📚 Year 10 SQA Statistics: Core Concepts Review | SQA 十年级统计：核心知识点梳理
This comprehensive guide covers the essential topics in the Year 10 SQA Statistics curriculum. From understanding data types and sampling methods to calculating measures of spread and probability, each section is designed to help you build a solid foundation for your National 5 Statistics assessments. Let's walk through the key concepts you must master.
本综合指南涵盖了 SQA 十年级统计课程的核心课题。从理解数据类型与抽样方法，到计算离散程度和概率，每个部分都旨在帮助你为 National 5 统计考试打下坚实基础。让我们逐一梳理你必须掌握的关键概念。
1. Types of Data | 数据类型
Data can be classified as either 'qualitative' or 'quantitative'. Qualitative data describes qualities or categories, such as eye colour or car brand. Quantitative data involves numbers that represent amounts, and it can be further split into 'discrete' data (countable, e.g. number of siblings) and 'continuous' data (measurable, e.g. height in cm). Understanding these types helps you choose the correct diagram and analysis later.
数据可分为 "定性数据" 与 "定量数据"。定性数据描述品质或类别，例如眼睛颜色或汽车品牌。定量数据涉及表示数量大小的数字，并可进一步分为 "离散数据"（可计数，如兄弟姐妹的数量）和 "连续数据"（可测量，如以厘米为单位的身高）。理解这些类型有助于你后续选择合适的图表和分析方法。
Within qualitative data, we sometimes distinguish between 'nominal' data (no natural order, e.g. movie genres) and 'ordinal' data (has a meaningful order, e.g. satisfaction ratings: poor, good, excellent). This distinction influences how we summarise the central tendency.
在定性数据内部，我们有时会区分 "名义数据"（无自然顺序，如电影类型）和 "有序数据"（有意义的顺序，如满意度评分：差、好、优秀）。这种区别会影响我们如何概括其集中趋势。
2. Sampling Methods | 抽样方法
In statistics, we often collect data from a sample rather than the entire population. A 'simple random sample' gives every member of the population an equal chance of being chosen, which reduces bias. 'Stratified sampling' divides the population into groups (strata) and takes a random sample from each stratum in proportion to its size. This ensures representation from all important subgroups.
在统计中，我们经常从样本而非整个总体中收集数据。"简单随机抽样" 使总体中的每个成员都有相等的被选中机会，从而减少偏差。"分层抽样" 将总体分成若干组（层），并按各层大小的比例从每一层中随机抽取样本。这确保了所有重要子群都能被代表。
Other methods include 'systematic sampling', where you select every k-th person from a list, and 'convenience sampling', which uses the easiest-to-reach individuals but often introduces bias. For SQA questions, you should be able to identify the sampling technique and comment on its advantages and disadvantages.
其他方法包括 "系统抽样"（从列表中每隔一定间隔选取一人）和 "便利抽样"（使用最容易接触到的人，但常常引入偏差）。对于 SQA 试题，你应该能够识别抽样技术并评述其优缺点。
3. Frequency Tables and Diagrams | 频率表与图表
Once data is collected, we use frequency tables to organise it. For discrete data, a simple frequency table lists each value alongside its count. For continuous data, we group values into intervals, creating a 'grouped frequency table'. An extra column can show cumulative frequency, which is the running total of frequencies up to that interval.
收集好数据后，我们使用频率表来整理数据。对于离散数据，简单的频率表列出每个数值及其计数。对于连续数据，我们将数值分组为区间，形成 "分组频率表"。额外的一列可以显示累积频率，即截至该区间的频率运行总数。
Common diagrams include bar charts for discrete or categorical data, and histograms for continuous data. In a histogram, the area of each bar is proportional to the frequency, so equal-width bars have heights that represent frequency; if widths vary, you must use 'frequency density'. Pie charts display proportions of a whole, and stem-and-leaf diagrams combine a table with a simple shape that shows the distribution without losing the original data.
常见图表包括用于离散或类别数据的条形图，以及用于连续数据的直方图。在直方图中，每个条形的面积与频率成正比，因此等宽条形的柱高代表频率；若宽度不同，你必须使用 "频率密度"。饼图显示整体的各部分比例，茎叶图则将表格与简单形状相结合，展示分布且不丢失原始数据。
4. Measures of Central Tendency | 集中趋势的度量
The three main measures are the mean, median, and mode. The 'mean' dividing by the count. For a dataset {3, 5, 5, 7, 10}, the mean is (3+5+5+7+10)/5 = 6. The 'median' is the middle value when data is ordered; here the median is 5. The 'mode' is the most frequent value, which is also 5. Each measure has its own strengths.
