📚 A-Level Cambridge Psychology: Formula & Theorem Quick Reference Handbook | A-Level剑桥心理学：公式定理速查手册
Welcome to the Cambridge A-Level Psychology Formula & Theorem Quick Reference Handbook. This guide consolidates essential statistical formulas and psychological theorems, such as measures of central tendency, inferential test statistics, and psychophysical laws, that you must recall for examinations. Each entry is presented with the formula, a brief explanation, and its application context.
欢迎使用剑桥A-Level心理学公式定理速查手册。本手册汇集了必须掌握的统计公式和心理学定理，包括集中趋势量度、推断性检验统计量以及心理物理定律，并附有简要说明与应用背景，助你高效备考。
1. Measures of Central Tendency | 集中趋势量度
The mean describes the arithmetic average and is the foundation of many parametric tests. It uses every score, so it is sensitive to outliers.
平均数描述算术平均值，是许多参数检验的基础。它使用每个分数，因此对异常值敏感。
x̄ = Σx / n
x̄ = Σx / n
The median is the middle score when data are ordered. Its position is determined without a full calculation formula, but it is often located at the (n+1)/2th value.
中位数是数据排序后位于中间的分数。其位置通常由第 (n+1)/2 个值确定，无需复杂计算式。
The mode is simply the most frequently occurring value in a set. It is the only measure of central tendency suitable for nominal data.
众数是一组数据中出现次数最多的值。它是唯一适用于称名数据的集中趋势量度。
2. Measures of Dispersion | 离差量度
The range gives a quick sense of spread but is heavily influenced by extreme scores.
全距可以快速反映分散程度，但极易受极端分数影响。
Range = max – min
全距 = 最大值 – 最小值
Variance and standard deviation quantify the average squared deviation from the mean. The sample variance uses n-1 as degrees of freedom correction.
方差与标准差量化了分数围绕平均数的平均平方离差。样本方差使用 n-1 作为自由度校正。
s2 = Σ (x – x̄)2 / (n – 1)
s2 = Σ (x – x̄)2 / (n – 1)
s = √[ Σ (x – x̄)2 / (n – 1) ]
s = √[ Σ (x – x̄)2 / (n – 1) ]
3. Spearman’s Rank Correlation | 斯皮尔曼等级相关
Spearman’s rho (rs) measures the strength and direction of association between two ranked variables. It is non-parametric and works with ordinal data or non-linear monotonic relationships.
斯皮尔曼等级相关系数 (rs) 测量两个等级变量之间的关联强度与方向。这是一种非参数检验，适用于顺序数据或单调非线性关系。
rs = 1 – (6 Σ d2) / [ n (n2 – 1) ]
rs = 1 – (6 Σ d2) / [ n (n2 – 1) ]
d is the difference between the ranks of each pair, and n is the number of pairs. The coefficient ranges from -1 to +1.
d 为每对数据的等级之差，n 为对数。系数范围在 -1 到 +1 之间。
4. Pearson’s r | 皮尔逊积差相关
Pearson’s r examines the linear association between two interval or ratio variables. The calculation uses raw scores and assumes normal distributions.
皮尔逊相关系数 r 检验两个等距或比率变量之间的线性关联。计算采用原始分数，并假设数据呈正态分布。
r = ( n Σxy – Σx Σy ) / √( [n Σx2 – (Σx)2] [n Σy2 – (Σy)2] )
r = ( n Σxy – Σx Σy ) / √( [n Σx2 – (Σx)2] [n Σy2 – (Σy)2] )
A value close to +1 indicates a strong positive linear relationship, while -1 indicates a strong negative linear relationship. 0 suggests no linear correlation.
接近 +1 表示强正线性相关，-1 表示强负线性相关，0 意味着没有线性相关。
5. Unrelated t-test (Independent Samples) | 独立样本t检验
The independent t-test compares the means of two separate groups. It requires homogeneity of variance and interval data. The pooled variance sp2 is used when variances are assumed equal.
独立样本t检验比较两组独立样本的平均数。需要方差齐性且为等距数据。当方差相同时使用合并方差 sp2。
sp2 = [ (n1-1)s12 + (n2-1)s22 ] / (n1 + n2 – 2)
sp2 = [ (n1-1)s12 + (n2-1)s22 ] / (n1 + n2 – 2)
t = (x̄1 – x̄2) / √( sp2 (1/n1 + 1/n2) )
t = (x̄1 – x̄2) / √( sp2 (1/n1 + 1/n2) )
The calculated t-value is compared with the critical value at a chosen significance level and degrees of freedom (n1+n2-2).
