Pre-U AQA Psychology: Formula & Theorem Quick Reference Guide | AQA 心理学：公式定理速查手册

📚 Pre-U AQA Psychology: Formula & Theorem Quick Reference Guide | AQA 心理学：公式定理速查手册

This quick reference guide compiles the essential statistical formulas and theorems required for AQA A-level Psychology. Use it to revise key calculations, ensure accurate application of inferential tests, and build confidence for your examinations.

这本速查手册汇编了 AQA A-level 心理学所需的基本统计公式和定理。用它来复习关键计算、确保推论检验的准确应用，并为考试建立信心。

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1. Descriptive Statistics: Central Tendency | 描述统计：集中趋势

The mean is the arithmetic average, calculated by summing all scores and dividing by the number of scores. It is the most commonly used measure of central tendency but can be influenced by extreme values.

平均值是算术平均数，通过将所有分数相加再除以分数的个数来计算。它是最常用的集中趋势测量指标，但可能受极端值的影响。

Mean (M) = Σx / N

The median is the middle value when data are arranged in order of magnitude. If the number of scores is even, the median is the average of the two middle scores.

中位数是将数据按大小顺序排列后位于中间的值。如果分数的个数为偶数，中位数则是中间两个分数的平均值。

The mode is the most frequently occurring value in a data set. A set of scores can have one mode, more than one mode (bimodal or multimodal), or no mode if all values are unique.

众数是数据集中出现频率最高的值。一组分数可以有一个众数、多个众数（双众数或多众数），或者如果所有值都独一无二则没有众数。

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2. Measures of Dispersion: Range, Variance & Standard Deviation | 离散度量：极差、方差与标准差

Dispersion measures indicate how spread out the scores are. The range is the simplest measure, defined as the difference between the highest and lowest scores.

离散度量表示分数的分布情况。极差是最简单的度量，定义为最高分与最低分之间的差。

Range = Maximum value – Minimum value

Variance measures the average squared deviation from the mean. For a sample, use N–1 to give an unbiased estimate.

方差衡量的是各数据与平均值之差的平方的平均数。对于样本，使用 N–1 可获得无偏估计。

Sample variance: s² = Σ(x – x̄)² / (N – 1)

The standard deviation is the square root of the variance and is expressed in the same units as the original data. It is the most informative measure of dispersion for normally distributed data.

标准差是方差的平方根，其单位与原始数据相同。对于正态分布的数据，这是最能提供信息的离散度量。

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

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3. Percentage Change | 百分比变化

Percentage change is used to compare the relative difference between an original value and a new value. It is frequently applied when reporting improvements or reductions in psychological studies.

百分比变化用于比较原始值和新值之间的相对差异。在心理学研究中，报告改善或下降时经常用到它。

% Change = (New value – Original value) / Original value × 100

A positive result indicates a percentage increase, while a negative result indicates a percentage decrease.

正值表示百分比增加，负值表示百分比减少。

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4. The Sign Test | 符号检验

The sign test is a non-parametric test used for repeated measures or matched pairs designs when the data are nominal (directional differences). It compares the number of positive and negative differences between two conditions.

符号检验是一种非参数检验，用于重复测量或配对设计，当数据为名义数据（方向性差异）时。它比较两种条件之间正差和负差的个数。

To calculate the sign test, ignore any pairs with zero difference. Count the number of plus signs (+) and minus signs (–). The observed value S is the smaller of these two counts.

计算符号检验时，忽略任何差值为零的对。数出正号 (+) 和负号 (–) 的数量。观测值 S 是这两个数量中较小的一个。

S = smaller number of + or – signs

Compare your obtained S value to a critical value from the binomial sign test table at your chosen significance level and N (number of non-zero differences). The result is significant if S is less than or equal to the critical value.

将你得到的 S 值与符号检验表中你选定的显著性水平及 N（非零差值的对数）下的临界值进行比较。如果 S 小于或等于临界值，结果就是显著的。

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5. Chi-Square Goodness of Fit | 卡方拟合度检验

The chi-square goodness of fit test determines whether observed frequencies differ significantly from expected frequencies for one categorical variable. It is used with independent groups and nominal data.

卡方拟合度检验用来判断一个分类变量的观测频率是否与期望频率有显著差异。它用于独立组和名义数据。

χ² = Σ ( (O – E)² / E )

Where O is the observed frequency and E is the expected frequency for each category. Expected frequencies may be based on theory or equal distribution.

其中 O 是每个类别的观测频率，E 是期望频率。期望频率可以基于理论或均匀分布。

Degrees of freedom: df = k – 1

k is the number of categories. Compare the calculated χ² to the critical value; if it exceeds the critical value, reject the null hypothesis.

k 是类别数。将计算出的 χ² 值与临界值比较；如果大于临界值，则拒绝零假设。

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6. Chi-Square Test of Independence | 卡方独立性检验

This test examines whether two categorical variables are associated. Data are arranged in a contingency table, and expected frequencies are calculated from row and column totals.

此检验考察两个分类变量是否有关联。数据被排列在列联表中，期望频率根据行与列的总和计算得出。

E = (Row total × Column total) / Grand total

The formula for χ² is the same, summing over all cells. Degrees of freedom are calculated differently.

χ² 公式相同，对所有单元格求和。自由度的计算方式不同。

χ² = Σ ( (O – E)² / E )

df = (Number of rows – 1) × (Number of columns – 1)

If the computed χ² is larger than the table critical value, the variables are significantly associated.

