A-Level Edexcel Psychology: Quick Reference Handbook of Formulas and Theorems | A-Level Edexcel 心理学：公式定理速查手册

📚 A-Level Edexcel Psychology: Quick Reference Handbook of Formulas and Theorems | A-Level Edexcel 心理学：公式定理速查手册

Psychology may not have theorems in the traditional sense, but statistical formulas are the logical backbone of psychological research. This handbook compiles every essential formula, decision rule, and key statistical concept required for the Edexcel A-Level Psychology specification, with a focus on research methods and inferential tests. Use it to refresh your memory or to check your calculations before tackling exam questions on data analysis.

心理学或许没有传统意义上的定理，但统计公式是心理科学研究的逻辑支柱。本速查手册汇集了 Edexcel A-Level 心理学所需的所有核心公式、决策规则和关键统计概念，重点涵盖研究方法和推断检验。在应对数据分析相关的考试题目之前，不妨用这份手册来巩固记忆、核对你的计算过程。

1. Descriptive Statistics: Central Tendency | 描述统计：集中量数

The mean (X̄) is the arithmetic average of a data set, calculated as ΣX / N for a population or Σx / n for a sample. The median is the middle value when data are ordered; for an even number of scores it is the mean of the two middle values. The mode is the most frequently occurring score. These three measures summarise the ‘centre’ of a data distribution and are the first step in any data analysis.

平均值 (X̄) 是一组数据的算术平均数，总体均数计算为 ΣX / N，样本均数计算为 Σx / n。中位数是排序后位于中间的数；当有偶数个数据时，为中位两数的均值。众数是出现次数最多的数值。这三个指标概括了数据分布的“中心”，是任何数据分析的第一步。

Mean: X̄ = Σx / n

均值：X̄ = Σx / n

2. Variance and Standard Deviation | 方差与标准差

Variance and standard deviation measure the spread of data around the mean. For a sample, variance (s²) is the sum of squared deviations divided by (n – 1). The standard deviation (s) is the square root of variance. These are used to assess how widely scores are distributed and are crucial for parametric tests.

方差和标准差衡量数据围绕均值的离散程度。样本方差 (s²) 是离差平方和除以 (n – 1)。标准差 (s) 是方差的平方根。它们用于判断分数的分散程度，对于参数检验至关重要。

Sample variance: s² = Σ (x – X̄)² / (n – 1)

样本方差：s² = Σ (x – X̄)² / (n – 1)

Sample standard deviation: s = √ [ Σ (x – X̄)² / (n – 1) ]

样本标准差：s = √ [ Σ (x – X̄)² / (n – 1) ]

3. The Normal Distribution and z-Scores | 正态分布与 z 分数

Many inferential tests assume the data are normally distributed. The standard normal distribution has a mean of 0 and standard deviation of 1. A raw score can be converted to a z-score to see how many standard deviations it lies from the mean. This helps in understanding probabilities and is the basis of parametric testing.

许多推断检验假设数据服从正态分布。标准正态分布的均值为 0，标准差为 1。原始分数可以转换为 z 分数，以表示该分数距离均值有多少个标准差。这有助于理解概率，也是参数检验的基础。

z = (x – X̄) / s

In a normal distribution, approximately 68% of scores lie within ±1 SD, 95% within ±1.96 SD, and 99.7% within ±3 SD. These percentages are used to determine whether an observed result is extreme enough to be statistically significant.

在正态分布中，大约 68% 的分数落在均值 ±1 个标准差内，95% 落在 ±1.96 个标准差内，99.7% 落在 ±3 个标准差内。这些百分比用于判断观测结果是否极端到具有统计显著性。

4. Spearman’s Rank Correlation Coefficient (rₛ) | 斯皮尔曼等级相关系数 (rₛ)

Spearman’s rₛ measures the strength and direction of association between two ranked variables. It is a non-parametric test suitable for ordinal data or when the assumptions of Pearson’s r are violated. The test statistic is calculated from differences in ranks (D). The obtained rₛ value is compared with a critical values table.

斯皮尔曼等级相关系数衡量两个秩次变量之间关联的强度和方向。它是一种非参数检验，适用于顺序数据或当皮尔逊相关的假设不满足时。检验统计量根据秩次之差 (D) 计算得出，得到的 rₛ 值与临界值表进行比较。

rₛ = 1 – [ 6 Σ D² / (n (n² – 1)) ]

where D is the difference between each pair of ranks and n is the number of pairs. A perfect positive correlation gives rₛ = +1, a perfect negative correlation gives rₛ = -1. The null hypothesis of no correlation is rejected if the calculated rₛ (ignoring sign) is greater than or equal to the critical value.