计算出的 t 值与选定显著性水平及自由度 (n1+n2-2) 下的临界值比较。
6. Related t-test (Paired Samples) | 相关样本t检验
The related t-test is used when the same participants are tested twice (repeated measures) or pairs are matched. It analyses the difference scores (D) for each pair.
相关样本t检验适用于同一组被试前后测或配对匹配设计。它分析每对数据的差值分数 (D)。
t = ( ∑D ) / √( [ n ∑D2 – (∑D)2 ] / (n – 1) )
t = ( ∑D ) / √( [ n ∑D2 – (∑D)2 ] / (n – 1) )
Degrees of freedom = n – 1, where n is the number of pairs. This test controls for individual differences, increasing sensitivity.
自由度为 n – 1，n 是对数。该检验控制了个体差异，提高了检验敏感度。
7. Mann-Whitney U Test | 曼-惠特尼U检验
The Mann-Whitney U test is the non-parametric alternative to the independent t-test. It ranks all scores from both groups together and compares the rank sums.
曼-惠特尼U检验是独立样本t检验的非参数替代。它将所有分数混合编秩并比较秩和。
U1 = n1n2 + n1(n1+1)/2 – R1
U1 = n1n2 + n1(n1+1)/2 – R1
U2 = n1n2 – U1
U2 = n1n2 – U1
R1 is the sum of ranks for group 1. The smaller of U1 and U2 is compared with the critical U value. The test is appropriate for ordinal or skewed interval data.
R1 为第一组的秩和。取 U1 和 U2 中较小者与临界值比较。适用于顺序数据或偏态等距数据。
8. Wilcoxon Signed-Rank Test | 威尔科克森符号秩检验
This non-parametric test replaces the related t-test when differences are not normally distributed. It considers the magnitude and sign of difference scores.
当差值不符合正态分布时，此非参数检验替代相关样本t检验。它同时考虑差值的大小和符号。
T = sum of ranks for the less frequent sign
T = 较少符号的秩和
Steps: calculate difference D for each pair, exclude zeros, rank the absolute values of D, then sum the ranks for positive and negative differences separately. The smaller sum is T, compared with the critical T value.
步骤：计算每对差值D，剔除零，将|D|编秩，分别求正、负差的秩和。较小的秩和为T，与临界T值比较。
9. Chi-Squared Test (χ2) | 卡方检验
The chi-squared test assesses whether there is a significant association between two categorical variables, or goodness of fit to an expected distribution. Observed frequencies (O) are compared with expected frequencies (E).
卡方检验考查两个类别变量间是否显著关联，或观测频数是否符合期望分布。比较观测频数(O)与期望频数(E)。
χ2 = Σ (O – E)2 / E
χ2 = Σ (O – E)2 / E
E for each cell = (row total × column total) / grand total. Degrees of freedom = (rows-1) × (columns-1). A large χ2 value suggests a significant difference.
每个单元格的E = (行合计 × 列合计) / 总计。自由度 = (行数-1) × (列数-1)。χ2值越大表明差异越显著。
10. Signal Detection Theory (d’) | 信号检测论
Signal detection theory separates sensory sensitivity from response bias. The key measure d’ (d-prime) quantifies how easily a signal can be distinguished from background noise.
信号检测论将感觉敏感性与反应偏向分开。关键指标 d’ (d prime) 量化了信号从噪音中辨别的难易程度。
d’ = z(Hit rate) – z(False Alarm rate)
d’ = z(击中率) – z(虚报率)
Hit rate is the proportion of ‘yes’ responses when a signal is present; false alarm rate is the proportion of ‘yes’ responses when only noise is present. Higher d’ indicates better discrimination.
击中率是有信号时回答“是”的比例；虚报率是仅有噪音时回答“是”的比例。d’ 越大表示辨别力越强。
11. Psychophysical Laws: Weber & Fechner | 心理物理定律：韦伯与费希纳
Weber’s Law states that the just noticeable difference (JND) between two stimuli is proportional to the magnitude of the original stimulus. The constant k is the Weber fraction.
韦伯定律指出，两个刺激的最小可觉差 (JND) 与初始刺激的强度成正比。常数k为韦伯分数。
ΔI / I = k
ΔI / I = k
Fechner’s Law builds on Weber’s work, proposing that subjective sensation S is a logarithmic function of physical intensity I.
费希纳定律在韦伯的基础上提出，主观感觉S是物理强度I的对数函数。
S = k log (I / I0)
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