📚 Year 12 OCR Psychology: Formula & Theorem Quick Reference | OCR心理学公式定理速查手册
This quick reference handbook brings together all the essential formulas, statistical tests, and key numerical principles you need for Year 12 OCR Psychology. From measures of central tendency to non‑parametric tests and psychophysical laws, every formula is presented with clear notational definitions and worked examples where appropriate. Use this guide to revise efficiently and to check your calculations when practising research methods questions.
这本速查手册汇总了 Year 12 OCR 心理学中所有关键的公式、统计检验以及重要的定量原则。从集中趋势量到非参数检验，再到心理物理学定律，每一项公式都配有清晰的符号定义，并酌情提供计算示例。你可以利用这本指南高效复习，并在练习研究方法题目时核验自己的计算过程。
1. Measures of Central Tendency | 集中趋势的测量
Central tendency describes the most typical or representative value in a data set. The mean is calculated as the sum of all scores divided by the number of scores: Mean = Σx / n. The median is the middle value when scores are ordered from smallest to largest; for an even number of scores it is the average of the two middle values. The mode is simply the most frequently occurring score.
集中趋势描述数据集中最典型或最具代表性的值。均值是所有数值之和除以数值个数：均值 = Σx / n。中位数是将数据从小到大排序后位于正中间的值；若数据个数为偶数，则为中间两个值的平均数。众数则是出现频率最高的值。
For the data set 3, 5, 7, 7, 9, 12: Mean = (3+5+7+7+9+12) ÷ 6 = 43 ÷ 6 = 7.17; Median = 7 (average of two middle 7s); Mode = 7.
对于数据集 3, 5, 7, 7, 9, 12：均值 = (3+5+7+7+9+12) ÷ 6 = 43 ÷ 6 = 7.17；中位数 = 7（两个中间值 7 的平均）；众数 = 7。
2. Measures of Dispersion: Range & Interquartile Range | 离散度：全距与四分位距
Dispersion indicates how spread out the scores are. The range is the simplest measure: Range = highest score − lowest score. However, the range is easily distorted by outliers, so the interquartile range (IQR) is often preferred.
离散度反映数据的分散程度。全距是最简单的指标：全距 = 最大值 − 最小值。但全距容易被异常值拉大，因此常使用四分位距（IQR）。
Quartiles divide the ordered data set into four equal parts. Q₁ is the median of the lower half, Q₃ is the median of the upper half. IQR = Q₃ − Q₁. The IQR captures the middle 50% of scores and is more robust to extreme values.
四分位数将有序数据集分为四等份。第一四分位数（Q₁）是下半部分的中位数，第三四分位数（Q₃）是上半部分的中位数。四分位距 = Q₃ − Q₁。IQR 涵盖了中间 50% 的数据，在面对极端值时更为稳健。
3. Variance & Standard Deviation | 方差与标准差
Variance measures the average squared deviation from the mean, while standard deviation is the square root of the variance and is expressed in the original units of measurement. For a sample, the formulas are:
方差衡量各个数据与均值之差的平方的平均值，而标准差是方差的平方根，单位与原始数据相同。样本方差与标准差的计算公式为：
Variance s² = Σ (x − x̄)² / (n − 1)
Standard deviation s = √ [ Σ (x − x̄)² / (n − 1) ]
Here x̄ represents the sample mean and n is the number of scores. Subtracting 1 (Bessel’s correction) gives an unbiased estimate of the population variance.
其中 x̄ 表示样本均值，n 为数据个数。分母采用 n − 1（贝塞尔校正）可得到总体方差的无偏估计。
Steps: (1) calculate the mean, (2) subtract the mean from each score and square the result, (3) sum those squared deviations, (4) divide by (n−1), (5) take the square root for the standard deviation.
计算步骤：(1) 求均值；(2) 每个数据减去均值，然后平方；(3) 将所有平方差求和；(4) 除以 (n−1)；(5) 取平方根即得标准差。
4. Z‑scores & the Normal Distribution | Z分数与正态分布
A z‑score expresses how many standard deviations a raw score lies above or below the mean. The formula is: z = (x − μ) / σ, where μ is the population mean and σ is the population standard deviation. In a normal distribution, about 68% of scores fall within ±1σ, 95% within ±1.96σ, and 99.7% within ±3σ.
Z分数表示一个原始分数距离总体均值有多少个标准差。公式为：z = (x − μ) / σ，其中 μ 为总体均值，σ 为总体标准差。在正态分布中，约 68% 的数值落在 ±1σ 范围内，95% 落在 ±1.96σ 范围内，99.7% 落在 ±3σ 范围内。
Z‑scores allow researchers to compare scores from different distributions and to determine the probability of obtaining a given score under the null hypothesis.
