Year 13 OCR Psychology: Formula & Theorem Quick Reference | 公式定理速查手册

📚 Year 13 OCR Psychology: Formula & Theorem Quick Reference | 公式定理速查手册

This quick-reference handbook covers the essential formulas, statistical tests, and key laws you need for OCR A‑Level Psychology (Year 13). Use it to review descriptive statistics, inferential tests, effect size, psychophysical laws, and decision errors that appear in research methods and applied topics.

本速查手册涵盖了 OCR A‑Level 心理学（Year 13）所需的核心公式、统计检验和重要定律。可用于复习研究方法与应用主题中涉及的描述统计、推论检验、效应量、心理物理定律和决策错误。

1. Descriptive Statistics Formulas | 描述统计公式

Descriptive statistics summarise data sets. The mean is the arithmetic average, while the standard deviation measures the spread of scores around the mean.

描述统计用于概括数据集的整体情况。均值为算术平均数，标准差则度量分数围绕均值的离散程度。

Mean (M) = Σx / N

均值（M）= Σx / N

M = Σx / N

Sample standard deviation (s) = √[ Σ (x – M)² / (N – 1) ]

样本标准差（s）= √[ Σ (x – M)² / (N – 1) ]

s = √[ Σ (x – M)² / (N – 1) ]

The median is the middle score when data are ordered; the mode is the most frequent score; the range is the difference between the highest and lowest values.

中位数为排序后位于中间位置的数值；众数为出现频率最高的数值；全距为最大值与最小值之差。

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2. The Sign Test | 符号检验

The sign test is a non‑parametric test for repeated measures or matched‑pairs designs using nominal data. It compares the direction of differences (plus or minus) between pairs of scores.

符号检验是一种用于重复测量或配对组设计的名义数据的非参数检验，比较成对分数差异的方向（正或负）。

The test statistic S is the smaller count of positive signs or negative signs, ignoring any pairs where the difference is zero.

检验统计量 S 为正号数目与负号数目中的较小值，忽略差异为零的配对。

S = min( N₊ , N₋ )

Compare S against the critical value from the sign test table at the chosen significance level. If S is equal to or less than the critical value, the result is significant.

将 S 值与所选显著性水平下的符号检验临界值表进行比较。若 S 小于或等于临界值，则结果显著。

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3. Wilcoxon Signed‑Rank Test | 威尔科克森符号秩检验

Used for ordinal data in repeated measures or matched‑pairs designs. It considers both the direction and the magnitude of differences by ranking the absolute differences.

用于重复测量或配对组设计中的顺序数据。通过对绝对差值进行排序，同时考虑差异的方向与大小。

Calculate the differences for each pair, remove zeros, rank the absolute differences, then sum the ranks for positive differences (ΣR₊) and for negative differences (ΣR₋). The test statistic T is the smaller of these two sums.

计算每对的差值，剔除零值，对绝对差值编秩，然后分别求出正差值的秩和（ΣR₊）与负差值的秩和（ΣR₋）。检验统计量 T 取两者中的较小值。

T = min( ΣR₊ , ΣR₋ )

The observed T must be equal to or less than the critical value in the Wilcoxon table for the result to be significant.

观察到的 T 值必须小于或等于威尔科克森临界值表中的数值，结果才具显著性。

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4. Mann–Whitney U Test | 曼‑惠特尼 U 检验

This test compares two independent groups when the dependent variable is ordinal. It ranks all scores together regardless of group membership.

当因变量为顺序数据时，该检验用于比较两个独立组。它将两组的所有分数混合后进行统一编秩。

U is calculated for each group and the smaller value is used as the test statistic.

为每组计算 U 值，并以较小的 U 值作为检验统计量。

U_A = n_An_B + n_A(n_A+1)/2 – R_A

U_B = n_An_B – U_A

Observed U = min( U_A , U_B )

where R_A is the sum of ranks for group A, n_A and n_B are the sample sizes. Compare the observed U with the critical value; a significant result is obtained when U is equal to or smaller than the critical value.

其中 R_A 为 A 组秩和，n_A 与 n_B 为样本量。将观察 U 值与临界值比较，当 U 小于或等于临界值时，结果显著。

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5. Spearman’s Rank Correlation Coefficient | 斯皮尔曼等级相关系数

Spearman’s rho measures the strength and direction of association between two ordinal variables or ranked data.

斯皮尔曼等级相关系数衡量两个顺序变量或等级数据之间关联的强度与方向。

First rank both variables separately. Then compute the difference D between each pair of ranks, square these differences, and apply the formula.

首先分别对两个变量排序。然后计算每对排序的差值 D，求平方和，并代入公式。

r_s = 1 – (6 Σ D²) / (N (N² – 1))

The value of r_s ranges from –1 to +1. Compare the absolute value with the Spearman critical value to determine significance.

r_s 的取值范围为 –1 到 +1。将绝对值与斯皮尔曼临界值比较以判断显著性。

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6. Chi‑Square Tests | 卡方检验

Chi‑square is used for nominal data and tests whether observed frequencies differ significantly from expected frequencies.