如果计算出的 χ² 大于表中的临界值，则变量之间有显著关联。

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

Spearman’s rho is a non-parametric test for correlation using ordinal or ranked data. It assesses the strength and direction of a monotonic relationship between two variables.

斯皮尔曼等级相关系数是一种使用顺序或等级数据的非参数相关检验。它评估两个变量之间单调关系的强度和方向。

Rank the scores for each variable separately. For each pair of ranks, calculate the difference d, square it, and then apply the formula.

分别对每个变量的分数进行排序。对每对等级，计算差值 d，将其平方，然后套用公式。

rₛ = 1 – ( 6 Σd² ) / ( n(n² – 1) )

Where d is the difference between ranks for each participant, and n is the number of pairs. The obtained rₛ ranges from –1 (perfect negative) to +1 (perfect positive). Compare with the critical rₛ value.

其中 d 是每位参与者两个等级之间的差值，n 是配对数。得到的 rₛ 值范围从 –1（完全负相关）到 +1（完全正相关）。与临界 rₛ 值进行比较。

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

Pearson’s r is a parametric test measuring the linear correlation between two continuous variables. Data should be interval/ratio and approximately normally distributed.

皮尔逊积差相关系数是一种参数检验，度量两个连续变量之间的线性相关性。数据应为等距/等比且近似正态分布。

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

x̄ and ȳ represent the means of the x and y variables. The value of r also falls between –1 and +1, and its significance is checked against a Pearson critical value table using N – 2 degrees of freedom.

x̄ 和 ȳ 分别代表 x 变量和 y 变量的平均值。r 的值同样介于 –1 和 +1 之间，其显著性需要对照皮尔逊临界值表，使用 N – 2 作为自由度。

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9. 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 for comparing two independent groups when data are at least ordinal.

曼-惠特尼U检验是独立t检验的非参数替代方法，用于比较两个独立组，数据至少为顺序水平。

Combine scores from both groups and rank them. Sum the ranks for group 1 (R₁) and group 2 separately. Then calculate U for each group and take the smaller U as the test statistic.

将两组的分数合并并排序。分别求出第一组的秩和 (R₁) 和第二组的秩和。然后计算每组的 U 值，取较小的 U 作为检验统计量。

U₁ = n₁n₂ + n₁(n₁ + 1)/2 – R₁

U₂ = n₁n₂ – U₁

Observed U = min(U₁, U₂)

Compare the smaller U with the critical U value from tables. If your U is less than or equal to the critical value, the difference between groups is significant.

将较小的 U 与表中的临界 U 值进行比较。如果你的 U 值小于或等于临界值，则组间差异显著。

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10. 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. It is used for repeated measures or matched pairs with ordinal data.

威尔科克森符号秩检验是相关t检验的非参数等效方法。它用于重复测量或配对设计，数据为顺序水平。

Calculate the difference between each pair of scores, ignoring zeros. Rank the absolute values of the differences, then assign the signs of the original differences back to the ranks. Sum the positive ranks and the negative ranks separately.

计算每对分数之间的差值，忽略零。对差值的绝对值进行排序，然后将原始差值的正负号赋回这些秩。分别将所有正秩求和，所有负秩求和。

T = smaller of (sum of positive ranks, sum of negative ranks)

Compare your obtained T to the critical T value. The result is significant if T is equal to or less than the critical value.

将你得到的 T 值与临界 T 值比较。如果 T 等于或小于临界值，结果即为显著。

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11. Related t-test (Paired Samples t-test) | 相关t检验（配对样本）

The related t-test is a parametric test comparing two sets of scores from the same participants under two conditions. The data must be interval or ratio and normally distributed differences are assumed.

相关t检验是一种参数检验，比较同一批参与者在两种条件下的两组分数。数据必须是等距或等比数据，且差值需要满足正态分布假定。

Compute the difference scores D for each participant, then calculate the mean of these differences (D̄) and the standard deviation of the differences (s_D).

为每位参与者计算差值分数 D，然后求出这些差值的平均数 (D̄) 和差值的标准差 (s_D)。

t = D̄ / (s_D / √n)

Where n is the number of pairs. Degrees of freedom are n – 1. Compare the calculated t to the critical t value to determine significance.

其中 n 是配对数。自由度为 n – 1。将计算的 t 值与临界 t 值比较，以确定显著性。

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12. Unrelated t-test (Independent Samples t-test) | 独立t检验（独立样本）

The unrelated t-test is a parametric test used to compare the means of two different, independent groups. Assumptions include interval/ratio data, homogeneity of variance, and approximate normality.

独立t检验是一种参数检验，用于比较两个不同的独立组的平均值。假定包括等距/等比数据、方差齐性和近似正态分布。

First calculate the pooled variance (s_p²) from the two sample variances, then use it to estimate the standard error of the difference between means.

首先根据两个样本方差计算合并方差 (s_p²)，然后用它来估计均值差异的标准误差。

s_p² = ( (n₁ – 1)s₁² + (n₂ – 1)s₂² ) / (n₁ + n₂ – 2)

t = (M₁ – M₂) / √( s_p² (1/n₁ + 1/n₂) )

Degrees of freedom are df = n₁ + n₂ – 2. Check the t value against the critical value; if |t| is larger, reject the null hypothesis.

自由度 df = n₁ + n₂ – 2。将 t 值与临界值对比；如果 |t| 更大，则拒绝零假设。

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Published by TutorHao | Psychology Revision Series | aleveler.com

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