📚 Year 13 Cambridge Psychology: Quick Reference Handbook of Formulas and Theorems | 剑桥A2心理学公式定理速查手册
This handbook summarises the essential statistical formulas, psychophysical laws and key theorems that Year 13 learners must confidently apply in the Cambridge A2 Psychology course. Whether you are planning a research investigation, analysing data or interpreting published studies, having a quick reference to the correct equations, their assumptions and their meaning will strengthen both your evaluation and your exam performance.
本手册总结了剑桥A2心理学课程中必须熟练掌握的基本统计公式、心理物理定律以及核心定理。无论你正在设计研究方案、分析数据还是解读已发表的研究，能够快速查阅正确的方程式、前提条件及其含义，都会显著提升你的评估能力和考试成绩。
1. Measures of Central Tendency | 集中趋势度量
The mean (M) is the arithmetic average obtained by dividing the sum of all scores by the number of observations. It is the most commonly used measure of central tendency but is sensitive to extreme values.
M = ΣX / N
平均值 (M) 是通过将所有分数之和除以观测值个数得到的算术平均数。它是使用最广泛的集中趋势度量，但对极端值十分敏感。
The median is the middle value when scores are arranged in ascending order. For an odd number of data points, it is the central score; for an even number, it is the mean of the two central scores. The median is not distorted by outliers.
Median = middle score, or (n/2th + (n/2+1)th) / 2
中位数是将分数按升序排列后位于中间的值。数据个数为奇数时，选择最中心的分数；为偶数时，取中间两个分数的平均值。中位数不受异常值的扭曲。
The mode is the most frequently occurring value in a data set. A distribution may have one mode (unimodal), two modes (bimodal) or more.
Mode = most frequent score
众数是数据集中出现次数最多的值。分布可以是单峰、双峰或多峰的。
2. Measures of Dispersion | 离散度量
The range is the simplest measure of spread, calculated as the difference between the maximum and minimum scores. Although quick to compute, it ignores the distribution of values between the extremes.
Range = Xₘₐₓ − Xₘᵢₙ
极差是最简单的离中趋势指标，即最大值与最小值之差。虽然计算便捷，但忽略了极端值之间的分布状况。
Variance (σ² or s²) expresses the average of the squared deviations from the mean. For population data, divide by N; for a sample, use n − 1 (Bessel’s correction) to obtain an unbiased estimate.
σ² = Σ (X − μ)² / N s² = Σ (X − M)² / (n − 1)
方差 (σ² 或 s²) 表示离均差平方的平均值。总体数据除以 N，样本则用 n − 1（贝塞尔校正）以获得无偏估计。
Standard deviation (σ or s) is the positive square root of the variance. It returns the spread to the original units and is critical for calculating standard errors, z‑scores and effect sizes.
σ = √[ Σ (X − μ)² / N ] s = √[ Σ (X − M)² / (n − 1) ]
标准差 (σ 或 s) 是方差的正平方根，将离散度恢复为原始单位，对于计算标准误、z 分数和效应量至关重要。
3. Standard Scores and the Normal Distribution | 标准分数与正态分布
A z‑score indicates how many standard deviations a raw score lies above or below the mean. It allows comparison across different distributions and is fundamental to probability calculations in normally distributed data.
z = (X − μ) / σ
z 分数表示原始分数距离平均值多少个标准差，使不同分布之间能够进行对比，也是正态分布中概率计算的基础。
In a normal distribution, approximately 68% of data fall within ±1 SD, 95% within ±1.96 SD and 99.7% within ±3 SD. These percentages underpin most significance testing in psychology.
P(−1σ < z < 1σ) ≈ 68% P(−1.96σ < z < 1.96σ) ≈ 95%
在正态分布中，约 68% 的数据落在均值 ±1 个标准差之间，95% 落在 ±1.96 个标准差之间，99.7% 落在 ±3 个标准差之间。这些百分比是心理学显著性检验的基石。
The standard error of the mean (SE) estimates how much the sample mean is expected to vary from the population mean. It shrinks as sample size increases, reflecting greater precision.
SE = σ / √n (or SE = s / √n)
均值标准误 (SE) 估计样本均值与总体均值之间的预期波动幅度。样本量增大时标准误减小，反映更高的精度。
4. Parametric Tests: t-tests | 参数检验：t 检验
The independent‑samples t‑test compares the means of two unrelated groups. It assumes normal distribution within groups and homogeneity of variance. Use Levene’s test to check the latter assumption.
t = (M₁ − M₂) / √[ (sₚ²/n₁) + (sₚ²/n₂) ] where sₚ² = [ (n₁−1)s₁² + (n₂−1)s₂² ] / (n₁ + n₂ − 2)
独立样本 t 检验比较两个无关组的均值。前提条件为组内正态分布和方差齐性，可用 Levene’s 检验验证后者。
The related (paired) t‑test examines mean differences within the same participants measured twice or in matched pairs. It uses the mean of the difference scores (M_D) and its standard error.
