Year 12 SQA Psychology: Quick Reference Formula & Theorems Handbook | Year 12 SQA 心理学：公式定理速查手册

📚 Year 12 SQA Psychology: Quick Reference Formula & Theorems Handbook | Year 12 SQA 心理学：公式定理速查手册

This handbook provides a concise, at-a-glance summary of the essential statistical formulas and theorems you will encounter in the SQA Higher Psychology course. Whether you are analysing data from experiments, interpreting correlations, or determining the significance of your findings, these tools form the backbone of psychological research methods. Use this reference to support your revision and to ensure you can select and apply the correct formula in the appropriate context.

本手册为你提供 SQA Higher 心理学课程中核心统计公式与定理的速查摘要。无论你是在分析实验数据、解读相关关系，还是判断研究发现的显著性，这些工具都是心理学研究方法的基石。请将本参考用于复习巩固，并确保你能在恰当的语境下选择和应用正确的公式。

1. Mean, Median & Mode | 均值、中位数与众数

The mean is the arithmetic average and is the most commonly used measure of central tendency for interval or ratio data. It is computed by summing all scores (Σx) and dividing by the total number of scores (n). Formula: x̄ = Σx / n. The median is the middle value when data are arranged in ascending order; it is robust against extreme scores. The mode is the value that appears most frequently in a data set.

均值即算术平均数，是间隔或等比数据中最常用的集中趋势指标。计算方法为将所有分数求和(Σx)再除以个数(n)，公式为 x̄ = Σx / n。中位数是将数据按升序排列后的中间值，对极端值不敏感。众数是数据集中出现频率最高的值。

x̄ = Σx / n

Mean is suitable for symmetrical distributions; median preferred for skewed distributions.
均值适用于对称分布；中位数更适合偏态分布。

2. Range, Variance & Standard Deviation | 全距、方差与标准差

Measures of dispersion describe the spread of scores. The range is simply the difference between the highest and lowest scores: Range = max – min. A more precise measure is variance, which quantifies the average squared deviation from the mean. For a sample, variance (s²) = Σ(x – x̄)² / (n – 1). The standard deviation (s) is the square root of variance, giving a measure in the original units: s = √[ Σ(x – x̄)² / (n – 1) ].

离散程度指标描述分数的分散性。全距即最高分与最低分之差：Range = max – min。更精确的指标是方差，它量化了与均值偏差的平方平均。样本方差公式为 s² = Σ(x – x̄)² / (n – 1)。标准差(s)是方差的平方根，以原始单位呈现，公式为 s = √[ Σ(x – x̄)² / (n – 1) ]。

s² = Σ(x – x̄)² / (n – 1) and s = √s²

Use n-1 for sample statistics to provide an unbiased estimate of the population parameter.
使用 n-1 以得到总体参数的无偏估计。

3. Normal Distribution & Z-Scores | 正态分布与Z分数

Many psychological variables are assumed to follow a normal distribution, a bell-shaped curve where the mean, median and mode coincide. The standard normal distribution has a mean of 0 and a standard deviation of 1. A z-score indicates how many standard deviations a raw score is from the mean: z = (x – μ) / σ, or when using sample estimates, z = (x – x̄) / s. Approximately 68% of data fall within ±1 SD, 95% within ±1.96 SD, and 99.7% within ±3 SD.

许多心理变量被假设服从正态分布，呈钟形曲线，均值、中位数和众数重合。标准正态分布的均值为0，标准差为1。Z分数表示一个原始分数距离均值多少个标准差：z = (x – μ) / σ，或使用样本估计时 z = (x – x̄) / s。约68%的数据落在±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。

z = (x – μ) / σ or z = (x – x̄) / s

Z-scores allow comparison across different measures and identification of outliers.
Z分数支持跨不同测量工具的比较并用于识别异常值。

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

Spearman’s rho (rₛ) is a non-parametric test used to assess the strength and direction of association between two ordinal variables, or when data do not meet parametric assumptions. First, rank both sets of scores individually; then compute the difference (d) between each pair of ranks. The formula is rₛ = 1 – (6 Σd²) / [n(n² – 1)], where n is the number of pairs. The value ranges from -1 (perfect negative correlation) to +1 (perfect positive correlation).

