Year 9 SQA Statistics Formula & Theorem Quick Reference Guide | Year 9 SQA 统计：公式定理速查手册

📚 Year 9 SQA Statistics Formula & Theorem Quick Reference Guide | Year 9 SQA 统计：公式定理速查手册

This article serves as a concise, exam-ready reference for all the key formulas and theorems you will encounter in the Year 9 SQA Statistics course. Keep it handy for quick revision and to reinforce your problem-solving confidence.

本文是 Year 9 SQA 统计课程中所有核心公式与定理的简明速查手册，便于你随时翻阅和考前复习，巩固解题信心。

1. Measures of Central Tendency | 集中趋势的度量

The mean is the arithmetic average of a data set, calculated by summing all values and dividing by the number of observations. It is sensitive to extreme values but uses every piece of data.

平均数是数据集的算术平均值，通过总和所有数值再除以观测值个数求得。它对极端值敏感，但利用了每一个数据。

Mean (x̄) = Σx / n (for raw data)
平均值 (x̄) = Σx / n（原始数据）

The median is the middle value when data is sorted in order. For an odd number of values, it is the central one; for an even number, it is the average of the two middle values.

中位数是按顺序排列后位于中间的值。奇数个数据时取正中间的数，偶数个时取中间两个数的平均值。

The mode is the value that occurs most frequently. A data set can have no mode, one mode, or more than one mode (bimodal or multimodal).

众数是出现频率最高的值。一组数据可能没有众数，也可能有一个或多个众数（双众数或多众数）。

2. Measures of Spread | 离散程度的度量

The range measures the difference between the largest and smallest values. It gives a quick sense of spread but can be heavily affected by outliers.

极差衡量最大值与最小值之差，能快速反映数据的分散程度，但极易受异常值影响。

Range = Maximum – Minimum
极差 = 最大值 – 最小值

Quartiles divide a sorted data set into four equal parts. The lower quartile (Q1) is the median of the lower half, the upper quartile (Q3) is the median of the upper half, and the second quartile (Q2) is the median of the whole set.

四分位数将排序后的数据分为四个等份。下四分位数 (Q1) 是下半部分的中位数，上四分位数 (Q3) 是上半部分的中位数，而第二四分位数 (Q2) 即整组数据的中位数。

The interquartile range (IQR) measures the middle 50% of the data and is resistant to outliers:

四分位距 (IQR) 衡量中间 50% 数据的范围，不受异常值影响：

IQR = Q3 – Q1
四分位距 = Q3 – Q1

Semi-interquartile range is sometimes used as half of the IQR: (Q3 – Q1) / 2.

半四分位距有时也会用到，等于四分位距的一半：(Q3 – Q1) / 2。

3. Frequency Tables and Grouped Data | 频数表与分组数据

When data is presented in a frequency table, the mean is estimated using the midpoints of class intervals. The formula becomes:

当数据以频数表形式给出时，使用各组的组中值来估算平均数。公式为：

Estimated Mean = Σ(f × m) / Σf

估计平均数 = Σ(f × m) / Σf

where f is the frequency of the class, m is the midpoint of the class interval, and Σf is the total frequency.

其中 f 为该组的频数，m 为组中值，Σf 表示总频数。

The modal class is the interval with the highest frequency. For grouped data we cannot determine an exact mode, only a modal class.

众数所在组是频数最高的区间。分组数据无法确定精确的众数，只能指出众数所在组。

The median interval is found by locating the cumulative frequency position (n+1)/2, then reading off the corresponding class interval from a cumulative frequency table.

中位数所在组则由累积频数表确定：先计算位置 (n+1)/2，再从表中读出所在的组区间。

4. Box Plots | 箱形图

A box plot (or box-and-whisker diagram) displays the five-number summary: minimum, Q1, median, Q3, and maximum. It is drawn to scale and clearly shows the spread and symmetry of the data.

箱形图（箱线图）展示五数概括：最小值、Q1、中位数、Q3 和最大值。它按比例绘制，可清晰显示数据的分散程度和对称性。

To construct a box plot:

绘制箱形图的步骤：

Draw a horizontal/vertical scale and mark the five numbers.
绘制水平/垂直标度，标出五个关键数值。
Draw a box from Q1 to Q3 with a vertical line at the median.
从 Q1 到 Q3 画一个矩形，并在中位数处画一条竖线。
Extend whiskers from the box to the minimum and maximum, as long as there are no outliers.
从矩形两端向最小值和最大值画“须”，前提是无异常值。

Outliers, if present, are typically defined as values beyond 1.5 × IQR from the quartiles and are marked with a dot or asterisk.

如果有异常值，通常定义为超出四分位数 1.5 × IQR 范围的数据点，并用圆点或星号标出。

Term	Definition
Five-number summary	Min, Q1, Median, Q3, Max
IQR	Q3 – Q1
Outlier boundary	Q1 – 1.5×IQR and Q3 + 1.5×IQR

5. Probability Basics | 概率基础

Probability measures the chance of an event occurring and always lies between 0 and 1 inclusive. It can be expressed as a fraction, decimal, or percentage.

概率度量事件发生的可能性，取值总在 0 到 1 之间（含 0 和 1），可用分数、小数或百分数表示。

The theoretical probability of an event A is:

事件 A 的理论概率为：

P(A) = Number of favourable outcomes / Total number of possible outcomes

P(A) = 有利结果数 / 所有可能结果总数

The complement rule states that the probability of an event not happening is 1 minus the probability of it happening: P(not A) = 1 – P(A).

互补规则说明，事件不发生的概率等于 1 减去它发生的概率：P(非 A) = 1 – P(A)。

Relative frequency is an experimental approach: probability ≈ (frequency of event) / (total trials). As the number of trials increases, the relative frequency tends towards the theoretical probability.

