Year 8 OCR Statistics: Formula and Theorem Quick Reference | Year 8 OCR 统计：公式定理速查手册

📚 Year 8 OCR Statistics: Formula and Theorem Quick Reference | Year 8 OCR 统计：公式定理速查手册

This quick reference guide brings together all the key formulas, definitions and theorems you need for Year 8 OCR Statistics. Use it to check your understanding, to support homework and to revise efficiently before tests.

这份速查手册汇集了 Year 8 OCR 统计所需的所有关键公式、定义和定理。你可以用它来检查理解、辅助家庭作业，并在考试前高效复习。

1. Mean, Median, Mode and Range | 平均数、中位数、众数和极差

The mean is the sum of all the data values divided by the number of values. It is often called the average and is sensitive to extreme values (outliers).

平均数等于所有数据值的总和除以值的个数。它通常被称为平均值，并且对极端值（离群值）敏感。

Mean = Σx / n

平均数 = Σx ÷ n

The median is the middle value when the data are arranged in order. If n is odd, it is the value at position (n+1)/2. If n is even, it is the mean of the two middle values at positions n/2 and n/2 + 1.

中位数是数据按顺序排列后的中间值。如果 n 为奇数，中位数就是第 (n+1)/2 个位置的值。如果 n 为偶数，则是第 n/2 和 n/2+1 位置两个中间值的平均数。

The mode is the value that occurs most often. A set of data may have one mode, more than one mode (bimodal or multimodal), or no mode if all values appear with the same frequency.

众数是出现次数最多的值。一组数据可能有一个众数、多个众数（双众数或多众数），或者如果所有值出现的频率相同，则没有众数。

The range measures how spread out the data are. It is the difference between the largest and the smallest values.

极差衡量数据的分散程度。它是最大值与最小值之间的差。

Range = maximum value – minimum value

极差 = 最大值 – 最小值

Always remember to sort data before finding the median and to identify which average is most suitable: the mean uses all values, the median is not affected by outliers, and the mode shows the most common value.

务必记住在寻找中位数之前对数据进行排序，并确定哪种平均数最合适：平均数用到了所有值，中位数不受离群值影响，而众数则显示最常出现的值。

2. Calculating the Mean from a Frequency Table | 根据频率表计算平均数

When data are shown in a frequency table, the mean is found by multiplying each data value by its frequency, adding all these products, and then dividing by the total frequency.

当数据以频率表的形式呈现时，平均数是通过将每个数据值乘以其频率，再将所有乘积相加，然后除以总频率得到的。

Mean = Σ(fx) / Σf

平均数 = Σ(f × x) ÷ Σf

Here f is the frequency of each value and x is the data value. It is helpful to add an extra column in the table for f × x.

这里的 f 是每个值的频率，x 是数据值。在表格中添加一列 f × x 会很有帮助。

The total sum Σf is the number of data points, and Σ(fx) is the total of all values combined. This method avoids writing out long lists of repeated numbers.

总和 Σf 是数据点的个数，Σ(fx) 是所有值的总和。这种方法避免了将重复的数字写成很长的列表。

3. Estimated Mean for Grouped Data | 分组数据的估计平均数

When data are grouped into intervals, we do not know the exact values. We use the midpoint of each interval as an estimate for x.

当数据被分组为区间时，我们无法知道确切的值。我们使用每个区间的中点作为 x 的估计值。

Estimated mean ≈ Σ(f × midpoint) / Σf

估计平均数 ≈ Σ(f × 中点) ÷ Σf

The midpoint is calculated by adding the lower and upper boundaries of the interval and dividing by 2. For example, the midpoint of 10–19 is (10 + 19) / 2 = 14.5.

中点的计算是将区间的下边界和上边界相加再除以 2。例如，10–19 的中点就是 (10+19) ÷ 2 = 14.5。

This is an estimate because we assume all values within an interval are evenly spread, which may not be perfectly true. Nevertheless it gives a useful summary.