斯皮尔曼等级相关系数(rₛ)是一种非参数检验，用于评估两个顺序变量之间的关系强度和方向，或当数据不满足参数假设时使用。首先，分别对两组分数排序；然后计算每对等级之差(d)。公式为 rₛ = 1 – (6 Σd²) / [n(n² – 1)]，其中 n 为对数。其值介于-1（完全负相关）到+1（完全正相关）之间。

rₛ = 1 – (6 Σd²) / [n(n² – 1)]

Used with repeated measures or matched pairs designs when at least ordinal data are collected.
当收集到至少是顺序数据时，适用于重复测量或配对组设计。

5. Pearson’s Product-Moment Correlation | 皮尔逊积矩相关系数

Pearson’s r is a parametric test for the linear relationship between two interval/ratio variables. It assumes normally distributed data and homoscedasticity. The formula is r = Σ[(x – x̄)(y – ȳ)] / √[ Σ(x – x̄)² Σ(y – ȳ)² ], where x̄ and ȳ are the means of the two variables. The resulting coefficient also lies between -1 and +1, with 0 indicating no linear correlation.

皮尔逊积矩相关系数(r)是一种参数检验，用于考察两个间隔/等比变量之间的线性关系。其假设数据呈正态分布且方差齐性。公式为 r = Σ[(x – x̄)(y – ȳ)] / √[ Σ(x – x̄)² Σ(y – ȳ)² ]，x̄和ȳ是两个变量的均值。相关系数值同样处于-1到+1之间，0表示无线性相关。

r = Σ[(x – x̄)(y – ȳ)] / √[ Σ(x – x̄)² Σ(y – ȳ)² ]

Commonly used in correlational studies; does not imply causation.
常用于相关研究；不意味着因果关系。

6. Chi-Square Test for Independence | 独立性卡方检验

The chi-square (χ²) test for independence is a non-parametric procedure used to determine whether there is a significant association between two categorical variables. It compares observed frequencies (O) with expected frequencies (E) under the null hypothesis of no association. The test statistic is χ² = Σ[(O – E)² / E], where E = (row total × column total) / grand total. Degrees of freedom (df) = (number of rows – 1) × (number of columns – 1).

独立性卡方检验是一种非参数方法，用于确定两个分类变量之间是否存在显著关联。它比较观察频数(O)与零假设下无关联的期望频数(E)。检验统计量为 χ² = Σ[(O – E)² / E]，其中期望频数 E = (行合计 × 列合计) / 总计。自由度为 df = (行数 – 1) × (列数 – 1)。

χ² = Σ[(O – E)² / E]

Requires independent observations and expected frequencies of at least 5 in each cell.
需要独立观测值，且每个单元格的期望频数至少为5。

7. Mann-Whitney U Test | 曼-惠特尼U检验

The Mann-Whitney U test is a non-parametric alternative to the independent t-test, used when comparing two independent groups on an ordinal dependent variable. Combine the scores from both groups, rank them, and sum the ranks for each group (R₁ and R₂). The U statistics are calculated as U₁ = n₁n₂ + n₁(n₁+1)/2 – R₁ and U₂ = n₁n₂ + n₂(n₂+1)/2 – R₂. The smaller U value is compared against critical values to determine significance.

曼-惠特尼U检验是独立样本t检验的非参数替代，用于在顺序因变量上比较两个独立组。将两组分数合并并排秩，分别计算各组的秩和(R₁和R₂)。计算U值：U₁ = n₁n₂ + n₁(n₁+1)/2 – R₁，U₂ = n₁n₂ + n₂(n₂+1)/2 – R₂。取较小的U值与临界值比较以判断显著性。

U = n₁n₂ + n₁(n₁+1)/2 – R₁ or U = n₁n₂ + n₂(n₂+1)/2 – R₂

Suitable for independent groups designs when the assumption of normality is violated.
适用于违反正态性假设的独立组设计。

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

The Wilcoxon signed-rank test is the non-parametric equivalent of the related t-test, designed for repeated measures or matched pairs with ordinal data. Calculate the difference between each pair, rank the absolute differences ignoring zero differences, then assign the sign of the difference to each rank. Sum the positive ranks (T⁺) and negative ranks (T⁻). The test statistic T is the smaller of these two sums, compared to the critical Wilcoxon value.

威尔科克森符号秩检验是相关样本t检验的非参数等价形式，适用于重复测量或配对组且数据为顺序变量的情况。计算每对数据的差值，对绝对值进行排秩（忽略差值为零的对），再为各秩附上差值的符号。分别计算正秩和(T⁺)与负秩和(T⁻)。检验统计量T为二者中较小者，与威尔科克森临界值进行比较。

T = min(Σ positive ranks, Σ negative ranks)

Use when the differences are not normally distributed but data are at least ordinal.
当差值不服从正态分布但数据至少为顺序水平时使用。

9. Independent Samples t-Test | 独立样本t检验

The independent t-test is a parametric test that compares the means of two unrelated groups on an interval/ratio dependent variable. It assumes normality, homogeneity of variance, and independence of observations. When equal variances are assumed, the formula uses pooled variance (sₚ²): t = (x̄₁ – x̄₂) / √[ sₚ²(1/n₁ + 1/n₂) ], where sₚ² = [(n₁-1)s₁² + (n₂-1)s₂²] / (n₁+n₂-2). Compare the obtained t to the critical t-value with df = n₁+n₂-2.

