📚 Year 11 WJEC Psychology: Formula & Theorem Quick Reference Guide | Year 11 WJEC 心理学：公式定理速查手册
This quick reference guide brings together the essential statistical formulas and calculation rules you need to master for your Year 11 WJEC Psychology studies. Whether you are analysing data from a class experiment or tackling an exam question on research methods, these formulas will help you describe patterns, compare conditions, and draw evidence-based conclusions. Every formula is presented in plain language with a worked example, so you can apply it confidently under timed conditions.
这本速查手册汇集了你在 Year 11 WJEC 心理学学习中必须掌握的核心统计公式与计算规则。不论你是在分析课堂实验的数据，还是在应对研究方法相关的考试题目，这些公式都能帮助你描述数据模式、比较不同条件并得出基于证据的结论。每条公式都采用通俗易懂的语言并配以实例，让你在限时考试中也能自信运用。
1. Mean | 平均数
The mean is the arithmetic average. It is the most commonly used measure of central tendency in psychological research because it takes every score into account.
平均数就是算术平均值。它是心理学研究中最常用的集中趋势指标，因为它将每一个数据都纳入了计算。
Formula: Mean = Sum of all scores ÷ Number of scores
公式：平均数 = 所有得分之和 ÷ 得分个数
Example: In a memory test, five participants recall these numbers of words: 7, 9, 6, 8, 10. The sum is 40 and there are 5 scores, so Mean = 40 ÷ 5 = 8.
示例：在一项记忆测试中，五名参与者分别回忆出 7、9、6、8、10 个词语。总和为 40，共有 5 个数据，因此平均数 = 40 ÷ 5 = 8。
Watch out: The mean can be distorted by extreme outliers. When a data set contains very large or very small values, consider using the median instead.
注意：平均数可能被极端异常值拉偏。如果数据集中含有极大或极小的数值，应考虑改用中位数。
2. Median | 中位数
The median is the middle value when all scores are arranged in order from smallest to largest. It is not affected by extreme scores, making it a better summary of skewed data.
中位数是将所有数据按从小到大的顺序排列后处于中间位置的那个数值。它不受极端值的影响，更适合概括偏态分布的数据。
How to find it: Arrange the data in ascending order. If there is an odd number of scores, the median is the middle number. If there is an even number, find the mean of the two middle numbers.
如何确定中位数：将数据从小到大排列。若数据个数为奇数，中位数就是正中间的那个数。若为偶数，则取中间两个数的平均数。
Example for odd n: Scores: 4, 7, 9, 12, 15 → Median = 9.
奇数个数的示例：数据为 4、7、9、12、15 → 中位数 = 9。
Example for even n: Scores: 4, 7, 9, 12 → middle two are 7 and 9, so Median = (7+9) ÷ 2 = 8.
偶数个数的示例：数据为 4、7、9、12 → 中间两个数是 7 和 9，所以中位数 = (7+9) ÷ 2 = 8。
3. Mode | 众数
The mode is the most frequently occurring score in a data set. Some sets have one mode (unimodal), two modes (bimodal) or more. The mode is the only measure of central tendency that can be used with nominal (category) data.
众数是数据集中出现次数最多的数值。有些数据集只有一个众数（单峰），有的有两个（双峰），或更多。众数是唯一可用于称名（类别）数据的集中趋势指标。
To identify the mode: Count how many times each score appears. The score with the highest frequency is the mode.
如何找众数：统计每个数值出现的次数，出现频率最高的那个数值就是众数。
Example: Reaction times (ms): 250, 260, 250, 270, 250, 280. The number 250 appears three times, so Mode = 250 ms.
示例：反应时间（毫秒）：250、260、250、270、250、280。数字 250 出现了三次，所以众数 = 250 毫秒。
Remember: The mode is not necessarily a ‘typical’ score if the data are evenly distributed; always check for multiple modes.
请记住：如果数据分布均匀，众数未必能代表‘典型’数值；务必检查是否存在多个众数。
4. Range | 全距（范围）
The range is a simple measure of dispersion. It tells you how spread out the data are by subtracting the smallest score from the largest score.
全距是一种简单的离散程度指标。它通过最大值减去最小值来反映数据的分散程度。
Formula: Range = Highest score − Lowest score
公式：全距 = 最高分 − 最低分
Example: In a happiness survey, scores range from 12 to 48. Range = 48 − 12 = 36.