Z分数使研究者能够比较来自不同分布的数据，并确定在零假设下获得某个分数的概率。
5. Probability & Significance Levels | 概率与显著性水平
In inferential statistics, the p‑value is the probability of obtaining the observed result (or one more extreme) if the null hypothesis is true. The conventional significance level (α) in psychology is 0.05, meaning there is a 5% risk of concluding that an effect exists when it does not. If p ≤ 0.05, the result is statistically significant and the null hypothesis is rejected.
在推论统计中，p 值是指在零假设为真的情况下，获得当前观察结果（或更极端结果）的概率。心理学中常用的显著性水平（α）为 0.05，表示有 5% 的风险错误地认为有效应存在。如果 p ≤ 0.05，则结果具有统计显著性，拒绝零假设。
For a one‑tailed test, the 5% rejection region is placed entirely in one tail of the distribution; for a two‑tailed test, it is split into 2.5% in each tail. The choice of one‑ or two‑tailed test depends on whether the research hypothesis predicts a direction.
对于单尾检验，5% 的拒绝域全部置于分布的一侧尾部；对于双尾检验，两侧尾部各分配 2.5%。选择单尾还是双尾取决于研究假设是否预测了方向。
6. The Sign Test | 符号检验
The sign test is a non‑parametric test used with repeated measures or matched pairs designs when the data are nominal or ordinal. It simply counts the number of participants who show a change in one direction versus the other, ignoring the magnitude of change. The test statistic, S, is the smaller of the two counts.
符号检验是一种非参数检验，适用于重复测量或配对组设计，且数据类型为称名或顺序数据。它只统计朝一个方向变化与朝另一方向变化的人数，忽略变化量的大小。检验统计量 S 是两组计数中较小的那个。
To use the sign test: (1) record the direction of change for each participant (e.g. more or less); (2) tally the number of pluses and minuses, ignoring ties; (3) S = the smaller frequency; (4) compare S to the critical value from the sign test table (using N = number of non‑tie pairs). If S ≤ critical value, the result is significant.
使用符号检验的步骤：(1) 记录每名参与者的变化方向（如增加或减少）；(2) 分别统计加号和减号的个数，忽略无变化的情况；(3) S = 较小的频次；(4) 将 S 与符号检验表中的临界值比较（N 为有成对变化的人数）。若 S ≤ 临界值，则结果显著。
7. Chi‑Square Test for Association | 卡方关联检验
The chi‑square (χ²) test is used with independent groups designs and categorical (nominal) data to test for an association between two variables or a difference between expected and observed frequencies. The formula is:
卡方（χ²）检验适用于独立组设计及分类（称名）数据，用于检验两个变量之间的关联性，或观察频次与期望频次之间的差异。计算公式为：
χ² = Σ (O − E)² / E
where O = observed frequency, E = expected frequency. Expected frequencies are calculated assuming the null hypothesis is true: E = (row total × column total) / grand total.
其中 O 为观察频次，E 为期望频次。期望频次基于零假设为真的前提计算：E =（行合计 × 列合计）/ 总计。
Degrees of freedom (df) = (number of rows − 1) × (number of columns − 1). The computed χ² value is compared with the critical value in the χ²‑distribution table. If χ² ≥ critical value, the null hypothesis is rejected.
自由度 (df) = (行数 − 1) × (列数 − 1)。将算得的 χ² 值与卡方分布表中的临界值比较。若 χ² ≥ 临界值，则拒绝零假设。
8. Wilcoxon Signed‑Ranks Test | 威尔科克森符号秩检验
The Wilcoxon test is a non‑parametric alternative to the related t‑test, used with repeated measures or matched pairs when the data are at least ordinal. It takes into account both the direction and the magnitude of differences. The test statistic, T, is the smaller of the sum of positive ranks and the sum of negative ranks.
威尔科克森检验是配对 t 检验的非参数替代方法，适用于重复测量或配对设计且数据至少为顺序水平。它既考虑差异的方向，也考虑差异的大小。检验统计量 T 是正秩和与负秩和中较小的那个。
Procedure: (1) calculate the difference between each pair of scores, omit zeros; (2) rank the absolute differences from 1 (smallest) to N (largest); (3) attach the sign of each difference to its rank; (4) sum the positive ranks (W⁺) and negative ranks (W⁻); (5) T = the smaller of W⁺ and W⁻. Compare T with the Wilcoxon table critical value for the given N. The result is significant if T ≤ critical value.
步骤：(1) 计算每对分数的差值，剔除差值为零的情况；(2) 按差值的绝对值从小到大排序 (1 至 N)；(3) 将原始差值的正负号赋予相应的秩次；(4) 分别计算正秩和 (W⁺) 与负秩和 (W⁻)；(5) T = W⁺ 与 W⁻ 中的较小者。将 T 与威尔科克森表中对应 N 的临界值比较，若 T ≤ 临界值则显著。
9. Mann‑Whitney U Test | 曼‑惠特尼U检验
The Mann‑Whitney U test is the non‑parametric equivalent of the independent t‑test, used with independent groups designs and ordinal (or interval) data. It ranks all scores from both groups together and then compares the sum of ranks for each group. The test statistic U is calculated for both groups; the smaller U is used as the final test statistic.
