📚 Year 13 CCEA Psychology: Quick-Reference Formula & Theorem Handbook | Year 13 CCEA 心理学：公式定理速查手册
This handbook collects the essential formulas, statistical tests, and key decision rules you will meet across the Year 13 CCEA Psychology specification. Use it to review how to calculate measures of central tendency, dispersion, inferential statistics, and effect sizes, and to choose the correct test for a given research design.
本手册汇集了 Year 13 CCEA 心理学课程中必备的公式、统计检验与关键决策规则。你可以用它复习集中趋势量数、离散量数、推断统计和效应量的计算方法，并学会根据研究设计选择正确的检验。
1. Measures of Central Tendency | 集中趋势量数
The mean is the arithmetic average of all scores. It is most appropriate for interval/ratio data without extreme outliers.
均值是所有分数的算术平均数，最适用于没有极端异常值的等距/等比数据。
Mean (x̄) = Σx / n
where Σx is the sum of all scores and n is the number of scores.
其中 Σx 表示所有分数之和，n 是分数的个数。
The median is the middle score when data are ordered. It is used with ordinal data or when outliers distort the mean.
中位数是将数据排序后位于中间位置的分数，适用于顺序数据或均值受异常值影响时使用。
Median position = (n + 1) / 2
The mode is the most frequently occurring score; it is the only average suitable for nominal data.
众数是出现次数最多的分数，是唯一适用于称名数据的平均数。
2. Measures of Dispersion | 离散量数
The range is the simplest measure of spread. It is sensitive to extreme values.
极差是最简单的离散指标，但对极端值敏感。
Range = highest score − lowest score
Variance and standard deviation quantify how much scores deviate from the mean. Standard deviation is the square root of variance and is expressed in the original units.
方差和标准差量化了分数偏离均值的程度。标准差是方差的平方根，以原始单位表示。
Variance (s²) = Σ(x − x̄)² / (n − 1)
Standard deviation (s) = √[ Σ(x − x̄)² / (n − 1) ]
Here n − 1 is used for a sample estimate; for a population, divide by n. In CCEA psychology, the sample formula is standard.
此处使用 n − 1 作为样本估计；若为总体则除以 n。在 CCEA 心理学中，通常使用样本公式。
3. Normal Distribution and z‑Scores | 正态分布与 z 分数
The normal distribution is a bell-shaped curve where about 68% of scores fall within ±1 SD, 95% within ±1.96 SD, and 99.7% within ±3 SD.
正态分布是钟形曲线，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。
A z‑score converts a raw score into standard deviation units, telling you how many standard deviations the score lies above or below the mean.
z 分数将原始分数转换为标准差单位，表示该分数在均值之上或之下多少个标准差。
z = (x − x̄) / s
z‑scores are essential for comparing scores from different distributions and for determining probabilities using z‑tables.
z 分数对于比较不同分布的分数以及用 z 表确定概率至关重要。
4. Pearson’s r Correlation Coefficient | 皮尔逊积差相关系数
Pearson’s r measures the strength and direction of a linear relationship between two continuous variables. Values range from −1 (perfect negative) to +1 (perfect positive).
皮尔逊 r 衡量两个连续变量之间线性关系的强度和方向。数值范围从 −1（完全负相关）到 +1（完全正相关）。
r = [ Σ(x − x̄)(y − ȳ) ] / √[ Σ(x − x̄)² Σ(y − ȳ)² ]
An r of 0 indicates no linear correlation. The coefficient of determination, r², shows the proportion of shared variance.
r 为 0 表示无线性相关。决定系数 r² 表示共享方差的比例。
Spearman’s rho is the non‑parametric alternative used when data are ordinal or assumptions of normality are violated.
当数据为顺序数据或正态性假设被违反时，可使用非参数替代方法——斯皮尔曼等级相关。
5. Linear Regression | 线性回归
Simple linear regression predicts the value of one variable (Y) from another (X) using a line of best fit.
简单线性回归使用最佳拟合线，从一个变量 (X) 预测另一个变量 (Y) 的值。
Regression equation: Y' = a + bX
The slope b is calculated as b = r (sy / sx), and a = ȳ − b x̄, where sy and sx are the standard deviations of Y and X.
斜率 b 的计算公式为 b = r (sy / sx)，截距 a = ȳ − b x̄，其中 sy 和 sx 分别是 Y 和 X 的标准差。
6. Probability and Inferential Statistics Foundations | 概率与推断统计基础
Probability (p) is the likelihood of an event occurring, ranging from 0 to 1. In inferential statistics, p indicates the chance of obtaining the observed result if the null hypothesis were true.
概率 (p) 是事件发生的可能性，取值范围从 0 到 1。在推断统计中，p 值表示若零假设为真时获得当前观测结果的概率。
p = number of favourable outcomes / total number of outcomes
The null hypothesis (H₀) assumes no effect or no difference; the alternative hypothesis (H₁) states there is an effect or a difference.
零假设 (H₀) 假定没有效应或无差异；备择假设 (H₁) 则声称存在效应或差异。
lly set at α = 0.05. If p ≤ 0.05, the result is considered significant and H₀ is rejected.
统计显著性水平通常设定为 α = 0.05。若 p ≤ 0.05，则认为结果显著，拒绝 H₀。
7. Hypothesis Testing and Critical Values | 假设检验与临界值
A test statistic (e.g., t, r, U, χ²) is calculated and compared with a critical value from a statistical table. The comparison depends on degrees of freedom (df), the chosen alpha level, and whether the test is one‑tailed or two‑tailed.
计算出检验统计量（如 t、r、U、χ²）后，需与统计表中的临界值进行比较。比较时需考虑自由度 (df)、选定的显著水平以及检验是单尾还是双尾。
ult (for most tests, except when the observed value must be ≤ critical value, as with Mann‑Whitney U and Wilcoxon).
