📚 Year 13 Edexcel Psychology: Formula & Statistical Tests Quick Reference | Year 13 Edexcel 心理学：公式定理速查手册
In Year 13 Edexcel Psychology, mastering inferential statistics and research methods is essential for analysing experimental data. This quick reference compiles every formula, statistical test, and central theorem you need to know, from descriptive statistics to effect sizes, ensuring you are fully equipped for exam calculations and interpretations.
在 Year 13 Edexcel 心理学中，掌握推断统计和研究方法对于分析实验数据至关重要。这份速查手册汇集了你必须掌握的每一个公式、统计检验和核心定理——从描述统计到效应量，确保你为考试中的计算和解读做好充分准备。
1. Descriptive Statistics: Mean, Variance & Standard Deviation | 描述统计：均值、方差与标准差
In any psychological study, the first step is to summarise data. The mean (x̄) is the arithmetic average, calculated by summing all scores and dividing by the number of participants n.
在任何心理学研究中，第一步都是对数据进行总结。均值（x̄）是算术平均数，计算方法为将所有分数相加后除以参与者人数 n。
x̄ = Σx / n
Variance (s²) measures the spread of scores around the mean. For a sample, we use n – 1 to give an unbiased estimate of the population variance.
方差（s²）衡量的是分数围绕均值的离散程度。对于样本，我们使用 n – 1 以获得总体方差的无偏估计。
s² = Σ(x – x̄)² / (n – 1)
The standard deviation (s) is the square root of the variance, restoring the units to the original measurement. It is the most widely reported measure of dispersion.
标准差（s）是方差的平方根，将单位恢复到原始测量尺度。它是最广泛使用的离散程度指标。
s = √[ Σ(x – x̄)² / (n – 1) ]
Always remember: small standard deviation means data points cluster close to the mean; large s indicates high variability.
始终记住：标准差小意味着数据点紧密聚集在均值附近；s 大则表示变异性高。
2. Normal Distribution & Z-scores | 正态分布与 Z 分数
Many psychological variables follow a normal distribution, a symmetrical bell-shaped curve where the mean, median, and mode coincide. The total area under the curve equals 1, and fixed percentages of data lie within standard deviation bands (e.g., 68% within ±1 SD).
许多心理学变量服从正态分布，这是一种对称的钟形曲线，均值、中位数和众数重合。曲线下总面积为 1，固定百分比的数据落在标准差区间内（例如，±1 个标准差内包含 68% 的数据）。
A z-score converts an individual raw score into a standardised value, expressing how many standard deviations it is above or below the mean. This allows us to compare scores from different distributions.
Z 分数将个体原始分数转换为标准化数值，表示它高于或低于均值多少个标准差。这使我们能够比较来自不同分布的分数。
z = (x – μ) / σ (population) z = (x – x̄) / s (sample)
Once you have z, you can consult standard normal tables to find the proportion of scores below that value – a building block for hypothesis testing.
一旦获得 z 值，你就可以查阅标准正态分布表找到低于该值的比例——这是假设检验的基石。
3. Standard Error of the Mean | 均值标准误
When we draw a sample, the sample mean is an estimate of the population mean. The standard error of the mean (SEM) quantifies the precision of that estimate; it is the standard deviation of the sampling distribution of the mean.
当我们抽取样本时，样本均值是总体均值的估计。均值标准误（SEM）量化了这个估计的精确度；它是均值抽样分布的标准差。
SEM = s / √n
A larger sample size reduces the SEM, making our estimate more stable. SEM is used to construct confidence intervals and to perform inferential tests like the t-test.
样本量越大，SEM 越小，使我们的估计更稳定。SEM 用于构建置信区间以及进行 t 检验等推断检验。
4. Confidence Intervals | 置信区间
A confidence interval (CI) gives a range of values within which the true population mean is likely to fall. In psychology, we commonly use a 95% CI, meaning there is a 95% probability that the interval contains the population mean.
置信区间（CI）给出了一个真总体均值可能落入的数值范围。在心理学中，我们通常使用 95% CI，即区间包含总体均值的概率为 95%。
95% CI = x̄ ± 1.96 × (s / √n) (when n is large)
For smaller samples, the critical t-value replaces 1.96 from the z-table. Wider intervals signal less precision; narrow intervals suggest a reliable estimate.