t = M_D / (s_D / √n) where M_D = ΣD / n and s_D = √[ Σ(D − M_D)² / (n − 1) ]
相关（配对） t 检验考察同一组被试两次测量或配对设计下的均值差异。使用差异分数的平均值 (M_D) 及其标准误进行计算。
Degrees of freedom (df) for the independent t‑test are (n₁ + n₂ − 2); for the related t‑test, df = n − 1. The obtained t value is compared against a critical value from the t‑distribution table at the chosen α level.
df (independent) = n₁ + n₂ − 2 df (related) = n − 1
独立 t 检验的自由度为 n₁ + n₂ − 2，相关 t 检验的自由度为 n − 1。计算所得 t 值与 t 分布表中选定 α 水平的临界值进行比较。
5. Correlation Coefficients | 相关系数
The Pearson product‑moment correlation (r) measures the strength and direction of a linear relationship between two continuous variables. Values range from −1 (perfect negative) to +1 (perfect positive). Data must be interval‑level and normally distributed.
r = Σ [(X − M_X)(Y − M_Y)] / √[ Σ (X − M_X)² Σ (Y − M_Y)² ]
皮尔逊积矩相关系数 (r) 度量两个连续变量之间线性关系的强弱与方向，取值从 −1（完全负相关）到 +1（完全正相关）。数据要求为等距量表且正态分布。
Spearman’s rank‑order correlation (ρ or rₛ) is the non‑parametric equivalent, suitable for ordinal data or when assumptions of Pearson’s r are violated. It is based on the differences between ranked positions (d).
ρ = 1 − (6 Σ d²) / [n(n² − 1)]
斯皮尔曼等级相关系数 (ρ 或 rₛ) 是皮尔逊 r 的非参数对应，适用于顺序数据或当皮尔逊 r 的前提不满足时。它基于两个变量排序位置的差值 (d) 进行计算。
Both coefficients require a significance test. For Pearson’s r, degrees of freedom = n − 2. For Spearman’s ρ with small samples, critical values are obtained from a Spearman’s table.
df for r = n − 2
两个系数均需进行显著性检验。皮尔逊 r 的自由度为 n − 2，而小样本的斯皮尔曼 ρ 需查阅专门的临界值表。
6. Chi-Square Tests | 卡方检验
The chi‑square (χ²) test is used with categorical (nominal) data. The goodness‑of‑fit test compares observed frequencies to expected frequencies based on a theoretical distribution; the test of independence examines whether two categorical variables are associated.
χ² = Σ (O − E)² / E
卡方 (χ²) 检验用于类别（名义）数据。拟合优度检验比较观察频数与基于理论分布的期望频数；独立性检验则考察两个类别变量是否相关。
Expected frequencies (E) are calculated as (row total × column total) / grand total for the independence test. Each cell should have an expected frequency of at least 5 for the approximation to be reliable; otherwise, apply Yates’ correction or use Fisher’s exact test.
E = (Row Total × Column Total) / Grand Total
独立性检验中期望频数为（行总计 × 列总计）/ 总计。每个单元格的期望频数应至少为 5 以保证近似有效性；否则需应用 Yates 校正或 Fisher 精确检验。
Degrees of freedom for a goodness‑of‑fit test are (number of categories − 1). For a test of independence in a contingency table, df = (rows − 1) × (columns − 1).
df (goodness‑of‑fit) = k − 1 df (independence) = (R − 1)(C − 1)
拟合优度检验的自由度为类别数 k − 1；独立性检验中，df = (行数 − 1) × (列数 − 1)。
7. Non‑parametric Alternatives | 非参数替代检验
The Mann–Whitney U test is the non‑parametric counterpart of the independent t‑test. It ranks all scores from both groups together, then computes U₁ and U₂; the smaller U value is compared to a critical table.
U₁ = n₁n₂ + n₁(n₁ + 1)/2 − R₁ U₂ = n₁n₂ − U₁ U = min(U₁, U₂)
曼‑惠特尼 U 检验是独立 t 检验的非参数版本。它将两组所有分数统一排序，计算 U₁ 和 U₂，取较小的 U 值与临界值表进行比较。
The Wilcoxon signed‑rank test replaces the related t‑test when data are ordinal or the distribution of difference scores is non‑normal. Differences are ranked ignoring sign, then the sums of positive (T⁺) and negative (T⁻) ranks are computed; the smaller sum, T, is the test statistic.
T = min(T⁺, T⁻) where T⁺ = Σ ranks of positive differences, T⁻ = Σ ranks of negative differences
威尔科克森符号秩检验适用于顺序数据或差异分数非正态的情况，作为相关 t 检验的替代。对差异的绝对值排序，忽略符号，计算正秩和 (T⁺) 与负秩和 (T⁻)，较小的秩和即检验统计量 T。
Both tests require the use of critical‑value tables specific to the test. The null hypothesis is rejected if the obtained statistic is equal to or smaller than the tabled critical value (for U or T).
