📚 SQA Psychology Formula & Theorem Quick Reference Handbook | SQA心理学公式定理速查手册
Welcome to your essential reference for formulas and key theorems in SQA Year 11 Psychology. This handbook covers core statistical methods used in research, classic psychophysical laws, and influential psychological principles. Whether you are preparing for your exam or consolidating knowledge, use this guide to review definitions, notations, and practical applications. All formulas are presented with step‑by‑step explanations and paired bilingual insights.
欢迎使用 SQA Year 11 心理学的必备公式与定理速查手册。本手册涵盖研究方法中的核心统计工具、经典心理物理学定律以及影响深远的心理学原理。无论你是在备考还是巩固知识，都可以通过本指南重温定义、符号和实际应用。每个公式都配有逐步解释和双语要点，方便你快速查阅。
1. Mean (Average) Formula | 平均数公式
The mean is the sum of all raw scores divided by the number of scores. It is the most widely used measure of central tendency in psychological research.
平均数等于所有原始分数之和除以分数个数，是心理学研究中使用最广泛的集中趋势量数。
M = Σx / n
Here Σx represents the sum of all individual data points, and n is the total sample size. The mean is sensitive to extreme values (outliers), so it best represents normally distributed interval or ratio data.
其中 Σx 表示所有个体数据点之和，n 为样本总容量。平均数对极端值（离群值）敏感，因此最适合代表呈正态分布的等距或比率数据。
2. Standard Deviation (SD) Formula | 标准差公式
The standard deviation measures the average spread of scores around the mean. A larger SD indicates greater variability. For a sample, we usually use the formula that divides by (n – 1) to give an unbiased estimate of the population SD.
标准差衡量分数围绕平均数的平均离散程度；标准差越大，变异越大。计算样本标准差时通常除以 (n – 1)，以得到总体标准差的无偏估计。
SD = √[ Σ(x – M)² / (n – 1) ]
Each deviation (x – M) is squared to remove negative signs. The square root brings the value back to the original unit. In psychological studies, SD helps describe the consistency of behaviours, reaction times, or survey responses.
每个离差 (x – M) 平方后消除负号，再开方使数值恢复原单位。在心理学研究中，标准差可用于描述行为、反应时或问卷答案的一致程度。
3. Z-Score Formula | 标准分数（Z 分数）公式
A z-score transforms a raw score into a standardised value that indicates how many standard deviations it is above or below the mean. This allows meaningful comparisons across different distributions or measurement scales.
Z 分数将原始分数转化为标准化数值，表示该分数高出或低于平均数多少个标准差，从而可以在不同分布或测量尺度之间进行有意义的比较。
z = (x – M) / SD
For example, a z-score of +1.5 means the score is 1.5 standard deviations above the mean. In the standard normal distribution, about 68% of z-scores fall between –1 and +1. Psychologists use z-scores to interpret clinical scales and research outcomes.
例如，z 分数为 +1.5 表示该分数高于平均数 1.5 个标准差。在标准正态分布中，约 68% 的 z 分数落在 –1 至 +1 之间。心理学家用 z 分数解读临床量表和研究成果。
4. Chi-Square (χ²) Formula | 卡方检验公式
The chi-square test for independence assesses whether two categorical variables are associated. It compares observed frequencies (O) with expected frequencies (E) under the null hypothesis.
独立性卡方检验评估两个类别变量之间是否存在关联。它比较观察频数 (O) 与零假设下的期望频数 (E)。
χ² = Σ[(O – E)² / E]
l) / grand total. A larger χ² value suggests a significant deviation from chance. You must use a critical values table and check degrees of freedom (rows–1)(columns– nificance. SQA Higher candidates should be able to calculate χ² and interpret its meaning.
每格期望频数 = (行合计数 × 列合计数) / 总计。χ² 值越大，说明偏离随机程度的可能性越大。你需要查阅临界值表，并根据自由度 (行数–1)(列数–1) 判断显著性。SQA Higher 考生应能计算 χ² 并解释其含义。
5. Spearman’s Rank Correlation Coefficient | 斯皮尔曼等级相关系数
Spearman’s rho (ρ) measures the strength and direction of a monotonic relationship between two ranked (ordinal) variables. It is a non-parametric alternative to Pearson’s r.
斯皮尔曼 ρ 系数测量两个等级（顺序）变量之间单调关系的强度和方向，是皮尔逊 r 的非参数替代方法。
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. A ρ of +1 indicates a perfect positive correlation, –1 a perfect negative correlation, and 0 no correlation. You must rank the scores separately for each variable before computing d.
其中 d 是每对数据的秩次差，n 是对数。ρ 为 +1 表示完全正相关，–1 表示完全负相关，0 表示无相关。计算前需先分别对每个变量排序赋予等级。
6. Mann–Whitney U Test Formula | 曼–惠特尼 U 检验公式
The Mann–Whitney U test is a non-parametric test for differences between two independent groups. It uses rank orders rather than raw scores, making it suitable for ordinal data or data that violate parametric assumptions.
曼–惠特尼 U 检验是比较两个独立组差异的非参数检验。它基于秩次而非原始分数，适合顺序数据或不符合参数检验假设的数据。
U₁ = n₁n₂ + n₁(n₁ + 1)/2 – R₁
two groups; R₁ is the sum of ranks for group 1. The smaller of U₁ and U₂ (where U₂ = n₁n₂ – mpared with a critical value. If U is ≤ the critical value, the difference is significant.
n₁ 和 n₂ 分别为两组样本量，R₁ 为第 1 组的秩和。取 U₁ 和 U₂（U₂ = n₁n₂ – U₁）中较小的值与临界值比较。若 U 小于或等于临界值，则差异显著。
7. Wilcoxon Signed-Rank Test Formula | 威尔科克森符号秩检验公式
The Wilcoxon test compares two related samples or repeated measures. It looks at the difference scores, ranks them ignoring sign, then sums the positive and negative ranks separately.