曼‑惠特尼U检验是独立 t 检验的非参数等价方法，适用于独立组设计及顺序（或等距）数据。它将两组所有数据合并排序后求出各组的秩和，进而计算 U 统计量。最终取两个 U 值中较小的一个作为检验统计量。
Formula: U₁ = R₁ − n₁(n₁ + 1) / 2, where R₁ is the sum of ranks for group 1 and n₁ is the sample size of group 1. U₂ is similarly calculated. The smaller of U₁ and U₂ (Uₘᵢₙ) is compared with the Mann‑Whitney critical value table. If Uₘᵢₙ ≤ critical value, the difference between the groups is statistically significant.
计算公式：U₁ = R₁ − n₁(n₁ + 1) / 2，其中 R₁ 为第一组的秩和，n₁ 为第一组的样本容量。U₂ 同理计算。将 U₁ 与 U₂ 中的较小者 (Uₘᵢₙ) 与曼‑惠特尼临界值表对比。若 Uₘᵢₙ ≤ 临界值，则两组间差异显著。
10. Spearman’s Rank Correlation Coefficient | 斯皮尔曼等级相关系数
Spearman’s rho (ρ or rₛ) measures the strength and direction of a monotonic relationship between two ordinal variables. The formula based on rank differences is:
斯皮尔曼等级相关系数（ρ 或 rₛ）衡量两个顺序变量之间单调关系的强度与方向。基于秩次差值的计算公式为：
rₛ = 1 − [ 6 Σ d² / (n(n² − 1)) ]
where d = difference between the ranks of each pair, and n = number of pairs. The coefficient ranges from −1 (perfect negative correlation) to +1 (perfect positive correlation). The computed rₛ is compared with the Spearman’s rho critical value table using the appropriate n. If rₛ ≥ critical value (for a positive correlation) or rₛ ≤ −critical value (for a negative one), the correlation is significant.
其中 d = 每对数据的秩次差值，n = 数据对的个数。该系数取值范围从 −1（完全负相关）到 +1（完全正相关）。将计算出的 rₛ 与斯皮尔曼等级相关系数表中对应 n 的临界值比较。若 rₛ ≥ 临界值（正相关检验）或 rₛ ≤ −临界值（负相关检验），则相关具有显著性。
11. Effect Size: Cohen’s d | 效应大小：科恩d值
Cohen’s d is a standardised measure of the magnitude of a difference between two group means, independent of sample size. It is especially useful when reporting practical significance alongside statistical significance. The basic formula is:
科恩 d 值是一种标准化的效应量指标，用于衡量两组均值差异的大小，不受样本容量影响。在报告统计显著性的同时，它有助于判断实际意义。基本公式为：
d = (M₁ − M₂) / Sₚₒₒₗₑₔ
where M₁ and M₂ are the two group means, and Sₚₒₒₗₑₔ is the pooled standard deviation. A common approximation for pooled SD is √[(s₁² + s₂²) / 2] when the two sample sizes are equal or similar. Benchmarks: 0.2 = small, 0.5 = medium, 0.8 = large effect.
其中 M₁ 与 M₂ 是两组的均值，Sₚₒₒₗₑₔ 为合并标准差。当两样本量相等或相近时，合并标准差常用近似公式 √[(s₁² + s₂²) / 2]。效应大小的参考标准：0.2 为小效应，0.5 为中效应，0.8 为大效应。
12. Key Laws & Constants in Psychology | 心理学关键定律与常数
Several quantitative laws from psychophysics and cognitive psychology feature in Year 12 OCR topics. Weber’s Law states that the just‑noticeable difference (JND) for a stimulus is a constant proportion of the original intensity:
心理物理学与认知心理学中的几条定量定律在 Year 12 OCR 中有所涉及。韦伯定律指出，刚刚能察觉到的刺激差异（JND）与原始刺激强度之比是一个常数：
ΔI / I = k
where ΔI is the JND, I is the initial stimulus intensity, and k is the Weber fraction. Stevens’ Power Law extends this by proposing a power relationship between perceived magnitude (S) and physical intensity (I): S = k Iⁿ, where the exponent n depends on the sensory modality.
其中 ΔI 为最小可觉差，I 为初始刺激强度，k 为韦伯分数。史蒂文斯幂定律进一步提出感知量（S）与物理强度（I）之间呈幂函数关系：S = k Iⁿ，其中指数 n 取决于感觉通道。
In cognitive psychology, Miller’s Law describes the capacity of short‑term memory: the magical number 7 ± 2 refers to the number of chunks of information that can be held in working memory. While not a formula in the strict sense, it is an essential numerical reference for the multi‑store model of memory.
在认知心理学中，米勒定律描述了短时记忆的容量：神奇数字 7 ± 2 指的是工作记忆能够同时容纳的信息组块数量。虽然并非严格意义上的公式，却是记忆多储存模型必须掌握的关键数值参考。
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