这是一个估计值，因为我们假设区间内的所有值均匀分布，这在实际中可能并不完全正确。但它仍然能给出有用的汇总信息。

4. Basic Probability | 基本概率

Probability measures how likely an event is to happen. It can be written as a fraction, a decimal or a percentage, and always lies between 0 (impossible) and 1 (certain).

概率衡量一个事件发生的可能性有多大。它可以用分数、小数或百分数表示，并且总是在 0（不可能）和 1（必然）之间。

P(event) = number of favourable outcomes / total number of possible outcomes

概率 = 有利结果数 ÷ 所有可能结果总数

The sum of the probabilities of all possible mutually exclusive outcomes is always equal to 1. When outcomes are equally likely, we can simply count.

所有可能且互斥的结果的概率总和永远等于 1。当结果是等可能的时候，我们只需计数即可。

Remember to simplify fractions where possible and to express probability in its simplest form unless the question asks otherwise.

记住尽可能将分数化简，除非题目另有要求，否则用最简形式表达概率。

5. Complementary Events | 互补事件

The complement of an event A is the event that A does not happen. The probability of A not occurring is easy to find using the complement rule.

事件 A 的补事件是指 A 不发生的事件。利用互补法则可以很容易地求出 A 不发生的概率。

P(not A) = 1 – P(A)

非 A 的概率 = 1 – A 的概率

This rule is extremely useful when it is easier to calculate the probability of the opposite event. For example, the probability of not rolling a 6 on a fair die is 1 – 1/6 = 5/6.

当计算相反事件的概率更容易时，这个法则非常有用。例如，掷一枚均匀骰子不出现 6 点的概率是 1 – 1/6 = 5/6。

6. Mutually Exclusive Events and the Addition Rule | 互斥事件与加法法则

Two events are mutually exclusive if they cannot happen at the same time. When events A and B are mutually exclusive, the probability of either A or B occurring is the sum of their individual probabilities.

如果两个事件不可能同时发生，则它们是互斥的。当事件 A 和 B 互斥时，A 或 B 发生的概率就是它们各自概率之和。

For mutually exclusive events: P(A or B) = P(A) + P(B)

互斥事件：P(A 或 B) = P(A) + P(B)

If events are not mutually exclusive, this formula would count the overlap twice, but Year 8 mostly focuses on simple mutually exclusive situations such as drawing a red or a blue counter.

如果事件不是互斥的，这个公式会重复计算重叠部分，不过 Year 8 主要关注简单的互斥情况，比如抽出一个红色或蓝色筹码。

7. Experimental Probability and Expected Frequency | 实验概率与期望频率

Experimental probability (relative frequency) is based on actual trials or experiments: it is the number of successful trials divided by the total number of trials.

实验概率（相对频率）基于实际的试验或实验：它等于成功试验的次数除以试验总次数。

Experimental probability = number of successes / total number of trials

实验概率 = 成功次数 ÷ 试验总次数

Expected frequency tells us how many times we expect an event to occur in a given number of trials, using the theoretical probability.

期望频率告诉我们，在给定的试验次数中，根据理论概率，我们预期事件发生的次数。

Expected frequency = probability × number of trials

期望频率 = 概率 × 试验次数

The experimental probability gets closer to the theoretical probability as the number of trials increases, which is known as the law of large numbers.

随着试验次数的增加，实验概率会越来越接近理论概率，这被称为大数定律。

8. Scatter Graphs and Correlation | 散点图与相关性

A scatter graph is used to show the relationship between two sets of quantitative data. Each point represents a paired observation (x, y).

散点图用于展示两组定量数据之间的关系。每个点代表一对观测值 (x, y)。

Correlation describes the strength and direction of a linear relationship:

相关性描述线性关系的强度和方向：

Positive correlation: as x increases, y tends to increase.

