examines whether there is a significant association between two categorical variables. It compares observed frequencies with expected frequencies under the null hypothesis of no association.
卡方检验考察两个分类变量之间是否存在显著关联。它将观察频数与零假设（无关联）下的期望频数进行比较。
χ² = Σ [(O – E)² / E]
Expected frequency for each cell is calculated as (row total × column total) / grand total. Degrees of freedom for a contingency table is (rows – 1)(columns – 1).
每个单元格的期望频数计算公式为 (行合计 × 列合计) / 总计。列联表的自由度为 (行数 – 1)(列数 – 1)。
If the calculated χ² value exceeds the critical value at a given significance level (commonly p < 0.05), the null hypothesis is rejected.
如果计算出的 χ² 值超过给定显著性水平（通常 p < 0.05）下的临界值，则拒绝零假设。
7. Independent Samples t-Test | 独立样本 t 检验
An independent t-test compares the means of two unrelated groups to determine if they differ significantly. It assumes normality and homogeneity of variance.
独立 t 检验比较两个无关组的平均值，以确定它们是否存在显著差异。它假设正态性和方差齐性。
t = (x̄₁ – x̄₂) / √[sₚ²(1/n₁ + 1/n₂)]
Where sₚ² is the pooled variance: sₚ² = [(n₁–1)s₁² + (n₂–1)s₂²] / (n₁ + n₂ – 2). The degrees of freedom are n₁ + n₂ – 2.
其中 sₚ² 是合并方差：sₚ² = [(n₁–1)s₁² + (n₂–1)s₂²] / (n₁ + n₂ – 2)。自由度为 n₁ + n₂ – 2。
The paired samples (or related) t-test, used for repeated measures or matched pairs designs, relies on the mean of difference scores and uses a simpler formula: t = ȳd / (sd / √n).
配对样本（或相关）t 检验用于重复测量或配对设计，依赖差异分数的均值，使用更简单的公式：t = ȳd / (sd / √n)。
8. Mann-Whitney U Test & Wilcoxon Signed-Rank Test | 曼-惠特尼 U 检验与威尔科克森符号秩检验
When parametric test assumptions are violated, non-parametric equivalents are used. The Mann-Whitney U test is the alternative to the independent t-test, and the Wilcoxon test replaces the paired t-test.
当参数检验的假设被违背时，使用非参数等价方法。曼-惠特尼 U 检验是独立 t 检验的替代，威尔科克森检验取代配对 t 检验。
U = n₁n₂ + n₁(n₁+1)/2 – R₁
Here R₁ is the sum of ranks for group 1. The smaller U value is compared against critical values. For large samples, a normal approximation is used.
其中 R₁ 是第1组的秩和。较小的 U 值与临界值比较。对于大样本，使用正态近似。
For the Wilcoxon test, difference scores are ranked ignoring sign, and the test statistic T is the smaller sum of ranks of the less frequent sign.
对于威尔科克森检验，差异分数按绝对值排秩，检验统计量 T 是出现频率较低的符号的秩和中的较小值。
9. Signal Detection Theory Metrics | 信号检测论指标
Signal Detection Theory distinguishes between sensitivity (d’) and response bias (criterion). It is used in studies of perception, memory, and decision-making.
信号检测论区分感受性 (d’) 和反应偏向（判断标准）。它用于知觉、记忆和决策研究。
d’ = z(Hit) – z(False Alarm)
Sensitivity d’ reflects the ability to discriminate between signal and noise. A larger d’ means better discrimination.
感受性 d’ 反映了区分信号与噪音的能力。d’ 越大，意味着辨别力越好。
Criterion c = –[z(Hit) + z(False Alarm)] / 2
Criterion measures the participant’s bias towards responding ‘present’ or ‘absent’. A negative c indicates a liberal bias, a positive c a conservative bias.
判断标准衡量被试偏向于回答“有”或“无”的倾向。负 c 表示宽松偏向，正 c 表示保守偏向。
10. Weber’s Law and Fechner’s Law | 韦伯定律与费希纳定律
Weber’s Law states that the just noticeable difference (JND) between two stimuli is proportional to the magnitude of the initial stimulus. This is a foundational principle in psychophysics.
韦伯定律指出，两个刺激之间的最小可觉差 (JND) 与初始刺激的强度成正比。这是心理物理学的基石原则。
ΔI / I = k
Where ΔI is the JND, I is the stimulus intensity, and k is the Weber fraction constant for a given sensory modality.
其中 ΔI 是 JND，I 是刺激强度，k 是特定感觉通道的韦伯分数常数。
Fechner’s Law extends this by proposing that sensation magnitude grows logarithmically with stimulus intensity: S = k log I, where S is sensation and I is physical intensity.
费希纳定律对此进行扩展，提出感觉强度随刺激强度的对数增长：S = k log I，其中 S 是感觉量，I 是物理强度。
11. Effect Size: Cohen’s d | 效应量：科恩 d
While statistical significance tells you whether an effect exists, effect size quantifies the magnitude of the difference. Cohen’s d is frequently examined in SQA Higher Psychology courses.
虽然统计显著性告诉你效应是否存在，效应量则量化差异的大小。科恩 d 在 SQA 高阶心理学课程中常被考察。
d = (x̄₁ – x̄₂) / spooled
Cohen suggested benchmarks: d=0.2 (small), d=0.5 (medium), d=0.8 (large). Reporting effect size is a key requirement in modern psychological research to avoid over-reliance on p-values.
科恩提出基准：d=0.2（小），d=0.5（中），d=0.8（大）。报告效应量是现代心理学研究的关键要求，以避免过度依赖 p 值。
Alternatively, for correlation-based research, r itself serves as an effect size measure, with thresholds of 0.1, 0.3, and 0.5.
或者，对于基于相关的研究，r 本身可作为效应量度量，阈值分别为 0.1、0.3 和 0.5。
12. Probability and Hypothesis Testing Logic | 概率与假设检验逻辑
The logic of null hypothesis significance testing (NHST) requires clear understanding of Type I and Type II errors. A Type I error is rejecting a true null hypothesis (false positive), with probability α.
零假设显著性检验 (NHST) 的逻辑要求清晰理解 I 类和 II 类错误。I 类错误是拒绝正确的零假设（假阳性），概率为 α。
A Type II error is failing to reject a false null hypothesis (false negative), with probability β. Power is 1 – β, the chance of correctly detecting an effect.
II 类错误是未能拒绝错误的零假设（假阴性），概率为 β。统计功效为 1 – β，即正确检测到效应的机会。
p-value < α → reject H₀; p ≥ α → fail to reject H₀
Always interpret these outcomes in the context of the study, never concluding ‘prove’ the null or alternative hypothesis. The standard alpha level is α = 0.05 unless otherwise stated.
始终在研究语境中解释这些结果，永远不要得出“证明”了零假设或备择假设的结论。除非另有说明，标准 α 水平为 0.05。
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