独立样本t检验是一种参数检验，用于比较两个无关组在间隔/等比因变量上的均值。假定数据正态、方差齐性且观察独立。在假定方差齐性时，使用合并方差(sₚ²)计算公式：t = (x̄₁ – x̄₂) / √[ sₚ²(1/n₁ + 1/n₂) ]，其中 sₚ² = [(n₁-1)s₁² + (n₂-1)s₂²] / (n₁+n₂-2)。将所得t值与自由度为 df = n₁+n₂-2 的临界t值比较。

t = (x̄₁ – x̄₂) / √[ sₚ²(1/n₁ + 1/n₂) ]

Commonly used in experimental designs with an independent measures IV.
常用于自变量为独立测量的实验设计。

10. Related Samples t-Test | 相关样本t检验

Also known as the paired-samples or dependent t-test, this parametric test evaluates the mean difference between two related sets of scores (e.g., before vs. after, or matched pairs). Calculate the difference score d = x₂ – x₁ for each pair, then compute the mean difference (x̄_d) and its standard deviation (s_d). The t statistic is t = x̄_d / (s_d / √n), with degrees of freedom df = n – 1, where n is the number of pairs.

又称配对样本或相依t检验，该参数检验评估两组相关分数之间的均值差异（如前后测或配对组）。计算每对数据的差值 d = x₂ – x₁，再求差值的均值(x̄_d)及其标准差(s_d)。t统计量为 t = x̄_d / (s_d / √n)，自由度为 df = n – 1，其中 n 为配对数量。

t = x̄_d / (s_d / √n)

Reduces within-group variability, making it more powerful than the independent test when appropriate.
减少了组内变异，在适用时比独立检验效能更高。

11. Effect Size (Cohen’s d) | 效应量 (Cohen’s d)

While significance tests tell us whether an effect exists, effect size measures the magnitude of that effect. Cohen’s d is a commonly used index for comparing two means. For independent groups, d = (x̄₁ – x̄₂) / SD_pooled, where SD_pooled = √[((n₁-1)s₁² + (n₂-1)s₂²) / (n₁+n₂-2)]. By convention, d = 0.2 is small, 0.5 medium, and 0.8 large. For related designs, a variant using the standard deviation of the difference scores is used.

显著性检验告知我们效应是否存在，而效应量则衡量该效应的大小。Cohen’s d 是常用于比较两组均值的指标。对于独立组，d = (x̄₁ – x̄₂) / SD_pooled，其中 SD_pooled = √[((n₁-1)s₁² + (n₂-1)s₂²) / (n₁+n₂-2)]。依惯例，d = 0.2 为小效应，0.5 中等，0.8 为大效应。在相关设计中则使用基于差值标准差的变体公式。

d = (x̄₁ – x̄₂) / SD_pooled

Report effect size alongside p-values to convey practical significance.
应同时报告效应量和 p 值以传达实际意义。

12. Significance, Probability & Critical Values | 显著性水平、概率与临界值

In SQA Psychology, you must decide whether an observed result is statistically significant. The significance level (alpha, α) is typically set at 0.05 (5%), meaning there is a 5% risk of a Type I error – rejecting a true null hypothesis. Each statistical test has its own table of critical values; you compare your calculated test statistic to the critical value at the appropriate degrees of freedom and alpha level. If the calculated value exceeds the critical value, the result is significant and you reject the null hypothesis. The p-value represents the probability of obtaining the observed result, or one more extreme, if the null hypothesis is true.

在SQA心理学中，你需要判断观测结果是否具有统计显著性。显著性水平(α)通常设为0.05(5%)，意味着有5%的风险犯I类错误——即拒绝了一个真实的零假设。每种统计检验都有其专属的临界值表；你需要根据适当的自由度和α水平，将计算所得的检验统计量与临界值进行比较。若计算值超过临界值，则结果显著，你拒绝零假设。p值表示若零假设为真时，得到当前结果或更极端结果的概率。

If test statistic ≥ critical value, reject H₀; p ≤ α indicates significance.

Always check degrees of freedom and whether the test is one-tailed or two-tailed.
务必检查自由度以及检验是单尾还是双尾。

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Year 12 SQA Psychology: Quick Reference Formula & Theorems Handbook | Year 12 SQA 心理学：公式定理速查手册