📚 Year 11 SQA Statistics: Core Knowledge Summary | Year 11 SQA 统计：核心知识点梳理
This article provides a structured review of the essential topics in SQA National 5 Statistics. It covers data types, collection methods, visual displays, central tendency, dispersion, the normal distribution, and probability – all tailored to the Year 11 curriculum. Use this as a revision checklist and quick reference.
本文系统梳理了 SQA National 5 统计的核心考点，涵盖数据类型、收集方法、图表展示、集中趋势、离散程度、正态分布与概率等模块，贴合 Year 11 教学大纲，适合作为复习清单与快速参考。
1. Types of Data | 数据类型
Data can be qualitative (categorical) or quantitative (numerical). Quantitative data splits into discrete data, which takes exact whole‑number values, and continuous data, which can take any value within a range.
数据可分为定性（分类）数据和定量（数值）数据。定量数据又分为离散数据（只能取精确整数值）和连续数据（在某个范围内可取任意值）。
Examples: favourite colours are qualitative; shoe sizes are discrete; heights and times are continuous.
例如：最喜欢的颜色是定性数据；鞋码是离散数据；身高和时间是连续数据。
2. Collecting Data: Populations and Samples | 数据收集：总体与样本
A population includes every member of the group being studied. A sample is a subset of the population used to make inferences. A census collects data from the entire population, but is often impractical.
总体包含研究对象的每一个成员；样本是总体的一个子集，用于推断总体信息。普查会收集整个总体的数据，但往往难以实现。
Samples should be random and representative to avoid bias. Common sampling methods include simple random sampling, stratified sampling, and systematic sampling.
样本应随机且具有代表性以避免偏差。常见的抽样方法包括简单随机抽样、分层抽样和系统抽样。
3. Displaying Data: Charts and Graphs | 数据展示：图表与图形
Bar charts are used for qualitative or discrete data, with gaps between bars. Histograms display continuous data in frequency density form: the area of each bar is proportional to frequency.
条形图用于定性或离散数据，柱间留有空隙。直方图以频率密度形式展示连续数据，每个柱的面积与频数成正比。
Other key diagrams include stem‑and‑leaf plots, which preserve original values, and line graphs for trends over time. Pie charts show proportions but lose detail.
其他重要图形包括茎叶图（保留原始数值）和展示时间趋势的折线图。饼图显示比例但会丢失细节。
4. Measures of Central Tendency | 集中趋势的度量
The mean is the arithmetic average: x̄ = Σx / n. It uses all values but is sensitive to outliers.
均值是算术平均数：x̄ = Σx / n。它使用了所有数值，但易受异常值影响。
The median is the middle value when data are ordered. For n values, position is (n + 1)/2. It is robust to outliers.
中位数是排序后位于中间的值。n 个数据的中位数位置为 (n + 1)/2。它对异常值不敏感。
The mode is the most frequent value and can be used for qualitative data.
众数是出现次数最多的值，可用于定性数据。
5. Measures of Dispersion: Range and Interquartile Range | 离散程度：极差与四分位距
The range = maximum − minimum. It is quick to find but affected by extreme values.
极差 = 最大值 − 最小值。计算简便，但易受极端值影响。
The interquartile range (IQR) = Q₃ − Q₁. It measures the spread of the middle 50% of the data and is more resistant to outliers.
四分位距 (IQR) = Q₃ − Q₁。它衡量中间 50% 数据的分散程度，对异常值更具抗干扰性。
Quartiles are found by ordering data: Q₁ is the median of the lower half, Q₃ is the median of the upper half.
四分位数通过排序得出：Q₁ 是下半部分的中位数，Q₃ 是上半部分的中位数。
6. Standard Deviation | 标准差
Standard deviation measures how much individual values deviate from the mean. The sample standard deviation formula is:
标准差衡量各数值与均值的偏离程度。样本标准差公式为：
s = √( Σ(x − x̄)² / (n − 1) )
Steps: find the mean, subtract the mean from each value, square the results, sum them, divide by (n − 1), and take the square root. Variance = s².
计算步骤：求均值，各值减均值后平方，求和，除以 (n − 1)，再开平方。方差即为 s²。
A lower standard deviation indicates data clustered near the mean; a higher one shows greater spread.
标准差越小表示数据越集中在均值附近；越大则越分散。
7. Box Plots and Five‑Number Summaries | 箱线图与五数概括
A box plot shows minimum, Q₁, median, Q₃, and maximum on a scale. It visualises the spread and highlights outliers.
箱线图在坐标轴上显示最小值、Q₁、中位数、Q₃ 和最大值，直观呈现数据分布并突出异常值。
Outliers are often defined as values below Q₁ − 1.5 × IQR or above Q₃ + 1.5 × IQR.
异常值通常定义为小于 Q₁ − 1.5 × IQR 或大于 Q₃ + 1.5 × IQR 的数值。
Box plots are excellent for comparing two or more data sets side by side.
箱线图非常适合并排比较多组数据。
8. Scatter Graphs and Correlation | 散点图与相关性
A scatter graph plots paired bivariate data. The pattern reveals the type and strength of correlation: positive, negative, or none.
散点图绘制成对的二元数据，模式揭示相关性的类型和强度：正相关、负相关或无相关。
A line of best fit can be drawn by eye, passing through (x̄, ȳ) when sensible. The equation of the line can be used to make estimates.
最佳拟合线可通过目测画出，合理情况下应经过 (x̄, ȳ)。该直线方程可用于估计。
Be aware that extrapolation beyond the data range is unreliable.
注意，超出数据范围的外推是不可靠的。
9. Introduction to Probability | 概率基础
Probability is a number between 0 and 1 that measures how likely an event is. It can be expressed as a fraction, decimal, or percentage.
概率是 0 到 1 之间的一个数，衡量事件发生的可能性，可用分数、小数或百分数表示。
For equally likely outcomes: P(event) = number of favourable outcomes / total number of outcomes.
等可能结果下：P(事件) = 有利结果数 / 总结果数。
Relative frequency can estimate probability from experimental data: Relative frequency = frequency / total number of trials.
可用试验数据中的相对频率估计概率：相对频率 = 频数 / 试验总次数。
Expected frequency = probability × number of trials. This helps check if an actual frequency is surprising.
期望频数 = 概率 × 试验次数。这有助于判断实际频数是否异常。
10. The Normal Distribution | 正态分布
The normal distribution is a symmetric, bell‑shaped curve defined by its mean (μ) and standard deviation (σ). About 68% of data lies within 1σ of the mean, and 95% within 2σ.
正态分布是以均值 (μ) 和标准差 (σ) 定义的一种对称的钟形曲线。约 68% 的数据落在均值 ±1σ 内，95% 落在 ±2σ 内。
Many natural measurements approximate a normal distribution when taken in large samples.
许多自然界的测量数据在大样本下近似正态分布。
You may be asked to comment on whether a data set is approximately normal by examining its shape or comparing percentages within key intervals.
可能需要通过观察图形或比较关键区间内的百分比，判断一组数据是否近似正态。
Published by TutorHao | Statistics Revision Series | aleveler.com
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