📚 Year 7 SQA Statistics: Formula & Theorem Quick Reference Handbook | Year 7 SQA 统计：公式定理速查手册
This handbook brings together all the essential formulas and key ideas you need for Year 7 statistics under the Scottish SQA curriculum. Keep it handy when you practise questions, revise for tests or complete homework.
本手册汇集了 SQA 苏格兰课程 Year 7 统计所需的所有基本公式和关键概念。在练习题目、复习备考或完成作业时，随时查阅本手册。
1. The Mean (Average) | 平均数
The mean is one way to describe the centre of a data set. To calculate the mean, add up all the values and then divide by the number of values.
平均数是描述数据集中心位置的一种方法。计算平均数时，先将所有数值相加，然后再除以数值的个数。
Mean = (Sum of all values) ÷ (Number of values)
If you have the numbers 4, 7, 9, the mean is (4+7+9) ÷ 3 = 20 ÷ 3 ≈ 6.67.
例如，如果你有一组数 4, 7, 9，平均数为 (4+7+9) ÷ 3 = 20 ÷ 3 ≈ 6.67。
2. The Median | 中位数
The median is the middle value when the data is arranged in order from smallest to largest. If there is an even number of values, the median is the mean of the two middle numbers.
中位数是将数据从小到大排列后处于中间位置的数值。如果数值的个数为偶数，中位数是中间两个数的平均数。
Example: For 2, 3, 7, 9, 11, the median is 7. For 2, 3, 7, 9, the median is (3+7) ÷ 2 = 5.
示例：对于 2, 3, 7, 9, 11，中位数为 7。对于 2, 3, 7, 9，中位数为 (3+7) ÷ 2 = 5。
Remember, you must always order the data first before finding the median.
请记住，求中位数前必须先将数据按顺序排列好。
3. The Mode | 众数
The mode is the value that appears most often in a data set. A set of data can have one mode, more than one mode (bimodal or multimodal), or no mode if all values appear the same number of times.
众数是数据集中出现次数最多的数值。一组数据可以有一个众数、多个众数（双众数或多众数），如果所有数值出现的次数相同，则没有众数。
Example: In 3, 5, 5, 7, 9, the mode is 5. In 2, 2, 3, 3, 6, the modes are 2 and 3.
示例：在 3, 5, 5, 7, 9 中，众数是 5。在 2, 2, 3, 3, 6 中，众数是 2 和 3。
4. The Range | 范围（极差）
The range measures the spread of the data. It is calculated by subtracting the smallest value from the largest value.
范围（极差）衡量数据的分散程度。它是最大值减去最小值得到的差。
Range = Largest value − Smallest value
For the data set 4, 8, 15, 3, the range is 15 − 3 = 12. A small range means the data is close together; a large range shows it is more spread out.
对于数据集 4, 8, 15, 3，范围为 15 − 3 = 12。较小的范围意味着数据比较集中；较大的范围表明数据更分散。
5. Frequency Tables | 频率表
A frequency table shows how many times each value occurs. It helps to organise data before calculating statistics.
频率表显示每个数值出现的次数。它有助于在计算统计量之前整理数据。
You can use tally marks to count frequencies. The table should include a total frequency.
你可以用计数符号（划记）来统计频率。频率表应包含总频数。
| Value | Tally | Frequency |
|---|---|---|
| 1 | ||| | 3 |
| 2 | |||| | 4 |
| 3 | || | 2 |
Total frequency = 3 + 4 + 2 = 9. Always label your table clearly.
总频数 = 3 + 4 + 2 = 9。务必清晰地给表格添加标签。
6. Bar Charts | 条形图
Bar charts use rectangular bars to represent data. The height or length of each bar shows the frequency or value for each category.
条形图使用矩形条来表示数据。每个条形的高度或长度表示每个类别的频数或数值。
Key features: categories go on the horizontal axis, frequency on the vertical axis; bars are of equal width and do not touch (for discrete categories). Label both axes and give the chart a title.
关键特征：类别在水平轴上，频数在垂直轴上；条形宽度相等且不接触（对于离散类别）。给两个坐标轴添加标签，并为图表加上标题。
Bar charts are useful for comparing different groups or categories. When interpreting a bar chart, always read the scale carefully.
条形图对于比较不同的组或类别很有用。解读条形图时，一定要认真读取刻度。
7. Pie Charts | 饼图
A pie chart shows proportions of a whole. Each slice represents a category, and the angle of the slice corresponds to its frequency relative to the total.
饼图显示整体的各部分比例。每个扇形代表一个类别，扇形的角度与其频数相对于总数的比例对应。
Angle per category = (Category frequency ÷ Total frequency) × 360°
To draw a pie chart, you calculate the angle for each category, draw a circle, and use a protractor to mark the angles. Always check that the angles sum to 360°.
绘制饼图时，先计算每个类别的角度，画一个圆，然后用量角器标出角度。务必检查角度之和等于 360°。
When analysing a pie chart, the largest slice represents the most frequent category.
分析饼图时，最大的扇形代表频数最高的类别。
8. Basic Probability | 基本概率
Probability measures the chance that an event will happen. It always lies between 0 and 1. A probability of 0 means the event is impossible; a probability of 1 means it is certain.
概率衡量事件发生的可能性。它始终介于 0 和 1 之间。概率为 0 表示事件不可能发生；概率为 1 表示事件一定发生。
Probability (event) = Number of favourable outcomes ÷ Total number of possible outcomes
For example, when rolling a fair six-sided die, the probability of rolling a 3 is 1/6.
例如，掷一个均匀的六面骰子时，掷出 3 的概率是 1/6。
Probabilities can be written as fractions, decimals or percentages. The probability of an event not happening is 1 minus the probability of it happening.
概率可以用分数、小数或百分比表示。事件不发生的概率 = 1 − 事件发生的概率。
9. Calculating the Mean from a Frequency Table | 从频率表计算平均数
When data is in a frequency table, the mean is found by multiplying each value by its frequency, adding these products, and then dividing by the total frequency.
当数据以频率表的形式呈现时，计算平均数的方法是：用每个数值乘以其频数，将这些乘积相加，然后除以总频数。
Mean = (Σ value × frequency) ÷ (Σ frequency)
Example: For value 1 (freq 3), value 2 (freq 4), value 3 (freq 2). Sum of (value × freq) = (1×3)+(2×4)+(3×2) = 3+8+6 = 17. Total frequency = 3+4+2 = 9. Mean = 17 ÷ 9 ≈ 1.89.
示例：数值 1（频数 3），数值 2（频数 4），数值 3（频数 2）。 (数值×频数) 之和 = (1×3)+(2×4)+(3×2) = 3+8+6 = 17。总频数 = 9。平均数 = 17 ÷ 9 ≈ 1.89。
10. Finding the Mode and Median from a Frequency Table | 从频率表中找到众数和中位数
The mode in a frequency table is the value with the highest frequency. You just look for the largest number in the frequency column and pick the corresponding value.
频率表中的众数是频数最高的数值。你只需在频数列中找到最大的那个数字，并找出它对应的数值。
The median from a frequency table is found by locating the middle data point. Start by adding all the frequencies to find the total. Then find the position of the median: (total frequency + 1) ÷ 2 if the number of values is odd. For an even total, the median will lie between
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