示例：在一项幸福感调查中，分数最低为 12，最高为 48。全距 = 48 − 12 = 36。
Limitation: The range only uses two values and ignores the rest of the data. In WJEC Psychology, you should comment on how outliers might inflate the range.
局限性：全距只用到两个数值，忽略了其余所有数据。在 WJEC 心理学考试中，你应当说明异常值可能如何夸大全距。
5. Percentage Change | 百分比变化
Percentage change is used to compare a ‘before’ and ‘after’ score, such as improvement in a memory task or change in anxiety rating after an intervention.
百分比变化用于比较‘之前’和‘之后’的数值，例如记忆任务中的进步情况或干预后焦虑评级的改变。
Formula: Percentage Change = ((New score − Original score) ÷ Original score) × 100
公式：百分比变化 = (（新数值 − 原数值） ÷ 原数值) × 100
Example: Original recall score = 20 words, new recall score = 26 words. Change = (26 − 20) ÷ 20 × 100 = 6 ÷ 20 × 100 = 30% increase.
示例：原回忆量为 20 个词语，新回忆量为 26 个词语。变化 = (26 − 20) ÷ 20 × 100 = 6 ÷ 20 × 100 = 上升 30%。
If the new score is lower than the original, the result will be a negative percentage, indicating a decrease.
如果新数值低于原数值，结果将为负的百分比，表示下降。
6. Converting Fractions to Percentages | 分数转百分比
Psychologists often need to express part of a sample as a percentage, for example ’18 out of 30 participants showed a priming effect’. This conversion is a straightforward calculation.
心理学家经常需要把样本中的一部分表示为百分比，例如‘30 名参与者中有 18 名表现出启动效应’。这种转换是一个简单的计算。
Formula: Percentage = (Part ÷ Whole) × 100
公式：百分比 = (部分 ÷ 整体) × 100
Example: In a class of 25 students, 16 achieved Grade 7 or above. Percentage = 16 ÷ 25 × 100 = 64%.
示例：一个班级有 25 名学生，其中 16 人达到等级 7 或以上。百分比 = 16 ÷ 25 × 100 = 64%。
Tip: Always check that the ‘Whole’ represents the total number of participants in that condition, not the overall sample if you are only analysing a subset.
小贴士：务必确认‘整体’是该条件下的参与者总数，而非总体样本数（如果你只分析某个子集）。
7. Estimating the Mean from Grouped Data | 从分组数据估算平均数
When data are presented in a grouped frequency table (e.g. test marks in bands of 0–9, 10–19), you cannot calculate the exact mean, but you can estimate it using midpoints.
当数据以分组频数表呈现（例如考试成绩按 0–9、10–19 的区间分组）时，你无法计算精确的平均数，但可以用组中点进行估算。
Steps: (1) Find the midpoint of each group (lower bound + upper bound, divided by 2). (2) Multiply each midpoint (x) by its frequency (f). (3) Sum all these ‘fx’ values. (4) Divide by the total frequency.
步骤：(1) 求出每组的组中点（下限 + 上限，再除以 2）。(2) 每个组中点 (x) 乘以该组频数 (f)。(3) 将所有‘fx’值相加。(4) 除以总频数。
Formula: Estimated Mean = Σ(f × x) ÷ Σf
公式：估算平均数 = Σ(f × x) ÷ Σf
Example: Group 0–4 (f=3, midpoint=2), Group 5–9 (f=5, midpoint=7). Σ(f × x) = (3×2) + (5×7) = 6 + 35 = 41, Σf = 8. Estimated mean = 41 ÷ 8 = 5.125.
示例：组 0–4（频数=3，中点=2），组 5–9（频数=5，中点=7）。Σ(f × x) = (3×2)+(5×7)=6+35=41，总频数=8。估算平均数 = 41 ÷ 8 = 5.125。
8. Interquartile Range (IQR) | 四分位数间距
The interquartile range is a more robust measure of spread than the range because it focuses on the middle 50% of the data and is not influenced by outliers.