Reject H₀ if U ≤ critical value or T ≤ critical value
两个检验均需查阅专用的临界值表。若计算所得统计量 U 或 T 小于或等于表中临界值，则拒绝零假设。
8. Effect Size and Confidence Intervals | 效应量与置信区间
Coefficient of determination (r²) indicates the proportion of variance in one variable that can be explained by the other. It moves beyond mere significance to practical importance.
r² = (r)²
决定系数 (r²) 表示一个变量的方差中能被另一个变量解释的比例，超越单纯的显著性而评价实际重要性。
Cohen’s d is a standardised measure of effect size for comparing two independent means. Small, medium and large effects are conventionally defined as d = 0.2, 0.5 and 0.8 respectively.
d = (M₁ − M₂) / sₚ where sₚ = √[ ((n₁−1)s₁² + (n₂−1)s₂²) / (n₁ + n₂ − 2) ]
Cohen’s d 是用于比较两个独立均值的标准化效应量。小、中、大效应的惯例界值分别为 d = 0.2、0.5 和 0.8。
A confidence interval (CI) provides a range of values around the sample mean within which the population mean is likely to fall. The 95% CI is calculated using the critical t value for n−1 degrees of freedom.
95% CI = M ± (t_crit × SE) where SE = s / √n
置信区间 (CI) 给出样本均值周围的一个范围，总体均值有一定的概率落在该范围内。95% CI 使用自由度为 n−1 时的 t 临界值计算。
9. Psychophysical Laws | 心理物理学定律
Weber’s Law states that the just noticeable difference (JND) between two stimuli is a constant proportion (k) of the original stimulus intensity. It holds true for moderate stimulus ranges in many sensory modalities.
ΔI / I = k
韦伯定律指出，两个刺激之间的最小可觉差 (JND) 与原始刺激强度之比为一个常数 (k)。该定律在许多感觉通道的中等强度范围内成立。
Fechner’s Law extends Weber’s work by proposing that subjective sensation (S) increases with the logarithm of physical intensity (I). Each equal increment in sensation corresponds to a multiplying step in intensity.
S = k log I
费希纳定律在韦伯研究的基础上提出，主观感觉量 (S) 随物理强度 (I) 的对数增长。感觉的每个等量增加对应物理强度的一次翻倍。
Stevens’ Power Law offers a more flexible relationship: subjective magnitude (S) is proportional to the physical intensity raised to a power (n). The exponent n varies with the sensory continuum, explaining why brightness, loudness and pain grow at different rates.
S = k Iⁿ
史蒂文斯幂定律给出了一种更灵活的关系：主观量 (S) 与物理强度的某个幂次 (n) 成正比。指数 n 因感觉连续体而异，从而解释了明度、响度和痛觉的不同增长速率。
10. Probability and Statistical Significance | 概率与统计显著性
The p‑value represents the probability of obtaining the observed results, or more extreme ones, if the null hypothesis is true. In psychology, the conventional α level is .05, though stricter levels (e.g., .01) are sometimes used.
If p ≤ α → reject H₀
p 值表示在零假设为真的条件下，获得当前结果或更极端结果的概率。在心理学中，惯例的 α 水平为 .05，但有时也会采用更严格的水平（如 .01）。
Type I error (false positive) occurs when H₀ is wrongly rejected; its probability equals α. Type II error (false negative) occurs when H₀ is retained although there is a real effect; its probability is β. Power is 1 − β and increases with sample size and effect size.
α = P(Type I error) β = P(Type II error) Power = 1 − β
I 类错误（假阳性）是将正确的零假设错误拒绝，其概率等于 α。II 类错误（假阴性）是保留了事实上存在效应的零假设，其概率为 β。统计检验力为 1 − β，随样本量和效应量的增大而提高。
When multiple comparisons are made, the family‑wise error rate increases. A simple but conservative correction is the Bonferroni adjustment, where the nominal α is divided by the number of tests.
Adjusted α = α / m (where m = number of comparisons)
在进行多重比较时，家族误差率会上升。一种简单但保守的校正是 Bonferroni 调整，即将名义 α 除以检验的个数。
Directional (one‑tailed) tests are used when a hypothesis predicts the direction of an effect; non‑directional (two‑tailed) tests are used when the hypothesis merely predicts a difference. For the same α, a one‑tailed test has more power but must be justified a priori.
One‑tailed critical value < two‑tailed critical value
方向性（单尾）检验用于假设预测了效应的方向时；非方向性（双尾）检验用于仅预测存在差异时。在相同 α 下，单尾检验的检验力更高，但必须在实验前做出合理说明。
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