卡方检验用于名义数据，检验观察频次与期望频次之间是否存在显著差异。

χ² = Σ [ (O – E)² / E ]

For a goodness‑of‑fit test, expected frequencies (E) are based on theory or chance; degrees of freedom (df) = number of categories – 1.

对于拟合优度检验，期望频次（E）基于理论或随机概率；自由度 df = 类别数 – 1。

For a test of independence in a contingency table, calculate expected frequencies as (Row total × Column total) / Grand total. Degrees of freedom are (number of rows – 1) × (number of columns – 1).

对于列联表中的独立性检验，期望频次的计算公式为（行合计 × 列合计）/ 总计。自由度为（行数 – 1）×（列数 – 1）。

E = (Row total × Column total) / Grand total

df = (r – 1) × (c – 1)

If the calculated χ² exceeds the critical value, the result is statistically significant, indicating an association or a poor fit.

若计算出的 χ² 值大于临界值，则结果具有统计学显著性，表明存在关联或拟合不佳。

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7. Effect Size – Cohen’s d | 效应量 – 科恩 d

Cohen’s d expresses the magnitude of the difference between two group means in standard deviation units. It helps evaluate practical significance.

科恩 d 以标准差为单位表示两组均值差异的大小，有助于评估实际意义。

d = (M₁ – M₂) / s_pooled

The pooled standard deviation s_pooled is estimated from both samples.

合并标准差 s_pooled 由两个样本共同估计得出。

s_pooled = √[ ((n₁−1)s₁² + (n₂−1)s₂²) / (n₁+n₂−2) ]

Conventionally, d = 0.2 is small, 0.5 is medium, and 0.8 or above is large.

按照常规，d = 0.2 为小效应，0.5 为中等效应，0.8 及以上为大效应。

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8. Weber’s Law & 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 original stimulus.

韦伯定律指出，两个刺激间的恰可察觉差异（JND）与原始刺激的强度成正比。

ΔI / I = k

where ΔI is the increment needed for detection, I is the standard stimulus intensity, and k is the Weber fraction.

其中 ΔI 为检出所需增量，I 为标准刺激强度，k 为韦伯分数。

Fechner’s Law builds on this, proposing that subjective sensation scales logarithmically with physical intensity.

费希纳定律在此基础上提出，主观感觉量与物理强度呈对数关系。

S = k log I

These psychophysical laws are fundamental for understanding perception thresholds in biological psychology topics.

这些心理物理定律是理解生物心理学课题中感知阈限的基础。

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9. Yerkes–Dodson Law | 耶基斯‑多德森定律

The Yerkes–Dodson Law describes the relationship between arousal and performance as an inverted‑U curve: performance improves with moderate arousal but deteriorates when arousal is too low or too high.

耶基斯‑多德森定律将唤醒水平与表现之间的关系描述为一条倒 U 曲线：中等唤醒时表现最佳，而唤醒过低或过高时表现下降。

For complex tasks the optimal arousal level is lower, while simple tasks can be performed well under higher arousal. This principle is widely applied in sports psychology and stress management.

复杂任务的最佳唤醒水平较低，简单任务则在较高唤醒下表现良好。该原理广泛应用于运动心理学与压力管理。

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10. Errors in Hypothesis Testing | 假设检验中的错误类型

In inferential testing, a Type I error occurs when the null hypothesis is true but is wrongly rejected (a false positive). Its probability is set by the significance level, α (typically .05 or .01).

在推论检验中，I 型错误指零假设为真却被错误拒绝（假阳性），其概率由显著性水平 α 设定（通常为 .05 或 .01）。

A Type II error happens when a false null hypothesis is not rejected (a false negative), and its probability is denoted β. Power of a test is 1 – β, reflecting the chance of correctly detecting a real effect.

II 型错误则指零假设为假却未被拒绝（假阴性），其概率记作 β。检验功效为 1 – β，反映正确侦测到真实效应的机会。

Using a larger sample or a more sensitive statistical test reduces Type II error. The balance between Type I and Type II errors influences the choice of significance level and research design.

使用更大样本或更灵敏的统计检验可降低 II 型错误。I 型与 II 型错误之间的平衡影响着显著性水平的选择与研究设计。

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11. Parametric t‑Tests | 参数 t 检验

Parametric tests assume interval‑level data and normally distributed populations. The correlated t‑test is used for repeated measures or matched‑pairs designs.

参数检验要求数据为等距水平且总体呈正态分布。相关 t 检验适用于重复测量或配对组设计。

t = D̄ / (s_D / √n)

where D̄ is the mean of difference scores, s_D is the standard deviation of difference scores, and df = n – 1.

其中 D̄ 为差值均值，s_D 为差值的标准差，自由度 df = n – 1。

The independent‑samples t‑test compares two separate groups.

独立样本 t 检验用于比较两个独立组。

t = (M₁ – M₂) / √( s_p²/n₁ + s_p²/n₂ )

s_p² = [ (n₁−1)s₁² + (n₂−1)s₂² ] / (n₁+n₂−2)

where s_p² is the pooled variance.

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