观测值 ≥ 临界值 → 结果显著（大多数检验如此；但曼‑惠特尼 U 检验和威尔科克森检验要求观测值 ≤ 临界值才算显著）。
In psychology, a Type I error is rejecting a true null hypothesis (false positive); a Type II error is failing to reject a false null hypothesis (false negative).
在心理学中，一类错误是拒绝了真实的零假设（假阳性）；二类错误是未能拒绝错误的零假设（假阴性）。
8. Student’s t‑Test | 学生 t 检验
The independent‑samples t‑test compares the means of two unrelated groups. The formula uses the difference between sample means divided by the standard error of the difference.
独立样本 t 检验比较两个无关组的均值。公式为样本均值之差除以该差值的标准误。
t = (x̄₁ − x̄₂) / √[ (sp² / n₁) + (sp² / n₂) ]
where sp² is the pooled variance: sp² = [ (n₁−1)s₁² + (n₂−1)s₂² ] / (n₁ + n₂ − 2). Degrees of freedom = n₁ + n₂ − 2.
其中 sp² 为合并方差，公式如上；自由度为 n₁ + n₂ − 2。
The related (paired) t‑test compares two scores from the same participant or matched pairs. It analyses difference scores.
相关（配对）t 检验比较同一被试或配对的两个分数，分析的是差异分数。
t = d̄ / (sd / √n)
d̄ = mean of the difference scores, sd = standard deviation of the difference scores, n = number of pairs. df = n − 1.
d̄ 为差异分数的均值，sd 为差异分数的标准差，n 为对数，自由度为 n − 1。
9. Chi‑Square (χ²) Tests | 卡方检验
The chi‑square test for goodness of fit examines whether observed frequencies differ from expected frequencies in one categorical variable.
卡方拟合优度检验考察一个分类变量的观测频次是否与期望频次存在差异。
χ² = Σ [ (O − E)² / E ]
O = observed frequency, E = expected frequency. df = number of categories − 1.
O 为观测频次，E 为期望频次；自由度为类别数量减 1。
The chi‑square test for independence (contingency table) examines the association between two categorical variables.
卡方独立性检验（列联表）考察两个分类变量之间是否存在关联。
χ² = Σ [ (O − E)² / E ]
uencies are E = (row total × column total) / grand total. df = (rows − 1) × (columns − 1).
此时期望频次 E = (行合计 × 列合计) / 总计；自由度为 (行数−1) × (列数−1)。
Both tests require expected frequencies of at least 5 in each cell; otherwise, Fisher’s exact test or a likelihood ratio may be used.
两种检验都要求每个单元格的期望频次至少为 5；否则需使用费舍尔精确检验或似然比。
10. Effect Size: Cohen’s d and Partial Eta‑Squared | 效应量：科恩 d 与偏 η²
Effect size quantifies the magnitude of a difference or relationship, independent of sample size. It is required in CCEA psychology to complement significance testing.
效应量用来量化差异或关系的程度，与样本量无关。在 CCEA 心理学中，要求将效应量与显著性检验结合报告。
Cohen’s d = (x̄₁ − x̄₂) / spooled
Where spooled ² ) / (n₁ + n₂ − 2) ]. Values of 0.2, 0.5, 0.8 represent small, medium, and large effects.
其中 spooled 为合并标准差。d 值为 0.2、0.5、0.8 分别表示小、中、大效应。
For ANOVA or factorial designs, partial eta‑squared (ηp²) is used: ηp² = SSeffect / (SSeffect + SSerror).
对于方差分析或因子设计，使用偏 η²：ηp² = SS效应 / (SS效应 + SS误差)。
11. Non‑Parametric Tests | 非参数检验
When data are ordinal or assumptions of parametric tests are violated, psychologists use rank‑based alternatives.
当数据为顺序数据或违反参数检验假设时，心理学家使用基于秩次的替代方法。
Mann‑Whitney U test
he sum of ranks for group 1. The observed U must be ≤ the critical value for significance.
用于独立组设计。检验统计量 U 取 U₁ 和 U₂ 中的较小者；其中 U₁ = R₁ − [ n₁(n₁+1) / 2 ]，R₁ 为组 1 的秩和。观测 U 必须 ≤ 临界值才算显著。
Wilcoxon signed‑ranks test
sed for repeated measures / matched pairs. The test statistic T is the smaller sum of like‑signed ranks. The observed T must be ≤ the critical value.
用于重复测量或配对设计。检验统计量 T 是较小符号秩次之和。观测 T 必须 ≤ 临界值。
Spearman’s rho (ρ or rs)
rs where d is the difference in ranks for each pair. Used for correlations with ordinal data.
rs = 1 − [ 6 Σd² / n(n²−1) ]，其中 d 为每对数据的秩次差。用于顺序数据的相关性分析。
12. Confidence Intervals and Standard Error | 置信区间与标准误
A confidence interval gives a range of values within which the true population parameter is likely to lie. The 95% CI is the most commonly reported.
置信区间给出了真实总体参数可能落入的取值范围。最常报告的是 95% 置信区间。
95% CI = statistic ± (critical value × standard error)
For a mean, the 95% CI is x̄ ± (tcrit × s / √n), where tcrit depends on df and α. For a proportion, use the z‑distribution.
对于均值，95% CI 为 x̄ ± (t临界 × s / √n)，t临界 取决于自由度和 α 水平。对于比例，则使用 z 分布。
The standard error of the mean (SEM) is s / √n. It decreases as sample size increases, improving the precision of the mean estimate.
均值标准误 (SEM) 为 s / √n。它随着样本量增大而减小，从而提高了均值估计的精确度。
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