📚 AS OCR Psychology: Formula & Theorem Quick Reference Handbook | AS OCR心理学：公式定理速查手册
This article provides a concise revision handbook for AS OCR Psychology students, focusing on the essential formulas, statistical tests, and key principles required for research methods and data analysis. Every psychologist-in-training must be confident in selecting, computing, and interpreting descriptive and inferential statistics. The material is aligned with the OCR specification and is designed to be a quick reference before examinations.
本文为AS OCR心理学学生提供简明速查手册，重点涵盖研究方法与数据分析所必需的基本公式、统计检验和关键原理。每一位心理学学习者都必须熟练掌握描述统计与推论统计的选择、计算与解释。内容严格对应OCR考试大纲，旨在成为考前快速查阅的实用工具。
1. Measures of Central Tendency | 集中趋势测量
The three principal measures of central tendency summarise a data set with a single representative value. The mean is the arithmetic average, the median is the middle score when data are ordered, and the mode is the most frequently occurring value.
三种主要的集中趋势测量值用一个代表性数值概括数据集。均值是算术平均数，中位数是将数据排序后位于中间的数值，众数是出现频率最高的数值。
The mean is calculated by summing all raw scores and dividing by the total number of scores. It is the most sensitive measure because it includes every data point, but it is easily distorted by extreme outliers.
均值通过将所有原始分数相加再除以分数总个数计算得出。它是最灵敏的测量值，因为它包含了每一个数据点，但极易受到极端异常值的影响。
Mean = Σx / n
均值公式：Mean = Σx / n，其中 Σx 表示所有分数之和，n 表示样本容量。
The median is preferred when data are skewed or when outliers are present, as it divides the distribution into two equal halves. The mode is the only measure suitable for nominal (categorical) data and can indicate the most typical category.
当数据分布偏斜或存在异常值时，宜采用中位数，它能把分布分割为相等的两半。众数是唯一适用于名义（分类）数据的量数，能够指示最典型的类别。
2. Measures of Dispersion | 离散程度测量
Dispersion describes the spread or variability within a data set. The simplest measure is the range, but variance and standard deviation provide far more informative summaries of how scores deviate from the mean.
离散程度描述数据集内部的散布或变异情况。最简单的量数是全距，而方差和标准差能够更充分地反映各分数围绕均值的偏离程度。
Range = Maximum value − Minimum value
全距公式：全距 = 最大值 − 最小值。
Variance is the average of the squared deviations from the mean. For a sample, the denominator is (n − 1) to give an unbiased estimate of the population variance.
方差是各分数与均值离差平方的平均数。计算样本方差时，分母使用 (n − 1) 以给出总体方差的无偏估计。
s² = Σ(x − x̄)² / (n − 1)
样本方差：s² = Σ(x − x̄)² / (n − 1)，其中 x̄ 表示样本均值。
Standard deviation is the square root of the variance, returning the measure to the original units. It is central to the normal distribution and to many inferential tests.
标准差是方差的平方根，将量数恢复为原始单位。它在正态分布和许多推论检验中居于核心地位。
s = √[ Σ(x − x̄)² / (n − 1) ]
样本标准差：s = √[ Σ(x − x̄)² / (n − 1) ]。
3. Normal Distribution and Skewness | 正态分布与偏态
The normal distribution is a symmetrical, bell-shaped curve where the mean, median, and mode coincide at the centre. About 68% of values fall within one standard deviation of the mean, and roughly 95% fall within two standard deviations.
正态分布是一种对称的钟形曲线，均值、中位数和众数重合于中心。大约68%的数值落在均值的一个标准差范围内，约95%落在两个标准差范围内。
Skewness occurs when data are asymmetrical. A positive skew has a long tail on the right, and the mean is pulled above the median. A negative skew has a long tail on the left, and the mean falls below the median.