式中 D 是每对数据的秩次差，n 是数据对的数目。完全正相关时 rₛ = +1，完全负相关时 rₛ = -1。若计算出的 rₛ 的绝对值大于或等于临界值，则拒绝无相关的零假设。

5. Pearson’s Product-Moment Correlation Coefficient (r) | 皮尔逊积差相关系数 (r)

Pearson’s r is a parametric test of linear correlation between two continuous variables, assuming interval/ratio data and normally distributed scores. The formula uses raw scores, and it returns a value between -1 and +1. The significance is usually tested against a critical value of r for df = n – 2.

皮尔逊 r 是一种参数检验，用于两连续变量之间的线性相关，要求等距/等比数据和正态分布。该公式使用原始分数，返回介于 -1 和 +1 之间的值。显著性通常通过自由度为 n – 2 的 r 临界值表来检验。

r = [ n Σxy – (Σx)(Σy) ] / √ [ (n Σx² – (Σx)²)(n Σy² – (Σy)²) ]

The numerator is the covariance, and the denominator is the product of standard deviations. This formula is more sensitive to linear relationships but is heavily influenced by outliers.

分子是协方差，分母是标准差的乘积。这一公式对线性关系更敏感，但受异常值影响很大。

6. Chi-Square (χ²) Tests: Goodness of Fit and Independence | 卡方 (χ²) 检验：拟合优度与独立性

The chi-square test is a non-parametric test for frequency data. The goodness-of-fit version compares observed frequencies (O) to expected frequencies (E) in one categorical variable. The test of independence examines the association between two categorical variables in a contingency table.

卡方检验是针对频数数据的非参数检验。拟合优度检验将单个类别变量的观测频数 (O) 与期望频数 (E) 进行比较。独立性检验则分析列联表中两个类别变量之间的关联。

χ² = Σ [ (O – E)² / E ]

Degrees of freedom: for goodness of fit, df = number of categories – 1; for independence, df = (rows – 1) * (columns – 1). The calculated χ² must be compared against the critical value for the chosen significance level and df.

自由度：拟合优度检验中 df = 类别数 – 1；独立性检验中 df = (行数 – 1) * (列数 – 1)。计算出的 χ² 值必须与所选显著性水平和 df 下的临界值进行比较。

7. Mann-Whitney U Test | 曼-惠特尼 U 检验

The Mann-Whitney U test is a non-parametric test of difference between two independent groups. It is used when data are at least ordinal and the assumption of normality for an independent t-test is violated. The test statistic U is based on the sum of ranks for each group. The smaller U is compared with a critical value.

曼-惠特尼 U 检验是一种非参数检验，用来比较两个独立组之间的差异。当数据至少是顺序水平且不能满足独立 t 检验的正态假设时使用。检验统计量 U 基于每一组的秩和，较小的 U 值与临界值进行比较。

U₁ = n₁ n₂ + n₁(n₁+1)/2 – R₁

U₂ = n₁ n₂ + n₂(n₂+1)/2 – R₂

where n₁ and n₂ are the sizes of the two groups, and R₁ and R₂ are the respective rank sums. The obtained U is the smaller of U₁ and U₂. If U ≤ critical value, the difference is significant.

其中 n₁ 和 n₂ 是两组样本量，R₁ 和 R₂ 是各自的秩和。所得 U 值取 U₁ 和 U₂ 中较小的一个。若 U ≤ 临界值，差异显著。

8. Wilcoxon Signed-Rank Test | 威尔科克森符号秩检验

The Wilcoxon signed-rank test is a non-parametric alternative to the repeated-measures t-test. It assesses whether there is a significant difference between two sets of paired scores. Absolute differences are ranked, and ranks are signed (+ or -). The test statistic T is the smaller of the sums of positive and negative ranks.

威尔科克森符号秩检验是重复测量 t 检验的非参数替代方法，用于评估两组配对分数之间是否存在显著差异。对差值绝对值编秩，并赋予正负号。检验统计量 T 是正秩和与负秩和中较小的一个。

T = smaller of Σ R⁺ and Σ R⁻

T = 取 Σ R⁺ 与 Σ R⁻ 中较小的值

Zero differences are typically discarded and n is adjusted. The calculated T is compared with a critical values table; the result is significant if T is less than or equal to the critical value.

零差值通常被排除，n 相应调整。得到的 T 值与临界值表对比；若 T ≤ 临界值，则结果显著。

9. The Sign Test | 符号检验

The sign test is the simplest non-parametric test of difference for repeated measures. It looks only at the direction of change (+, -) and ignores the magnitude of differences. The test statistic S is the number of pluses or minuses, whichever is smaller. It is used when data are nominal (differences categorised as positive or negative).

