📚 Descriptive Statistics Core Concepts | IB数学：描述性统计核心内容梳理
Descriptive statistics is the branch of statistics that summarises and describes the main features of a data set without making inferences beyond the data. In IB Mathematics, this topic forms the foundation for data analysis, probability, and statistical testing.
描述性统计是统计学中在不进行超越数据本身推断的前提下，对数据集的主要特征进行概括和描述的分支。在IB数学课程中，该主题是数据分析、概率和统计检验的基础。
1. Types of Data and Variables | 数据类型与变量
Data can be classified as categorical (qualitative) or numerical (quantitative). Numerical data is further divided into discrete data, which takes countable values, and continuous data, which can take any value within an interval.
数据可分为分类数据（定性）和数值数据（定量）。数值数据又进一步分为离散数据和连续数据：离散数据取可数的值，连续数据可在某个区间内取任意值。
- Discrete data: number of students in a class, number of goals scored.
- 离散数据：班级中的学生人数、进球数。
- Continuous data: height, weight, time, temperature.
- 连续数据：身高、体重、时间、温度。
- Categorical data: favourite colour, type of car.
- 分类数据：最喜欢的颜色、汽车类型。
Understanding the type of data determines which measures of central tendency and dispersion are appropriate.
理解数据类型决定了应选用哪些集中趋势和离散程度的度量。
2. Frequency Distributions and Grouped Data | 频数分布与分组数据
A frequency table organises raw data by listing each value (or class interval) together with its frequency. For continuous data, data is often grouped into intervals such as 10–20, 20–30, etc.
频数表通过列出每个数值（或组区间）及其频数来整理原始数据。对于连续数据，通常将数据分组为10–20、20–30等区间。
| Height (cm) | Frequency |
| 150–160 | 5 |
| 160–170 | 12 |
| 170–180 | 8 |
When calculating statistics from grouped data, the midpoint of each interval is used as a representative value.
根据分组数据计算统计量时，通常使用每组区间的组中值作为代表值。
3. Measures of Central Tendency | 集中趋势度量
The mean, median, and mode summarise the centre of a data set. The mean is the arithmetic average, the median is the middle value when data is ordered, and the mode is the most frequent value.
均值、中位数和众数概括了数据集的中心。均值是算术平均数，中位数是数据排序后的中间值，众数是出现次数最多的值。
Mean = (Σxᵢ)/n
For grouped data, the mean is estimated using midpoints: where f is frequency and x is midpoint.
对于分组数据，使用组中值估计均值：其中 f 为频数，x 为组中值。
Mean ≈ (Σf·x)/Σf
If the data set has an even number of values, the median is the average of the two middle values.
如果数据集有偶数个数值，中位数取中间两个值的平均数。
4. Quartiles and Percentiles | 四分位数与百分位数
Quartiles divide ordered data into four equal parts. The lower quartile Q₁ is the median of the lower half, and the upper quartile Q₃ is the median of the upper half. The median is Q₂.
四分位数将有序数据分成四个等份。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，中位数即 Q₂。
Percentiles indicate the value below which a given percentage of observations fall. For example, the 80th percentile is the value below which 80% of the data lie.
百分位数表示给定百分比的观测值落在其之下的数值。例如，第80百分位数是80%的数据都小于它的值。
Interquartile Range = Q₃ − Q₁
The interquartile range (IQR) measures the spread of the middle 50% of the data and is less affected by outliers than the range.
四分位距（IQR）衡量数据中间50%的离散程度，与极差相比受异常值影响较小。
5. Measures of Dispersion | 离散程度度量
The range, variance, and standard deviation describe how spread out the data are. The range is the difference between the maximum and minimum values.
极差、方差和标准差描述数据的离散程度。极差是最大值与最小值之差。
Range = Max − Min
Variance measures the average squared deviation from the mean. For a sample, the formula uses n − 1 as the denominator.
方差度量数据与均值之间的平均平方偏差。对于样本，公式使用 n − 1 作为分母。
Sample variance s² = Σ(xᵢ − x̄)²/(n − 1)
The standard deviation is the square root of the variance and is expressed in the same units as the original data.
标准差是方差的平方根，与原始数据具有相同的单位。
s = √(Σ(xᵢ − x̄)²/(n − 1))
For grouped data, the variance is approximated using midpoints.
对于分组数据，使用组中值近似方差。
6. Box and Whisker Plots | 箱线图
A box plot displays the minimum, lower quartile, median, upper quartile, and maximum of a data set. It provides a clear visual summary of the centre, spread, and skewness.
箱线图显示数据集的最小值、下四分位数、中位数、上四分位数和最大值。它清晰地可视化了数据的中心、离散程度和偏态。
- The box spans from Q₁ to Q₃, with the median marked inside.
- 箱体从 Q₁ 延伸到 Q₃，内部标记中位数。
- Whiskers extend to the smallest and largest values within 1.5 × IQR from the quartiles.
- 须线延伸到距四分位数 1.5 × IQR 范围内的最小值和最大值。
- Points beyond the whiskers are considered outliers.
- 须线之外的点被视为异常值。
Box plots are especially useful for comparing two or more data sets.
箱线图特别适合比较两个或多个数据集。
7. Outliers | 异常值
An outlier is a data point that differs significantly from the rest of the data. A common rule in IB is to define an outlier as a value that is either less than Q₁ − 1.5 × IQR or greater than Q₃ + 1.5 × IQR.