对于小样本，临界 t 值会替换 z 表中的 1.96。区间越宽表示精确度越低；区间越窄则表明估计值可靠。
5. Levels of Significance & Probability | 显著性水平与概率
In hypothesis testing, we start with a null hypothesis (H₀) stating there is no effect or no difference. The significance level (α) is the probability of rejecting H₀ when it is true – typically set at 0.05 (5%) in psychology.
在假设检验中，我们以零假设（H₀）出发，假定没有效应或没有差异。显著性水平（α）是指在 H₀ 为真时拒绝 H₀ 的概率——心理学中通常设定为 0.05（5%）。
If the calculated probability (p-value) is less than α, we reject H₀ and conclude that the result is statistically significant. The p-value is the probability of obtaining the observed result (or one more extreme) if H₀ were true.
如果计算出的概率（p 值）小于 α，我们拒绝 H₀ 并得出结论：结果具有统计学显著性。p 值是在 H₀ 为真的条件下，获得当前结果（或更极端结果）的概率。
| p ≤ 0.05 | Significant – reject H₀ |
|---|---|
| p > 0.05 | Not significant – fail to reject H₀ |
Always remember: a non-significant result does not prove H₀; it simply lacks sufficient evidence to reject it.
始终记住：不显著的结果并不能证明 H₀；它只是缺乏足够证据去拒绝它。
6. Chi-Square Test | 卡方检验
The chi-square test is used with categorical, frequency-based data to determine whether there is a significant association between two independent variables (test of independence) or whether observed frequencies fit an expected distribution (goodness of fit).
卡方检验用于分类、频次型数据，以判断两个独立变量之间是否存在显著关联（独立性检验），或观察到的频次是否符合预期分布（拟合优度检验）。
χ² = Σ (O – E)² / E O = observed frequency, E = expected frequency
For a test of independence in a contingency table, the degrees of freedom are (rows – 1) × (columns – 1). Always check that no expected frequency is below 5, as this would violate test assumptions.
对于列联表的独立性检验，自由度为（行数 − 1）×（列数 − 1）。务必检查所有期望频次均不低于 5，否则会违反检验假设。
7. t-Test: Independent & Repeated Measures | t 检验：独立样本与重复测量
A t-test compares two sets of scores to see if they differ significantly. The independent (unrelated) t-test is used when two separate groups have experienced different conditions.
t 检验比较两组分数以判断是否存在显著差异。独立（无关）t 检验用于两个独立组分别经历了不同条件的情况。
t = (x̄₁ – x̄₂) / √( s²p (1/n₁ + 1/n₂) )
The pooled variance s²p combines the variances of both groups, weighted by degrees of freedom:
合并方差 s²p 按自由度加权合并两组的方差：
s²p = [ (n₁–1)s₁² + (n₂–1)s₂² ] / (n₁ + n₂ – 2)
The repeated measures (related) t-test is applied when the same participants are tested in both conditions. We first compute a difference score d for each participant.
重复测量（相关）t 检验适用于同一批参与者接受两种条件测试的情况。我们首先为每位参与者计算差异分数 d。
t = d̄ / (sd / √n) d̄ = mean of differences, sd = standard deviation of differences
For both tests, compare the calculated t with the critical t from the table at the chosen significance level and appropriate degrees of freedom.
两种检验都要将计算出的 t 值与选定显著性水平及适当自由度下的表格临界 t 值相比较。
8. Mann-Whitney U Test | 曼–惠特尼 U 检验
When the assumptions of an independent t-test are violated (e.g., ordinal data or non-normal interval data), the non-parametric Mann-Whitney U test is the appropriate alternative. It ranks all scores from both groups together and compares the sum of ranks between groups.
当独立 t 检验的假设被违反时（例如顺序数据或非正态的等距数据），非参数曼–惠特尼 U 检验是合适的替代方法。它将两组的全部分数一起排序，并比较两组间的秩和。
U₁ = n₁n₂ + n₁(n₁ + 1) / 2 – R₁
Where R₁ is the sum of ranks for group 1. Calculate both U₁ and U₂ (using R₂) and take the smaller U value. Compare this smaller U to the critical value from Mann-Whitney tables.