当数据不对称时就会出现偏态。正偏态右尾较长，均值被拉向中位数右侧；负偏态左尾较长，均值降至中位数左侧。
Recognising skewness is important because it influences which measure of central tendency is most appropriate, and it determines whether parametric tests (which assume normality) can be legitimately applied.
识别偏态十分重要，因为它会影响选用何种集中趋势量数最为合适，并决定参数检验（要求正态分布假设）是否可以合理使用。
4. Probability and Significance Levels | 概率与显著性水平
In inferential statistics, probability (p) expresses the likelihood that an observed result occurred by chance. The conventional significance level in psychology is p ≤ 0.05, meaning there is a 5% or lower probability that the result is due to random variation.
在推论统计中，概率 (p) 表示观察到的结果由偶然因素造成的可能性。心理学通用的显著性水平是 p ≤ 0.05，意味着结果由随机变异所致的概率不大于5%。
If the calculated test statistic exceeds the critical value for a given p level, the null hypothesis (H₀) is rejected in favour of the alternative hypothesis (H₁). This indicates a statistically significant finding.
如果计算得到的检验统计量超过了给定 p 水平的临界值，则拒绝零假设 (H₀)，接受备择假设 (H₁)。这表明结果具有统计显著性。
The 0.05 level is a convention, not a magic boundary. Researchers must also consider whether a more stringent level, such as p ≤ 0.01, is warranted to reduce the risk of a false positive.
0.05 水平只是约定俗成，并非绝对界限。研究者还应根据需要，考虑采用更严格的标准（如 p ≤ 0.01），以降低假阳性风险。
5. Choosing a Statistical Test | 统计检验的选择
Selecting the correct inferential test depends on three questions: (1) Is the research looking for a difference or a correlation? (2) What is the level of measurement (nominal, ordinal, or interval/ratio)? (3) What is the experimental design (independent groups or repeated measures/matched pairs)?
选择合适的推论检验取决于三个问题：(1) 研究在寻找差异还是相关？(2) 数据测量水平是什么（名义、顺序还是等距/等比）？(3) 实验设计是什么（独立组还是重复测量/配对组）？
For tests of difference with nominal data, the Chi-squared test is appropriate. For ordinal data, the Mann-Whitney U test is used for independent groups and the Wilcoxon signed-rank test for repeated measures. For a correlation between two ordinal variables, Spearman’s rho is the standard choice.
对于名义数据的差异检验，选用卡方检验。对于顺序数据，独立组设计采用曼-惠特尼U检验，重复测量设计采用威尔科克森符号秩检验。若要检验两个顺序变量之间的相关，斯皮尔曼秩相关系数是标准选择。
At AS level, students are not required to calculate parametric tests such as the t-test, but they must be able to justify the choice of an appropriate non-parametric test based on the design and data type.
在AS阶段，不要求计算t检验等参数检验，但学生必须能够根据实验设计和数据类型，论证为何选用某种合适的非参数检验。
6. Chi-Squared Test (χ²) | 卡方检验
The Chi-squared test assesses whether there is a significant association between two categorical (nominal) variables, or whether observed frequencies differ from expected frequencies. It uses data placed in a contingency table.
卡方检验评估两个分类（名义）变量之间是否存在显著关联，或观测频次是否与期望频次存在差异。它使用列联表中的数据。
χ² = Σ((O − E)² / E)
卡方计算公式：χ² = Σ((O − E)² / E)，其中 O 代表观测频次，E 代表期望频次。
Expected frequencies are calculated by (row total × column total) ÷ grand total. The degrees of freedom (df) for a test of independence are given by (number of rows − 1) × (number of columns − 1).
期望频次计算公式为：(行总和 × 列总和) ÷ 总频次。独立性检验的自由度 (df) 等于 (行数 − 1) × (列数 − 1)。
df = (r − 1) × (c − 1)
卡方自由度：df = (r − 1) × (c − 1)。
The obtained χ² value is compared with a critical value from the Chi-squared distribution table. If χ² is larger than the critical value at p ≤ 0.05, the result is significant and the null hypothesis is rejected.