异常值是与其余数据显著不同的数据点。IB中常用的规则是将小于 Q₁ − 1.5 × IQR 或大于 Q₃ + 1.5 × IQR 的值定义为异常值。
Lower boundary = Q₁ − 1.5 × IQR
Upper boundary = Q₃ + 1.5 × IQR
Outliers can arise from measurement errors, data entry mistakes, or genuine rare events. They should be investigated before being removed.
异常值可能来自测量误差、录入错误或真实的稀有事件。在删除之前应进行调查。
8. Cumulative Frequency | 累积频数
For grouped data, a cumulative frequency curve (ogive) plots cumulative frequency against the upper class boundary. This curve can be used to estimate the median, quartiles, and percentiles.
对于分组数据，累积频数曲线（ogive）以累积频数对组上界作图。该曲线可用于估计中位数、四分位数和百分位数。
To construct a cumulative frequency table, add frequencies successively. Then plot points at the upper boundary of each interval.
构建累积频数表时，依次累加频数。然后在每个区间的上界处描点。
- Median is estimated at the value corresponding to n/2 on the cumulative frequency axis.
- 中位数对应累积频数轴上 n/2 处的值。
- Q₁ corresponds to n/4, and Q₃ corresponds to 3n/4.
- Q₁ 对应 n/4，Q₃ 对应 3n/4。
The curve also helps identify whether data are symmetrically distributed or skewed.
该曲线还有助于判断数据是对称分布还是偏态分布。
9. Skewness | 偏度
Skewness describes the asymmetry of a data distribution. A distribution is positively skewed (right-skewed) if the tail extends to the right, and negatively skewed (left-skewed) if the tail extends to the left.
偏度描述数据分布的不对称性。若尾部向右延伸，则分布为正偏（右偏）；若尾部向左延伸，则为负偏（左偏）。
For a moderately skewed distribution, the relationship between mean, median, and mode can indicate the direction of skew:
对于中等偏斜的分布，均值、中位数和众数的关系可以指示偏斜方向：
- Mean > Median > Mode: positive skew.
- 均值 > 中位数 > 众数：正偏。
- Mean < Median < Mode: negative skew.
- 均值 < 中位数 < 众数：负偏。
- Mean ≈ Median ≈ Mode: symmetric distribution.
- 均值 ≈ 中位数 ≈ 众数：对称分布。
In a box plot, positive skew appears as a longer right whisker, while negative skew appears as a longer left whisker.
在箱线图中，正偏表现为右侧须线更长，负偏表现为左侧须线更长。
10. Using Technology | 技术工具的使用
IB Mathematics examinations often allow the use of a graphic display calculator (GDC). Students should be able to enter data into lists, calculate one-variable statistics, and draw box plots using their GDC.
IB数学考试通常允许使用图形计算器（GDC）。学生应能在计算器中输入数据、计算单变量统计量并绘制箱线图。
- Use the STAT mode to enter data and run 1-Var Stats.
- 使用 STAT 模式输入数据并运行 1-Var Stats。
- Identify mean, median, Q₁, Q₃, and standard deviation from the output.
- 从输出结果中识别均值、中位数、Q₁、Q₃ 和标准差。
- Use box plot settings to display data visually.
- 使用箱线图设置以可视化显示数据。
Efficient use of technology saves time and reduces arithmetic errors, but understanding the underlying concepts remains essential.
高效使用技术可以节省时间并减少计算错误，但理解基本概念仍然至关重要。
11. Choosing the Right Measure | 选择合适的统计量
The mean is the most commonly used measure of central tendency, but it is sensitive to outliers. The median is robust and preferable for skewed distributions. The mode is useful for categorical data.
均值是最常用的集中趋势度量，但它对异常值敏感。中位数具有稳健性，适用于偏态分布。众数对分类数据有用。
Similarly, the standard deviation is informative when the data are symmetric, while the IQR is better when outliers are present.
类似地，当数据对称时，标准差更有信息量；当存在异常值时，四分位距更适用。
Data set A: 5, 6, 7, 8, 9 → Mean = 7, Median = 7
Data set B: 5, 6, 7, 8, 50 → Mean = 15.2, Median = 7
In set B, the median better represents the central location because the outlier inflates the mean.
在数据集 B 中，由于异常值抬高了均值，中位数能更好地代表中心位置。
12. Common Exam Pitfalls | 常见考试误区
Students often confuse the population variance with the sample variance, forget to use n − 1 for samples, or mistakenly use class midpoints incorrectly. Another common error is reading the median from a cumulative frequency graph without interpolating accurately.
学生常混淆总体方差与样本方差，忘记样本使用 n − 1，或错误地使用组中值。另一个常见错误是从累积频数图读取中位数时没有进行准确内插。
- Always check whether the data are a population or a sample.
- 始终检查数据是总体还是样本。
- Use the correct formula: σ² for population, s² for sample.
- 使用正确的公式：总体用 σ²，样本用 s²。
- For grouped data, use midpoints, not boundaries, when estimating the mean.
- 对于分组数据，估计均值时使用组中值，而不是边界值。
- When finding quartiles from a cumulative frequency graph, read the horizontal axis at the correct cumulative frequency.
- 从累积频数图查找四分位数时，应在正确的累积频数处读取横轴。
A thorough understanding of definitions and formulas, together with careful calculator use, will help students succeed in descriptive statistics questions.
透彻理解定义和公式，加上仔细使用计算器，将帮助学生在描述性统计题目中取得好成绩。
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