其中 R₁ 为第 1 组的秩和。同时计算 U₁ 和 U₂（利用 R₂），取较小的 U 值。将较小的 U 与曼–惠特尼表中的临界值进行比较。
9. 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 repeated measures designs. It works with difference scores, ignoring the sign initially to assign ranks to the absolute differences.
威尔科克森符号秩检验是用于重复测量设计的非参数检验。它利用差异分数，先不考虑符号而对绝对差值排秩。
Steps: (1) Calculate the difference d for each pair. (2) Rank the absolute values of d from smallest to largest, ignoring zero differences. (3) Sum the ranks for positive differences (R⁺) and for negative differences (R⁻) separately. The test statistic T is the smaller of these two sums.
步骤：（1）计算每对数据的差异 d。（2）将 d 的绝对值从小到大排秩，忽略零差异。（3）分别计算正差异的秩和（R⁺）与负差异的秩和（R⁻）。检验统计量 T 为两者中的较小值。
T = min(R⁺, R⁻)
Compare T with the Wilcoxon critical value for N (number of non-zero differences); if T is less than or equal to the critical value, the result is significant.
将 T 与针对 N（非零差异的个数）的威尔科克森临界值比较；若 T 小于或等于临界值，则结果显著。
10. Spearman’s Rank Correlation Coefficient | 斯皮尔曼等级相关系数
Spearman’s rho (rs) measures the strength and direction of association between two ranked variables. It is used when data are ordinal or when the relationship is monotonic but non-linear.
斯皮尔曼等级相关系数（rs）测量两个排序变量之间关联的强度和方向。当数据为顺序数据，或关系单调但非线性的情况下使用。
First, rank each variable separately; then calculate the difference d between the ranks of each pair.
首先，分别对每个变量排秩；然后计算每对数据等级之间的差 d。
rs = 1 – [ 6 Σd² / (n(n² – 1)) ]
rs ranges from –1 to +1. Use a Spearman significance table to determine whether the observed rs is statistically significant for the given n.
rs 的范围在 –1 到 +1 之间。使用斯皮尔曼显著性表判断对于给定的 n，观测到的 rs 是否具有统计学显著性。
11. Pearson’s Product-Moment Correlation | 皮尔逊积差相关
Pearson’s r is the parametric equivalent of Spearman’s and assumes interval-level data and a linear relationship. It measures the covariance of x and y divided by the product of their standard deviations.
皮尔逊 r 是斯皮尔曼相关的参数版本，要求等距数据且呈线性关系。它测量 x 与 y 的协方差除以它们标准差的乘积。
r = Σ( (x – x̄)(y – ȳ) ) / √[ Σ(x – x̄)² Σ(y – ȳ)² ]
Alternatively, this can be expressed using the covariance formula. The value of r lies between –1 and +1, with +1 indicating a perfect positive linear correlation and –1 a perfect negative linear correlation.
或者，可以用协方差公式表示。r 值介于 –1 与 +1 之间，+1 表示完全正线性相关，–1 表示完全负线性相关。
After computing r, square it to obtain the coefficient of determination (r²), which indicates the proportion of variance in one variable shared with the other.
计算 r 之后，平方得到决定系数（r²），它表示一个变量的方差中能被另一个变量解释的比例。
12. Effect Size (Cohen’s d) | 效应量（科恩 d 值）
Statistical significance does not tell us about the magnitude of an effect. Cohen’s d is a widely used measure of effect size for comparing two means, expressing the difference in standard deviation units.
统计显著性并不能告诉我们效应的大小。科恩 d 是广泛使用的衡量两个均值差异的效应量指标，以标准差为单位表达差异。
d = (x̄₁ – x̄₂) / spooled
Where spooled is the pooled standard deviation. If only group standard deviations are available, compute spooled similarly to the independent t-test procedure. Guideline benchmarks: 0.2 = small effect, 0.5 = medium effect, 0.8 = large effect.
其中 spooled 为合并标准差。如果只有各组的标准差，可用与独立 t 检验类似的程序计算 spooled。参考标准：0.2 = 小效应，0.5 = 中等效应，0.8 = 大效应。
Reporting effect size has become a requirement in many psychology journals because it conveys practical significance, helping readers understand whether a result is meaningful beyond p-values.
报告效应量已成为许多心理学期刊的要求，因为它传达了实际显著性，帮助读者理解结果是否超越了 p 值而具有真实意义。
Published by TutorHao | Psychology Revision Series | aleveler.com
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