相对频率是一种实验方法：概率 ≈ (事件发生的次数) / (总试验次数)。随着试验次数增加，相对频率会趋近于理论概率。

6. Tree Diagrams and Combined Events | 树状图与组合事件

Tree diagrams help list all possible outcomes for two or more events in sequence. Each branch is labelled with its probability; the final outcomes are found by multiplying along the branches.

树状图能清晰地列出两个或多个相继事件的所有可能结果。每条分支标有其概率，最终结果的概率由沿着分支相乘得到。

For independent events A and B, the probability that both occur is:

对于独立事件 A 和 B，两者同时发生的概率为：

P(A and B) = P(A) × P(B)

P(A 且 B) = P(A) × P(B)

For mutually exclusive events A and B, the probability that either occurs is:

对于互斥事件 A 和 B，发生其中任一事件的概率为：

P(A or B) = P(A) + P(B)

P(A 或 B) = P(A) + P(B)

When events are not mutually exclusive, we must subtract the overlap: P(A or B) = P(A) + P(B) – P(A and B).

当事件不是互斥时，需要减去重叠部分：P(A 或 B) = P(A) + P(B) – P(A 且 B)。

Tree diagrams also handle conditional probability, but at Year 9 the focus is on independent events and simple combined events.

树状图也可处理条件概率，但 Year 9 阶段主要聚焦于独立事件和简单的组合事件。

7. Expected Value | 期望值

The expected value (or expectation) is the long-run average outcome of a random trial. It is calculated by multiplying each possible value by its probability and summing the products.

期望值是随机试验在长期重复下的平均结果，通过将每个可能值乘以其概率再求和来计算。

E(X) = Σ [x · P(X = x)]

期望值 E(X) = Σ [x · P(X = x)]

For a simple game or scenario, expected value helps decide whether a game is fair. A fair game has E(X) = 0 when viewed from the player’s net gain.

在简单的游戏或情景中，期望值可帮助判断游戏是否公平。从玩家的净收益看，公平游戏的期望值应为 0。

If a spinner has outcomes 2, 5, 10 with probabilities 0.5, 0.3, 0.2, then E(X) = 2×0.5 + 5×0.3 + 10×0.2 = 1 + 1.5 + 2 = 4.5.
若一个转盘的结果为 2、5、10，概率分别为 0.5、0.3、0.2，则 E(X) = 2×0.5 + 5×0.3 + 10×0.2 = 1 + 1.5 + 2 = 4.5。

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

A scatter graph displays the relationship between two quantitative variables. Each point represents a paired observation (x, y).

散点图展示两个数量变量之间的关系，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：x 增加，y 也倾向增加。
Negative correlation: as x increases, y tends to decrease.
负相关：x 增加，y 倾向减小。
No correlation: no clear pattern is visible.
无相关：无明显规律。

Strength is judged by how closely the points follow a straight line. Strong correlation means points are tightly clustered around a line; weak correlation means they are widely scattered.

强度取决于点围绕一条直线的紧密程度。强相关表示点紧密聚集在直线周围，弱相关则分散较开。

Correlation does not imply causation – a common trap in statistical interpretation.

相关并不意味着因果关系——这是统计解释中的常见陷阱。

9. Line of Best Fit | 最佳拟合线

When a scatter graph shows a roughly linear trend, we can draw a line of best fit (also called a trend line) by eye. It should pass as close as possible to all points, with roughly equal numbers of points above and below the line.

当散点图呈现大致线性的趋势时，我们可以凭目测画一条最佳拟合线（也称为趋势线）。它应尽可能靠近所有的点，且线上方和线下方的点数大致相等。

The line can be used to estimate unknown values:

该直线可用于估计未知值：

Interpolation: estimating a value within the range of the given data. This is generally reliable.
内插法：在已有数据范围内进行估计，通常较为可靠。
Extrapolation: estimating a value outside the range of the data. This can be unreliable because the trend may not continue.
外推法：估计数据范围以外的值。这种做法可能不可靠，因为趋势未必延续。

The equation of the line of best fit can be written in slope-intercept form y = mx + c, where m is the gradient and c is the y-intercept. In Year 9, you may simply read values from the drawn line rather than calculate the equation, but it is useful to understand the concept.

最佳拟合线的方程可写成斜截式 y = mx + c，其中 m 为斜率，c 为 y 轴截距。Year 9 阶段强调从图中读取数值，但理解直线方程亦有助于深化概念。

10. Sampling Methods | 抽样方法

Sampling is selecting a subset of individuals from a population to estimate characteristics of the whole population. The sample should be representative and unbiased.

抽样是从总体中选取一部分个体，以估计总体的特征。样本应具备代表性和无偏性。

Common sampling methods:

常见的抽样方法：

Simple random sampling: every member has an equal chance of being selected, e.g. using a random number generator.
简单随机抽样：每个成员都有相等的机会被选中，如使用随机数生成器。
Systematic sampling: selecting every k-th member from a list, after a random start.
系统抽样：在随机起点后，从名单中每隔 k 个选取一个成员。
Stratified sampling: the population is divided into groups (strata) and a random sample proportional to size is taken from each stratum.
分层抽样：总体被分为若干层，按各层所占比例从每层中随机抽取样本。
Convenience sampling: choosing individuals who are easy to reach – often biased and not recommended.
便利抽样：选取最容易接触到的个体——常产生偏差，不推荐使用。

The sample size affects reliability: larger samples generally give more precise estimates, but the key is to avoid bias in selection.

样本量影响可靠性：大样本通常能得到更精确的估计，但关键是避免选择偏差。

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Year 9 SQA Statistics Formula & Theorem Quick Reference Guide | Year 9 SQA 统计：公式定理速查手册