📚 OCR Psychology Year 12 Formula & Theorem Quick Reference | OCR 心理学 Year 12 公式定理速查手册
This quick reference handbook brings together every essential formula, statistical test and decision rule you need for OCR AS Psychology (Year 12) Research Methods. It covers levels of measurement, descriptive statistics, inferential tests and the logic of significance testing. Use it to revise step‑by‑step calculations and to make confident choices when designing or evaluating investigations.
这本速查手册汇总了OCR AS心理学（Year 12）研究方法中所有必备的公式、统计检验与判断规则。内容涵盖测量层次、描述统计、推断检验以及显著性检验的逻辑，帮助你按步骤复习计算过程，并在设计或评估研究时作出自信的选择。
1. Levels of Measurement | 测量层次
The level of measurement determines which descriptive statistics are meaningful and which inferential test is appropriate. The three levels recognised in OCR are nominal, ordinal and interval/ratio.
测量层次决定了哪些描述统计有意义以及应该选用哪种推断检验。OCR课程中认可的三种层次为：定类、定序以及定距/定比。
Nominal data are frequencies in categories that have no numerical order (e.g. hair colour, yes/no responses). You can only report the mode.
定类数据是落在无顺序的类别中的频次（如头发颜色、是/否回答），只能报告众数。
Ordinal data can be ranked but the intervals between ranks are not equal (e.g. Likert scale ratings, positions in a race). The median and range are appropriate.
定序数据可以排序，但等级之间的间距不相等（如李克特量表评分、比赛名次）。适合使用中位数和极差。
Interval/ratio data have equal intervals and, for ratio, a true zero (e.g. time, weight, test scores). The mean and standard deviation can be used.
定距/定比数据具有相等的单位间距，定比还存在绝对零点（如时间、重量、测验分数），可以使用平均值和标准差。
2. Measures of Central Tendency | 集中趋势度量
The mean is the arithmetic average. It is sensitive to extreme values and is only appropriate for interval/ratio data.
M = ΣX / N
平均值是算术平均数，对极端值敏感，仅适用于定距/定比数据。
The median is the middle score when data are ordered. It is unaffected by outliers and is suitable for ordinal and interval/ratio data.
中位数是将数据排序后位于中间位置的数值，不受极端值影响，适用于定序和定距/定比数据。
The mode is the most frequently occurring score. It can be used with any level of measurement, including nominal data.
众数是出现次数最多的数值，可用于任何测量层次，包括定类数据。
3. Measures of Dispersion | 离散程度度量
The range is the simplest measure of spread: the difference between the highest and lowest scores. It is sensitive to outliers.
Range = Max − Min
极差是最简单的离散度量，为最大值与最小值之差，易受极端值影响。
The variance and standard deviation capture how far scores deviate from the mean. The sample standard deviation (SD) uses N−1 to give an unbiased estimate.
Variance = Σ(X − M)² / (N − 1)
方差和标准差测量分数与平均值的偏离程度。样本标准差使用N−1以提供无偏估计。
SD = √[Σ(X − M)² / (N − 1)]
标准差是方差的平方根，单位与原始数据一致。
4. Probability, Significance and Errors | 概率、显著性与错误
In inferential testing 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, is set at 0.05 (5%).
在推断检验中，p值是在零假设成立时获得当前结果（或更极端结果）的概率。常规显著性水平α设为0.05（5%）。
istically significant and the null hypothesis is rejected. If p > 0.05 the result is not significant and the null hypothesis is retained.
若p ≤ 0.05，结果被认为具有统计显著性，拒绝零假设；若p > 0.05则结果不显著，保留零假设。
Type I error is a false positive: rejecting a true null hypothesis. Type II error is a false negative: failing to reject a false null hypothesis. The probability of a Type I error equals α.
第I类错误为误报（弃真错误）：拒绝了真的零假设。第II类错误为漏报（存伪错误）：未能拒绝假的零假设。第I类错误的概率等于α。
5. Binomial Sign Test | 二项符号检验
The binomial sign test is used with a repeated measures or matched‑pairs design when the data are nominal (dichotomous categories). It assesses whether there is a significant difference in the direction of the two conditions.
二项符号检验适用于重复测量或配对设计，且数据为定类（二分类型）的情况，用于判断两种条件的方向是否存在显著差异。
The null hypothesis states that there is no difference between the conditions; the calculated S value is the count of the less frequent sign (plus or minus).
S = smaller of count of '+' and count of '–'
零假设为两种条件无差异；计算出的S值为加号或减号中出现次数较小的那个频次。
Compare the observed S with the critical value from the binomial sign test table for the chosen significance level (usually 0.05) and the number of non‑tied pairs (N). If S is less than or equal to the critical value, the result is significant.
将观察到的S值与二项符号检验表中的临界值（基于选定的显著性水平，通常0.05，以及非平局对数N）进行比较。若S ≤ 临界值，则结果显著。
6. Chi‑Square Test for Independence | 卡方独立性检验
The chi‑square test is used with an independent groups design and nominal (frequency) data to test whether two categorical variables are associated.
卡方检验用于独立组设计和定类（频次）数据，以检验两个类别变量是否相关。
χ² = Σ((O − E)² / E)
其中O为观察频次，E为期望频次。
The expected frequency for each cell is calculated from the row and column totals:
E = (Row Total × Column Total) / Grand Total
每个单元格的期望频次由行合计和列合计计算得出。
(number of columns − 1). The obtained χ² is compared with the critical value from the chi‑ e calculated χ² is greater than or equal to the critical value, the result is significant.
自由度 = (行数 − 1) × (列数 − 1)。将计算得到的χ²值与卡方表中的临界值比较。若计算χ² ≥ 临界值，则结果显著。
7. Mann‑Whitney U Test | 曼‑惠特尼 U 检验
The Mann‑Whitney U test is used with an independent groups design and ordinal (or interval) data to test whether there is a significant difference between two conditions.
曼‑惠特尼U检验用于独立组设计和定序（或定距）数据，以检验两种条件之间是否存在显著差异。
All scores are ranked together, then the ranks are summed for each group (R₁ and R₂). The U statistic is calculated for both groups, and the smaller U is used as the test statistic.
U₁ = n₁n₂ + n₁(n₁ + 1)/2 − R₁
U₂ = n₁n₂ + n₂(n₂ + 1)/2 − R₂
U = min(U₁, U₂)
将所有分数统一排序，分别计算两组的秩和R₁和R₂。计算两组各自的U值，取较小的U作为检验统计量。
Compare the smaller U with the critical value from the Mann‑Whitney table. The result is significant if the observed U is less than or equal to the critical value.
将较小的U值与曼‑惠特尼表中的临界值比较。若观测U ≤ 临界值，则结果显著。
8. Wilcoxon Signed‑Rank Test | 威尔科克森符号秩检验
The Wilcoxon signed‑rank test is used with a repeated measures or matched‑pairs design and ordinal (or interval) data to detect a significant difference between two conditions.
威尔科克森符号秩检验用于重复测量或配对设计，且数据为定序（或定距）的情况，以检测两种条件之间的显著差异。
Calculate the difference between each pair, rank the absolute differences (ignoring zeros), attach the original signs to the ranks, and sum the positive ranks (ΣR⁺) and negative ranks (ΣR⁻). The test statistic T is the smaller of these two sums.
T = min(ΣR⁺, ΣR⁻)
计算每对数据的差值，对差值的绝对值进行排序（忽略零差），将原来的符号附加给秩次，然后分别求正秩和与负秩和。检验统计量T取两者中较小的一个。
Compare T with the critical value in the Wilcoxon table for the given N (number of non‑zero pairs
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