正相关：当 x 增加时，y 也趋于增加。
Negative correlation: as x increases, y tends to decrease.

负相关：当 x 增加时，y 趋于减少。
No correlation: there is no clear pattern between x and y.

无相关：x 和 y 之间没有明显的模式。

Outliers are points that do not fit the general pattern. They can affect how we interpret the strength of correlation and must be examined carefully.

离群值是不符合整体模式的点。它们可能影响我们对相关性强度的解读，因此必须仔细检查。

9. Pie Charts: Constructing and Interpreting | 饼图：构造与解读

In a pie chart, each category is shown as a sector of a circle. The angle of each sector is proportional to the frequency it represents.

在饼图中，每个类别都表示为一个扇形。每个扇形的角度与它所代表的频率成正比。

Sector angle = (frequency / total frequency) × 360°

扇形角度 = (频率 ÷ 总频率) × 360°

Always check that the angles sum to 360° before drawing. Interpreting a pie chart often involves comparing sector sizes or using the angle to find an unknown frequency.

在绘制之前一定要检查角度总和是否为 360°。解读饼图通常包括比较扇形的大小，或者利用角度求未知的频率。

10. Stem-and-Leaf Diagrams | 茎叶图

A stem-and-leaf diagram organises numerical data while keeping the original values visible. The stem usually represents the leading digit(s) and the leaf the final digit.

茎叶图在组织数值数据的同时保留了原始数据。茎通常代表前一位或几位数字，叶代表最后一位数字。

A key must always be provided, for example: 3 | 5 means 35. Data must be ordered in the leaf section and the diagram makes it easy to find the median and the range.

必须始终提供键，例如：3 | 5 表示 35。叶部分的数值必须按顺序排列，这样的图可以使寻找中位数和极差变得容易。

To find the median from a stem-and-leaf plot, count how many values there are (n). The median is the value at position (n+1)/2 if n is odd, or the mean of the two middle values when n is even.

从茎叶图中找中位数，要数出总共有多少个值 (n)。如果 n 为奇数，中位数就是第 (n+1)/2 个位置的值；如果 n 为偶数，则是中间两个值的平均数。

11. Types of Data | 数据类型

Data can be classified into two broad types: qualitative (categorical) data and quantitative (numerical) data.

数据可以大致分为两类：定性（分类）数据和定量（数值）数据。

Qualitative data describes qualities or categories that cannot be measured numerically, such as favourite colour or car brand.

定性数据描述无法用数值衡量的品质或类别，例如喜爱的颜色或汽车品牌。

Quantitative data is numerical and can be further split into discrete data (countable, e.g. number of books) and continuous data (measurable, e.g. height, mass).

定量数据是数值型数据，可进一步分为离散数据（可数的，例如书本的数量）和连续数据（可测量的，例如身高、质量）。

Choosing an appropriate graph or average often depends on the type of data you have.

选择合适的图表或平均数往往取决于所拥有的数据类型。

12. Sampling and Data Collection | 抽样与数据收集

A population is the entire set of people or items you want to investigate. A sample is a smaller group selected from the population.

总体是你想要研究的全部个体或物品的集合。样本是从总体中选出的较小群体。

A random sample gives every member of the population an equal chance of being chosen, which helps to avoid bias.

随机样本使总体中的每个成员都有相等的被选中机会，这有助于避免偏差。

Bias occurs when a sample does not fairly represent the population, often because of a flawed selection method. For example, only asking friends is not representative.

当样本不能公平地代表总体时，就会产生偏差，通常是因为选择方法有缺陷。例如，只询问朋友就不具有代表性。

When collecting data, you should think about whether to use a census (every member) or a sample, depending on time and resources.

在收集数据时，你应该根据时间和资源考虑是使用普查（调查每个成员）还是抽样。

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Year 8 OCR Statistics: Formula and Theorem Quick Reference | Year 8 OCR 统计：公式定理速查手册