三个主要度量是平均数、中位数和众数。"平均数"（通常称为均值）通过将所有值相加再除以个数来计算。对于数据集 {3, 5, 5, 7, 10}，平均数为 (3+5+5+7+10)/5 = 6。"中位数" 是数据排序后的中间值；这里中位数是 5。"众数" 是出现频率最高的值，这里也是 5。每种度量各有优势。
For grouped data, we can only estimate the mean using midpoints of intervals. The modal class is the interval with the highest frequency. The median can be found from a cumulative frequency graph or by identifying the interval containing the middle position.
对于分组数据，我们只能使用区间中点来估算平均数。众数组是频率最高的区间。中位数可以通过累积频率图或确定包含中间位置的区间来找到。
5. Quartiles and Box Plots | 四分位数与箱形图
alue, and the upper quartile (Q3) is the median of the upper half. For example, in the dataset 3, 4, 5, 6, 8, 9, 11, Q1 = 4, Q2 = 6, Q3 = 9. These five values (minimum, Q1, Q2, Q3, maximum) form a 'five-number summary' used to draw a box plot.
四分位数将有序数据分成四个相等的部分。下四分位数 (Q1) 是数据下半部分的中位数，中位数 (Q2) 是中间值，上四分位数 (Q3) 是上半部分的中位数。例如，在数据集 3, 4, 5, 6, 8, 9, 11 中，Q1 = 4，Q2 = 6，Q3 = 9。这五个值（最小值、Q1、Q2、Q3、最大值）构成用于绘制箱形图的 "五数概括"。
m within 1.5 × IQR from the quartiles. It is excellent for comparing distributions side by side.
箱形图直观地展示数据的散布和中心。箱子从 Q1 延伸到 Q3，并在中位数处画一条线；须线延伸至距四分位数 1.5×IQR 范围内的最小值和最大值。它在并列比较分布时非常有用。
6. Measures of Spread: Range and Interquartile Range | 离散程度的度量：极差与四分位距
The simplest measure of spread is the 'range': maximum value minus minimum value. However, it is sensitive to outliers. The 'interquartile range' (IQR) is more robust: IQR = Q3 − Q1. It tells you the spread of the middle 50% maller IQR means the central half of the data is more tightly clustered around the median.
最简单的离散程度度量是 "极差"：最大值减去最小值。然而，它对异常值敏感。"四分位距" (IQR) 更为稳健：IQR = Q3 − Q1。它告诉你中间 50% 数据的散布范围。较小的 IQR 意味着数据的中央一半更紧密地聚集在中位数周围。
When comparing data sets, always comment on both a measure of central tendency (mean or median) and a measure of spread (range or IQR). SQA exam questions frequently ask you to state which data set is more consistent, and the IQR is a key piece of evidence.
在比较数据集时，务必同时就集中趋势度量（平均数或中位数）和离散程度度量（极差或 IQR）发表评论。SQA 考试题目经常要求你陈述哪个数据集更一致，而 IQR 是关键的证据。
7. Standard Deviation | 标准差
Standard deviation measures how far, on average, each data value deviates from the mean. The formula for the 'sample standard deviation' (s) is:
s = √[ Σ(x − x̄)² / (n − 1) ]
where x̄ is the mean and n is the number of data points. A small standard deviation indicates the data points are close to the mean, while a large one shows they are spread out.
标准差衡量每个数据值与平均数的平均偏离程度。"样本标准差" (s) 的公式为：
s = √[ Σ(x − x̄)² / (n − 1) ]
其中 x̄ 为平均数，n 为数据点个数。较小的标准差表明数据点接近平均数，而较大的标准差则显示它们分布较散。
For a simple dataset 2, 4, 4, 4, 5, 5, 7, 9, the mean x̄ = 5. The sum of squared differences Σ(x−x̄ = (−3)² + (−1)² + (−1)² + (−1)² + 0² + 0² + 2² + 4² = 9+1+1+1+0+0+4+16 = 32. With n=8, s = √32/7 ≈ √4.571 ≈ 2.14. Always show your working step by step.