将计算出的 χ² 值与卡方分布表中的临界值进行比较。若 χ² 大于 p ≤ 0.05 时的临界值，则结果显著，拒绝零假设。
7. Mann-Whitney U Test | 曼-惠特尼U检验
The Mann-Whitney U test is a non-parametric test of difference used when two independent groups produce ordinal (ranked) data. It determines whether the scores from one condition are systematically higher or lower than those from the other.
曼-惠特尼U检验是一种非参数差异检验，用于两个独立组产生顺序（秩次）数据的情况。它判断一个条件下的分数是否系统地高于或低于另一条件。
All scores from both groups are combined and ranked from lowest to highest, with the lowest score receiving rank 1. Tied scores receive the average of the ranks that would have been assigned.
将两组合并后的所有分数由低到高排序，最低分赋予秩次1。相同分数则取它们本应获得秩次的平均值。
Next, the sums of ranks for each group (R₁ and R₂) are calculated. The U statistics for the two groups are computed as follows, and the smaller U is taken as the test statistic.
随后计算各组的秩和 (R₁ 与 R₂)。两组U统计量按以下公式计算，取较小的U作为检验统计量。
U₁ = n₁ n₂ + n₁(n₁ + 1)/2 − R₁
U₂ = n₁ n₂ + n₂(n₂ + 1)/2 − R₂
U = min(U₁, U₂)
U₁ = n₁ n₂ + n₁(n₁ + 1)/2 − R₁；U₂ = n₁ n₂ + n₂(n₂ + 1)/2 − R₂；检验统计量 U = min(U₁, U₂)。
The obtained U is compared with a critical U value from the Mann-Whitney table. For the result to be significant at the chosen p level, the obtained U must be equal to or less than the critical value.
将计算出的U值与曼-惠特尼U检验表中的临界U值比较。若要在选定的p水平上达到显著，获得的U值必须小于或等于临界值。
8. Wilcoxon Signed-Rank Test | 威尔科克森符号秩检验
The Wilcoxon signed-rank test is a non-parametric test for a repeated measures or matched pairs design, using ordinal data. It examines whether there is a significant difference between two related conditions.
威尔科克森符号秩检验是一种用于重复测量或配对组设计的非参数检验，处理顺序数据。它考察两个相关条件之间是否存在显著差异。
First, the difference between each pair of scores is calculated. Differences of zero are discarded, and the remaining differences are ranked by their absolute size, ignoring the sign.
首先计算每对分数的差值。剔除差值为零的数据，并将其余差值按绝对值大小排序，忽略正负号。
Ranks are then given the sign of the original difference. The sums of the positive ranks (T⁺) and negative ranks (T⁻) are computed. The test statistic T is the smaller of these two sums.
然后将秩次赋予原始差值的符号。计算正秩和 (T⁺) 与负秩和 (T⁻)。检验统计量T取两者中较小的值。
T = min(T⁺, T⁻)
检验统计量：T = min(T⁺, T⁻)。
The obtained T is compared against a critical T value from the Wilcoxon table. A significant result requires the obtained T to be equal to or less than the critical value at the chosen significance level.
将得到的T值与威尔科克森表中的临界T值比较。若获得的T值小于或等于选定显著性水平的临界值，结果即为显著。
9. Spearman’s Rho Correlation | 斯皮尔曼秩相关
Spearman’s rank-order correlation coefficient (rₛ) measures the strength and direction of a monotonic relationship between two ordinal variables. It is the non-parametric alternative to Pearson’s r.
斯皮尔曼秩相关系数 (rₛ) 衡量两个顺序变量之间单调关系的强度和方向。它是皮尔逊积差相关的非参数替代方法。
Each variable’s scores are ranked separately. The difference (d) between the ranks of each pair of observations is calculated, then squared and summed to give Σd².