符号检验是最简单的针对重复测量的非参数差异检验，只考虑变化的方向（+ 与 -），忽略差值大小。检验统计量 S 是正号或负号中数目较小的那个。当数据为称名水平（差值分为正向和负向）时使用。

S = smaller count of + or – signs

S = 正负号数目较小者

The null hypothesis states there is no difference, meaning P(+) = P(-) = 0.5. The observed S is compared with a binomial distribution table (or sign test table) at the chosen significance level and N (number of non-zero differences) to decide if the result is significant.

零假设为无差异，即 P(+) = P(-) = 0.5。将观测到的 S 值与在选定显著性水平和 N (非零差值数) 下的二项分布表 (或符号检验表) 对比，以判断结果是否显著。

10. Independent and Repeated Measures t-Tests | 独立样本与重复测量 t 检验

The independent t-test (unrelated t-test) compares the means of two separate groups. The repeated-measures t-test (related t-test) compares two means from the same group tested under two conditions. Both are parametric tests and assume normal distribution and homogeneity of variance (for independent).

独立样本 t 检验（非相关 t 检验）比较两个独立组的均值；重复测量 t 检验（相关 t 检验）比较同一组被试在两种条件下的均值。两者都是参数检验，需要数据正态分布，且独立 t 检验还要求方差齐性。

Independent t: t = (X̄₁ – X̄₂) / √ [ (s₁²/n₁) + (s₂²/n₂) ]

独立 t 检验：t = (X̄₁ – X̄₂) / √ [ (s₁²/n₁) + (s₂²/n₂) ]

Repeated-measures t: t = D̄ / (s_D / √n)

重复测量 t 检验：t = D̄ / (s_D / √n)

D̄ is the mean of the difference scores, and s_D is the standard deviation of the difference scores. Degrees of freedom: df = n – 1 for repeated measures; df = n₁ + n₂ – 2 for independent t. The obtained t is compared with the critical t-value from tables.

其中 D̄ 是差值的均值，s_D 是差值的标准差。自由度：重复测量中 df = n – 1；独立样本 t 中 df = n₁ + n₂ – 2。计算出的 t 值与 t 分布临界值表对比以判断显著性。

11. Effect Size: Cohen’s d | 效应量：科恩 d

Cohen’s d is a standardised measure of effect size, expressing the difference between two means in standard deviation units. It helps to evaluate the practical importance of a result, as statistical significance tells us only about the probability that an effect exists. A d of 0.2 is considered small, 0.5 medium, and 0.8 large.

科恩 d 是标准化的效应量，以标准差为单位表示两个均值之间的差异。它帮助评价结果的实用重要性，因为统计显著性仅告诉我们效应存在的概率。d = 0.2 视为小效应，0.5 为中等，0.8 为大效应。

d = (X̄₁ – X̄₂) / s_pooled

where s_pooled is the pooled standard deviation. For a repeated-measures design, a variant using the standard deviation of the difference scores may be used. Effect sizes are increasingly required in Edexcel exam discussions to strengthen the interpretation of inferential statistics.

其中 s_pooled 是合并标准差。对于重复测量设计，可使用差值标准差的一种变体。在 Edexcel 考试讨论中，越来越多的题目要求使用效应量来加强推断统计的解释。

12. Type I and II Errors, Power, and p-Value | I 型与 II 型错误、检验力及 p 值

A Type I error occurs when the null hypothesis is wrongly rejected (a false positive). The probability of a Type I error is set by the significance level (α), typically 0.05. A Type II error (β) happens when a false null hypothesis is not rejected (a false negative). The power of a test is 1 – β, the probability of correctly detecting an effect when it exists.

I 型错误发生在错误地拒绝零假设时（假阳性）。I 型错误的概率由显著性水平 (α) 设定，通常为 0.05。II 型错误 (β) 指未能拒绝一个错误的零假设（假阴性）。检验力为 1 – β，即当效应真实存在时正确检测出该效应的概率。

The p-value is the probability of obtaining the observed result, or one more extreme, assuming the null hypothesis is true. If p ≤ α, the result is statistically significant. Edexcel requires students to not only calculate test statistics but also to reach justified conclusions using critical values or p-values.

p 值是在零假设为真的条件下，得到当前观测结果（或更极端结果）的概率。若 p ≤ α，则结果具有统计显著性。Edexcel 要求考生不仅要计算检验统计量，还要使用临界值或 p 值得出有依据的结论。

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A-Level Edexcel Psychology: Quick Reference Handbook of Formulas and Theorems | A-Level Edexcel 心理学：公式定理速查手册