对于简单数据集 2, 4, 4, 4, 5, 5, 7, 9，平均数 x̄ = 5。偏差平方和 Σ(x−x̄)² = (−3)² + (−1)² + (−1)² + (−1)² + 0² + 0² + 2² + 4² = 32。由于 n=8，s = √32/7 ≈ 2.14。务必逐步展示你的计算过程。
8. Probability Basics | 概率基础
Probability is a number between 0 and 1 (or 0% to 100%) that represents how likely an event is to occur. The probability of an event A, denoted P(A), is given by:
P(A) = number of favourable outcomes / total number of possible outcomes
Probability 代表事件发生的可能性，是一个介于 0 和 1 之间的数字（或 0% 到 100%）。事件 A 的概率，记作 P(A)，由下式给出：
P(A) = 有利结果的数量 / 所有可能结果的总数
nnot happen together), P(A or B) = P(A) + P(B). If they are independent, P(A and B) = P(A) × P(B).
所有可能结果的概率之和总是 1。对于互补事件，P(非 A) = 1 − P(A)。若两个事件互斥（不能同时发生），P(A 或 B) = P(A) + P(B)。若它们相互独立，P(A 与 B) = P(A) × P(B)。
9. Tree Diagrams | 树状图
Tree diagrams are a powerful tool for handling multi-stage probability experiments. Each branch represents an outcome with its probability written along it. At the end of each path, you multiply the probabilities along the branches to find the probability of that combined outcome. For two stages, the sum of all end-point probabilities equals 1.
树状图是处理多阶段概率实验的强大工具。每个分支代表一个结果，旁边标注其概率。在每条路径的末端，你将沿分支的概率相乘，以得到该组合结果的概率。对于两个阶段，所有终点概率之和等于 1。
oved, the probability of red on the second draw becomes 2/4. The combined probability of two reds is (3/5) × (2/4) = 6/20 = 3/10. Always clearly label branches with probabilities that update after each selection.
例如，一个袋子里有 3 个红球和 2 个蓝球。不放回地抽取两个球。第一次抽到红球的概率为 3/5。若移走一个红球，第二次抽到红球的概率变为 2/4。两次都抽到红球的组合概率为 (3/5) × (2/4) = 6/20 = 3/10。务必清楚地在分支上标注随着每次选取而更新的概率。
10. Scatter Graphs and Correlation | 散点图与相关性
A scatter graph plots paired numerical data to reveal whether a relationship exists between two variables. 'Correlation' describes the strength and direction of this linear relationship. It can be positive (as one increases, the other tends to increase), negative (as one increases, the other tends to decrease), or zero (no linear pattern).
散点图通过绘制成对的数值数据来揭示两个变量之间是否存在关系。"相关性" 描述这种线性关系的强度和方向。它可以是正相关（一个增大，另一个也倾向于增大）、负相关（一个增大，另一个倾向于减小）或零相关（无线性模式）。
You can estimate the correlation visually. A strong correlation has points tightly clustered around a straight line; a weak correlation shows much scatter. It is important to remember that correlation does not imply causation: two variables may be associated without one causing the other.
你可以通过目测来估计相关性。强相关的点紧密聚集在一条直线周围；弱相关则显示很大的离散程度。重要的是要记住，相关性并不意味着因果关系：两个变量可能有关联，但并非一个导致另一个。
11. Line of Best Fit | 最佳拟合线
When a scatter graph shows a clear linear trend, you can draw a 'line of best fit' (also called a trend line). This line should pass as close as possible to all points, with roughly equal numbers of points above and below the line. A good practice in SQA Statistics is to ensure the line passes through the mean point (x̄, ȳ), where x̄ is the mean of the x-values and ȳ is the mean of the y-values.
当散点图显示出明显的线性趋势时，你可以绘制一条 "最佳拟合线"（也称为趋势线）。这条线应尽可能靠近所有点，线上方和下方的点数量大致相等。SQA 统计中的一个良好做法是确保该线通过均值点 (x̄, ȳ)，其中 x̄ 是 x 值的平均数，ȳ 是 y 值的平均数。
+ bx, where b is the gradient and a is the y-intercept. You can use the line of best fit to make predictions: read off a y-value for a given x-value within the range of the data (interpolation). Extrapolation beyond the data range is risky and should be treated with caution.
直线方程为 y = a + bx，其中 b 为斜率，a 为 y 轴截距。你可以利用最佳拟合线进行预测：根据给定的 x 值读取数据范围内的 y 值（内插法）。超出数据范围的外推法是有风险的，应谨慎对待。
Published by TutorHao | Statistics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图