SQA Year 12 Psychology: Formula & Theorem Quick Reference Handbook

📚 SQA Year 12 Psychology: Formula & Theorem Quick Reference Handbook | 公式定理速查手册

This quick reference handbook consolidates the essential formulae, statistical tests, and psychophysical theorems required for SQA Year 12 Higher Psychology. Whether you are analysing data from experiments or interpreting theoretical relationships, having these equations at your fingertips will strengthen your research methods and understanding of human behaviour. All symbols use standard notation for easy recall during revision and exams.

本速查手册汇集了 SQA Year 12 Higher 心理学所必需的核心公式、统计检验和心理物理学定理。无论你是在分析实验数据，还是在解释理论关系，随手查阅这些公式都能夯实你的研究方法和行为理解。所有符号均采用标准记法，便于复习和考试时快速回忆。

1. Measures of Central Tendency | 集中趋势量数

Central tendency summarises a dataset with a single representative value. The three main measures are the mean, median, and mode. The mean is the arithmetic average and is appropriate for interval or ratio data without extreme outliers.

集中趋势用一个代表性数值概括整个数据集。三种主要量数是均值、中位数和众数。均值是算术平均数，适用于无极端异常值的等距或等比数据。

Mean X̄ = ΣX / n

The median is the middle score when data are ordered; it is robust to outliers, used for ordinal data. The mode is the most frequently occurring value, useful for nominal data.

中位数是排序后居于中间位置的分数，对异常值稳健，适用于顺序数据。众数则是出现频次最高的数值，适用于名义数据。

2. Measures of Dispersion | 离散量数

Dispersion quantifies the spread of scores around the central tendency. The range is the simplest measure (maximum minus minimum). Variance and standard deviation are far more informative, showing average squared deviation and deviation in original units, respectively.

离散量数衡量分数围绕集中趋势的分散程度。全距是最简单的量数（最大值减最小值）。方差和标准差更具信息量，分别表示平均的平方离差和原单位下的离差。

Range = Xₘₐₓ – Xₘᵢₙ

Variance s² = Σ(X – X̄)² / (n – 1)

Standard deviation s = √[Σ(X – X̄)² / (n – 1)]

Here n is the sample size, and we divide by (n – 1) to give an unbiased estimate of the population variance. Standard deviation is essential for z-scores and inferential statistics.

其中 n 为样本量，除以 (n – 1) 是为了给出总体方差的无偏估计。标准差对计算 z 分数和推断统计至关重要。

3. Standard Error of the Mean | 均值的标准误

The standard error of the mean (SEM) indicates how far a sample mean is likely to be from the true population mean. It decreases as sample size increases, reflecting greater precision.

均值的标准误 (SEM) 表示样本均值可能偏离总体真实均值的程度。它随样本量增大而减小，反映更高的精确度。

SEM = s / √n

SEM is the standard deviation of the sampling distribution of the mean. It is used to construct confidence intervals and in t-tests.

SEM 是均值抽样分布的标准差，用于构建置信区间和进行 t 检验。

4. z-Scores and the Normal Distribution | z分数与正态分布

A z-score tells us how many standard deviations a raw score is from the mean. In a normal distribution, about 68% of values lie within ±1 SD, 95% within ±1.96 SD, and 99.7% within ±3 SD.

z分数表示一个原始分数距离均值有多少个标准差。在正态分布中，约68%的数值落在 ±1 个标准差内，95%在 ±1.96 个标准差内，99.7%在 ±3 个标准差内。

z = (X – μ) / σ

For samples, we estimate μ with X̄ and σ with s. z-scores allow comparison across different distributions and are fundamental to inferential tests.

对于样本，我们用 X̄ 估计 μ、用 s 估计 σ。z分数使得不同分布之间可以相互比较，是推断检验的基础。

5. Correlation Coefficients: Pearson’s r and Spearman’s rho | 相关系数：皮尔逊 r 与斯皮尔曼 ρ

Pearson’s r measures the strength and direction of a linear relationship between two interval/ratio variables. It ranges from –1 (perfect negative) to +1 (perfect positive), with 0 indicating no linear correlation.

皮尔逊 r 衡量两个等距/等比变量之间线性关系的强度和方向。其值从 –1（完全负相关）到 +1（完全正相关），0 表示无线性相关。

r = Σ[(X – X̄)(Y – Ȳ)] / √[Σ(X – X̄)² Σ(Y – Ȳ)²]

Spearman’s rho (ρ) is a non-parametric version for ranked data; it assesses monotonic relationships. Convert raw scores to ranks, then apply a simplified formula using the difference in ranks (D).

斯皮尔曼 ρ 是用于等级数据的非参数版本，评估单调关系。将原始分数转换为等级，再运用基于等级差 (D) 的简化公式。

ρ = 1 – [6 ΣD² / n(n² – 1)]

Always plot a scattergraph before interpreting r or ρ to check for curvilinear patterns.

在解读 r 或 ρ 之前，务必绘制散点图以检查是否存在曲线模式。

6. Linear Regression | 线性回归

Linear regression predicts scores on a dependent variable (Y) from an independent variable (X). The regression line is described by Y = a + bX, where b is the slope and a is the intercept.

线性回归根据自变量 (X) 预测因变量 (Y) 的分数。回归直线由 Y = a + bX 描述，其中 b 是斜率，a 是截距。

b = r (sʏ / sₓ)

a = Ȳ – b X̄

r is Pearson’s correlation, sʏ and sₓ are standard deviations. The coefficient of determination r² tells the proportion of shared variance.

r 为皮尔逊相关，sʏ 与 sₓ 为标准差。决定系数 r² 指示共享变异的比例。

7. t-Test for Independent Samples | 独立样本 t 检验

The independent-measures t-test compares the means of two separate groups. It assumes normal distributions, homogeneity of variance, and interval/ratio data. The formula evaluates whether the difference between means is greater than would be expected by sampling error.