威尔科克森检验比较两个相关样本或重复测量。它计算差值，忽略符号后进行秩次排序，再分别求正、负秩次之和。
T = the smaller sum of signed ranks
sample size. If T is ≤ the critical value, you reject the null hypothesis and conclude there is a significant difference.
检验统计量 T 取正秩和 (R⁺) 与负秩和 (R⁻) 中绝对值较小者。将 T 与对应样本量的临界值比较，若 T ≤ 临界值，则拒绝零假设，判定差异显著。
8. Weber’s Law | 韦伯定律
Weber’s Law describes the relationship between the magnitude of a physical stimulus and the just noticeable difference (JND). It states that the JND is a constant proportion of the original stimulus intensity.
韦伯定律描述物理刺激强度与最小可觉差 (JND) 的关系，指出 JND 是原始刺激强度的一个恒定比例。
ΔI / I = k
ΔI is the increment needed to detect a change, I is the initial stimulus intensity, and k is the Weber fraction. For example, the Weber fraction for lifted weights is about 0.02, meaning you can just notice a 2 g change on a 100 g weight. Weber’s Law holds for moderate stimulus ranges but breaks down at extremes.
ΔI 为觉察变化所需增量，I 为初始刺激强度，k 是韦伯分数。例如，提举重量的韦伯分数约为 0.02，意味着 100 克重量增加 2 克即可察觉。韦伯定律适用于中等刺激范围，在极端值处失效。
9. Fechner’s Law | 费希纳定律
Fechner extended Weber’s Law by proposing a logarithmic relationship between physical intensity and perceived sensation. As stimulus intensity multiplies, sensation increases in additive steps.
费希纳扩展了韦伯的工作，提出物理强度与心理感觉量之间呈对数关系：刺激强度成倍增加时，感觉量以等差方式增长。
S = k log I
S is the subjective sensation magnitude, I is the physical stimulus intensity, and k is a scaling constant. This law explains why a light that is twice as bright physically does not appear twice as bright perceptually. Fechner’s Law is historically important but often replaced by Stevens’ Power Law in modern psychophysics.
S 是主观感觉量，I 是物理刺激强度，k 为比例常数。这解释了一盏物理亮度两倍的灯，感知上并不觉得亮两倍。费希纳定律在历史上意义重大，但在现代心理物理学中常被史蒂文斯幂定律取代。
10. Stevens’ Power Law | 史蒂文斯幂定律
Stevens proposed a more flexible psychophysical function: sensation magnitude grows as a power function of stimulus intensity. The exponent n varies across sensory modalities.
史蒂文斯提出了更灵活的心理物理函数：感觉量按刺激强度的幂函数增长，指数 n 因感觉道而异。
S = k Iⁿ
For brightness, n is about 0.33 (compressive); for electric shock, n is about 3.5 (expansive). This means a small increase in shock intensity produces a large jump in perceived pain. The law is widely used in perception, ergonomics, and human‑factors design.
亮度的指数 n 约为 0.33（压缩函数）；电击的指数 n 约为 3.5（扩展函数），即电击强度小幅增加会导致痛觉大幅上升。该定律广泛用于知觉、人机工程和人因设计。
11. Yerkes–Dodson Law | 耶基斯–多德森定律
This law describes the relationship between arousal and performance. It states that performance increases with arousal up to an optimal point, after which further arousal leads to deteriorating performance – an inverted‑U shape.
该定律描述唤醒水平与表现之间的关系：表现随唤醒提高而提升，达到最佳点后，进一步唤醒导致表现下降，呈倒 U 型曲线。
The optimal level of arousal depends on task complexity: simple or well‑practised tasks benefit from higher arousal, while complex or novel tasks require lower arousal. This principle helps explain examination anxiety and sports performance. The law emerged from experiments in which mice learned to discriminate light at different shock intensities.
最佳唤醒水平取决于任务难度：简单或熟练的任务在较高唤醒下表现更好，复杂或新异任务则需要较低唤醒。这一原理有助于解释考试焦虑和运动表现。该定律源自老鼠在不同电击强度下学习光线辨别任务的实验。
12. Signal Detection Theory – d' (d‑prime) | 信号检测论 d′ 统计量
Signal detection theory separates sensitivity from response bias. The index d' (d‑prime) measures an observer’s ability to discriminate between signal‑present and signal‑absent trials, independent of their criterion.
信号检测论将感受性从反应偏向中分离出来。指标 d′ 衡量观察者区分信号出现与信号缺失试验的能力，且不受判断标准影响。
d′ = z(Hit Rate) – z(False Alarm Rate)
Hit Rate is the proportion of “yes” responses when a signal is present; False Alarm Rate is the proportion of “yes” responses when no signal is present. The z‑transformation converts these proportions to standard normal deviates. A higher d′ indicates better sensitivity. Understanding d′ is crucial for interpreting studies on memory, perception, and clinical diagnosis.
击中率为信号出现时回答“是”的比例；虚报率为无信号时回答“是”的比例。z 变换将这些比例转化为标准正态离差。d′ 越大，感受性越好。理解 d′ 对解读记忆、知觉及临床诊断研究至关重要。
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