分别对每个变量的分数进行排序。计算每一对观测值的秩次之差 (d)，将其平方后求和，得到 Σd²。
rₛ = 1 − (6 Σd²) / (n(n² − 1))
斯皮尔曼相关系数计算公式：rₛ = 1 − (6 Σd²) / (n(n² − 1))，其中 n 为配对观测值的数量。
The value of rₛ ranges from −1 to +1. A value of +1 indicates a perfect positive correlation, −1 a perfect negative correlation, and 0 indicates no monotonic relationship. The obtained rₛ is compared with a critical value table for Spearman’s rho; significance is reached when rₛ exceeds the critical value.
rₛ 的取值范围在 −1 到 +1之间。+1 表示完全正相关，−1 表示完全负相关，0 表示没有单调关系。将计算出的 rₛ 与斯皮尔曼秩相关系数临界值表进行比较；当 rₛ 大于临界值时达到显著。
10. Type I and Type II Errors | 第一类与第二类错误
A Type I error occurs when the null hypothesis is rejected when it is actually true – a false positive. The probability of making a Type I error is equal to the chosen significance level (α), typically 0.05.
当零假设实际为真却被错误拒绝时，发生第一类错误——即假阳性。犯第一类错误的概率等于选定的显著性水平 (α)，通常为 0.05。
A Type II error occurs when the null hypothesis is not rejected when it is actually false – a false negative. The probability of a Type II error is denoted by β, and it is inversely related to the power of the test.
当零假设实际为假却未被拒绝时，发生第二类错误——即假阴性。第二类错误的概率用 β 表示，它与检验的统计功效成反比。
These errors are a trade-off: reducing the risk of a Type I error by adopting a stricter p value (e.g., p ≤ 0.01) increases the risk of a Type II error, unless sample size is increased.
两种错误之间存在权衡：采用更严格的 p 值（如 p ≤ 0.01）可降低第一类错误风险，但会提高第二类错误风险，除非增大样本量。
11. Effect Size and Power | 效应量与统计功效
Statistical significance tells us whether a difference or relationship is likely to be genuine, but it does not reveal the size of the effect. Effect size remedies this by quantifying the magnitude of the finding.
统计显著性告诉我们差异或关系是否可能真实存在，但无法揭示效应的大小。效应量弥补了这一不足，它将研究结果的幅度加以量化。
At AS level, students may encounter Cohen’s criteria for correlation coefficients: r = 0.1 is a small effect, r = 0.3 a medium effect, and r = 0.5 a large effect. For group differences, a ‘standardised mean difference’ can be estimated, but detailed calculations are not required.
在AS阶段，学生可能会接触到相关系数的科恩标准：r = 0.1 为小效应，r = 0.3 为中等效应，r = 0.5 为大效应。对于组间差异，可以估算“标准化均值差”，但并不要求进行详细计算。
Statistical power is the probability that a test will correctly reject a false null hypothesis (1 − β). Power increases with larger sample sizes, larger effect sizes, and less stringent significance levels.
统计功效是指检验正确拒绝一个为假的零假设的概率 (1 − β)。增大样本量、增大效应量、放宽显著性水平，都会提高统计功效。
12. Reporting Statistical Findings | 报告统计结果
Psychological research must be reported following the conventions of the American Psychological Association (APA). The report includes the test statistic, degrees of freedom (if applicable), sample size, p value, and a statement of significance.
心理学研究报告应遵照美国心理学会 (APA) 的格式规范。报告须包含检验统计量、自由度（如适用）、样本量、p值，以及对显著性的陈述。
For a significant Mann-Whitney U test with independent groups, a typical sentence would be: “There was a significant difference in memory recall between the imagery and rehearsal groups, U = 18.5, p = 0.042 (one-tailed).” The exact p value should be reported when possible.
对于一项显著的
Published by TutorHao | AS Psychology Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导Cancel reply