四分位数间距比全距更具稳健性，因为它只关注中间 50% 的数据，不受异常值影响。
How to calculate IQR: (1) Rank the data from smallest to largest. (2) Find the median (Q2). (3) The lower quartile (Q1) is the median of the lower half of the data; the upper quartile (Q3) is the median of the upper half. (4) IQR = Q3 − Q1.
如何计算 IQR：(1) 将数据从小到大排序。(2) 找出中位数 (Q2)。(3) 下四分位数 (Q1) 是数据后半段的中位数；上四分位数 (Q3) 是数据前半段的中位数。(4) IQR = Q3 − Q1。
Example: Dataset: 3, 5, 7, 8, 9, 11, 15. Median (Q2) = 8. Lower half: 3,5,7 → Q1 = 5. Upper half: 9,11,15 → Q3 = 11. IQR = 11 − 5 = 6.
示例：数据集：3、5、7、8、9、11、15。中位数 Q2 = 8。下半部分：3,5,7 → Q1=5。上半部分：9,11,15 → Q3=11。IQR = 11 − 5 = 6。
In WJEC exams, quoting the IQR alongside the median shows a deeper understanding of data spread and resistance to outliers.
在 WJEC 考试中，将 IQR 与中位数同时报告能体现出你对数据分散性和抗异常值能力的深入理解。
9. Ratios | 比率
Ratios are used in psychology to express the relationship between two quantities, such as the number of male to female participants, or the frequency of two behaviours observed in a naturalistic observation.
比率在心理学中用来表达两个量之间的关系，例如男女参与者的数量比，或在自然观察中记录到的两种行为频率之比。
Formulaic approach: Divide both numbers by the greatest common factor to simplify the ratio to its simplest form. A ratio of X:Y means for every X units of one thing, there are Y units of the other.
公式化方法：将两个数同时除以最大公因数，将比率化为最简形式。比率 X:Y 表示每 X 个单位对应 Y 个单位的另一种事物。
Example: In a study, 24 participants were female and 16 were male. The ratio of females to males is 24:16. Divide both by 8 to simplify → 3:2.
示例：一项研究中有 24 名女性参与者和 16 名男性参与者。女性与男性的比率为 24:16。两边同除以 8 得到最简比率 → 3:2。
Always present ratios in the order stated in the question and, unless told otherwise, reduce them to whole numbers.
始终按照题目要求的顺序呈现比率，除非另有说明，否则应化为整数比。
10. Quick Reminders: Common Pitfalls and Best Practice | 常见误区与最佳实践速查
Calculating correctly is only half the battle; selecting the appropriate statistic and interpreting it carefully are essential skills in WJEC Psychology. Here are some key reminders distilled from examiners’ reports.
计算正确只是成功的一半；选择合适的统计量并审慎解读是 WJEC 心理学中的必备技能。以下是源自考官报告的关键提醒。
Always label your answers clearly: ‘Mean = 8’, ‘Median = 9’, ‘Range = 36’ — unlabelled numbers lose marks in research method questions.
始终清晰地标明答案：‘平均数 = 8’、’中位数 = 9’、’全距 = 36’ — 在研究方法题目中，没有标签的数字会丢分。
When comparing two sets of data, use at least one measure of central tendency AND one measure of dispersion. For example: ‘The mean recall was higher in the quiet condition (12 words) than in the noise condition (8 words), and the range was smaller (4 vs 10), suggesting less variability.’
在比较两组数据时，至少使用一种集中趋势量数与一种离散量数。例如：‘安静条件下平均数更高（12 个词语），噪声条件下较低（8 个词语），且安静组的全距更小（4 vs 10），表明数据变异性更小。’
Do not report a mean for categorical data. If participants chose from ‘chocolate, vanilla, strawberry’, the mode is the appropriate summary.
不要对类别数据报告平均数。如果参与者从’巧克力、香草、草莓’中做出选择，则应用众数来概括。
Percentage change is not the same as percentage difference. Always use the formula ((new − original) ÷ original) × 100 — do not just subtract percentages.
百分比变化不同于百分比差异。务必使用公式 ((新值 − 原值) ÷ 原值) × 100 — 不要简单地用百分比相减。
Work with your calculator efficiently. During practical investigations and exams, jot down the intermediate steps (sum of scores, n, sum of fx) so you can check for errors.
高效使用计算器。在实践探究和考试中，记下中间步骤（总分、样本量 n、fx 之和），以便检查错误。
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