独立样本 t 检验比较两个独立组的均值。它假定正态分布、方差齐性以及等距/等比数据。公式评估均值之差是否大于抽样误差的预期范围。

t = (M₁ – M₂) / √(s₁²/n₁ + s₂²/n₂)

Degrees of freedom df = n₁ + n₂ – 2. If the calculated t exceeds the critical value at a chosen alpha (typically p < 0.05), the null hypothesis is rejected.

自由度 df = n₁ + n₂ – 2。若计算所得 t 值大于选定 alpha 水平（通常 p < 0.05）下的临界值，则拒绝零假设。

8. Non-parametric Tests: Mann-Whitney U and Wilcoxon Signed-Rank | 非参数检验：曼-惠特尼 U 与威尔科克森符号秩检验

The Mann-Whitney U test is the non-parametric alternative to the independent t-test, used when data are ordinal or assumptions are violated. It ranks all scores from both groups together and compares the sum of ranks.

曼-惠特尼 U 检验是独立样本 t 检验的非参数替代方法，适用于顺序数据或假设不满足时。它将两组所有分数合并排序，并比较秩和。

U₁ = n₁n₂ + n₁(n₁+1)/2 – R₁

R₁ is the sum of ranks for group 1; U is the smaller of U₁ and U₂. The Wilcoxon signed-rank test handles repeated measures or matched pairs design, ranking the absolute differences and then assigning signs.

R₁ 为组一的秩和；U 取 U₁ 和 U₂ 中的较小者。威尔科克森符号秩检验用于重复测量或配对组设计，将差值的绝对值排序后赋予符号。

W = Σ(Signed Ranks)

The test statistic W is compared to critical values in Wilcoxon tables.

检验统计量 W 需与威尔科克森表中的临界值进行比较。

9. Chi-Square Test for Independence | 卡方独立性检验

The chi-square (χ²) test evaluates whether two categorical variables are associated. It compares observed frequencies (O) in a contingency table with expected frequencies (E) under the null hypothesis of independence.

卡方 (χ²) 检验评估两个分类变量是否相关。它将列联表中的观察频次 (O) 与基于独立性零假设的期望频次 (E) 进行比较。

χ² = Σ[(O – E)² / E]

Degrees of freedom df = (rows – 1) × (columns – 1). A large χ² suggests a significant association. This test is widely used in surveys and observational studies.

自由度 df = (行数 – 1) × (列数 – 1)。χ² 值越大越可能表明显著关联。该检验广泛用于问卷和观察研究。

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 a constant proportion of the original stimulus intensity. It forms the basis of psychophysics.

韦伯定律指出，两个刺激间的最小可觉差 (JND) 与原始刺激强度的比例恒定。它奠定了心理物理学的基础。

ΔI / I = k

Here I is stimulus intensity, ΔI is the increment needed to detect a change, and k is the Weber fraction. Fechner extended this by proposing that subjective sensation S grows logarithmically with physical intensity.

其中 I 为刺激强度，ΔI 为察觉变化所需的增量，k 为韦伯分数。费希纳进一步提出，主观感觉 S 随物理强度呈对数增长。

S = k log I

Fechner’s Law implies that equal ratios of intensity produce equal increments of sensation, explaining why we perceive brightness or loudness on a compressed scale.

费希纳定律表明相等的强度比产生相等的感觉增量，解释了为何我们对亮度或响度的感知呈压缩式尺度。

11. Signal Detection Theory: d’ and Criterion | 信号检测论：d’ 与判断标准

Signal detection theory (SDT) separates an observer’s sensitivity from their response bias. Sensitivity d’ (d-prime) measures the distance between signal and noise distributions in standard deviation units.

信号检测论 (SDT) 将观察者的感受性与反应偏向分离开来。感受性 d’ 表示信号与噪音分布之间的标准化距离。

d’ = z(Hit) – z(False Alarm)

A Hit is correctly saying ‘yes’ when a signal is present; a False Alarm is saying ‘yes’ when only noise is present. The criterion c (or β) reflects how liberal or conservative the observer is. SDT is vital in memory research, medical diagnosis, and eyewitness testimony.

“击中”是指信号出现时正确回答“是”；“虚报”是指仅有噪音时回答“是”。判断标准 c（或 β）反映观察者的宽松或保守程度。信号检测论在记忆研究、医疗诊断和目击证词中至关重要。

12. Effect Size: Cohen’s d and r² | 效应量：Cohen’s d 与 r²

Statistical significance tells us whether an effect exists; effect size tells us how large it is. Cohen’s d is a standardised measure of the difference between two means, independent of sample size.

统计显著性告诉我们效应是否存在；效应量则告诉我们效应有多大。Cohen’s d 是两均值差异的标准化量数，与样本量无关。

d = (M₁ – M₂) / SDₚₒₒₗₑₔ

The pooled standard deviation SDₚₒₒₗₑₔ is calculated from the two groups. Values of d around 0.2, 0.5, and 0.8 are considered small, medium, and large, respectively. For correlation, r² represents the proportion of variance in one variable accounted for by the other.

合并标准差 SDₚₒₒₗₑₔ 由两组数据计算得出。d 值约 0.2、0.5、0.8 分别对应小、中、大效应。在相关分析中，r² 表示一个变量的变异可被另一个变量解释的比例。

r² = (t²) / (t² + df)

Reporting effect size alongside p-values is now an expected standard in psychological research.

在报告 p 值的同时报告效应量已成为心理学研究中的规范要求。

Published by TutorHao | Psychology Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

SQA Year 12 Psychology: Formula & Theorem Quick Reference Handbook | 公式定理速查手册