Descriptive Statistics: Core Concepts and Exam Tips for IB Mathematics | IB数学：描述性统计核心概念与考点

📚 Descriptive Statistics: Core Concepts and Exam Tips for IB Mathematics | IB数学：描述性统计核心概念与考点

Descriptive statistics is a fundamental topic in IB Mathematics. It focuses on summarizing and visualizing data so that patterns and trends become clear. This article walks through the core concepts, common formulas, and exam-oriented techniques required for success in IB papers.

描述性统计是IB数学的基础专题。它主要通过对数据进行整理与可视化，使数据的特征和趋势变得清晰。本文将系统梳理核心概念、常用公式以及IB考试中必备的答题技巧。

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1. Types of Data | 数据类型

In statistics, data are first classified as categorical or numerical. Categorical data describe groups or qualities, such as colors or exam grades, while numerical data represent measurable quantities.

在统计学中，数据首先分为分类数据与数值数据。分类数据描述分组或性质，例如颜色或考试成绩等级；数值数据则表示可测量的数量。

Numerical data are divided into discrete and continuous types. Discrete data come from counting and take only distinct values, such as the number of students in a class. Continuous data come from measuring and can take any value in a range, such as height or time.

数值数据又分为离散型与连续型。离散型数据通过计数得到，只能取分散的数值，例如班级学生人数；连续型数据通过测量得到，可以在某个范围内取任意值，例如身高或时间。

IB exam questions often ask you to identify data types. Remember: discrete data are counted, continuous data are measured.

IB考试常要求判断数据类型。请记住：离散数据靠计数，连续数据靠测量。

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2. Frequency Distributions and Grouped Data | 频数分布与分组数据

A frequency distribution is a table that shows how often each value or class interval occurs in a data set. It is the first step in organizing raw data.

频数分布表用于展示每个取值或组距在数据集中出现的次数，是整理原始数据的第一步。

For ungrouped data, the table lists each possible value alongside its frequency. For grouped data, values are sorted into intervals, such as 0–9, 10–19, and only the interval frequencies are shown.

对于未分组数据，表中列出每个可能取值及其频数；对于分组数据，则将数据按区间归类，例如0–9、10–19，并显示各区间的频数。

When data are grouped, the midpoint of each interval is used as a representative value for calculations. The midpoint is found using (lower boundary + upper boundary) ÷ 2.

当数据分组后，每个区间的组中值被用作代表值参与计算。组中值通过（下界 + 上界）÷ 2 求得。

Frequency tables are important because they form the basis for calculating central tendency and dispersion from grouped data.

频数表非常重要，因为它是后续基于分组数据计算集中趋势和离散程度的基础。

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3. Measures of Central Tendency: Mean, Median, Mode | 集中趋势：均值、中位数、众数

The mean is the arithmetic average. For raw data, it is calculated by summing all values and dividing by the number of values.

均值是算术平均值。对于原始数据，通过将所有数据求和后除以数据个数得到。

x̄ = (Σx) / n

For grouped data, use the formula with frequencies and midpoints.

对于分组数据，则使用频数与组中值的公式。

x̄ = Σ(f × x) / Σf

The median is the middle value when data are arranged in order. If n is odd, the median is the (n + 1)/2-th value; if n is even, it is the average of the two middle values.

中位数是将数据按顺序排列后的中间值。若 n 为奇数，中位数是第 (n + 1)/2 个值；若 n 为偶数，则取中间两个值的平均数。

The mode is the value that occurs most often. A data set may have no mode, one mode, or several modes.

众数是出现次数最多的值。一个数据集可能没有众数、有一个众数，也可能有多个众数。

In IB questions, the choice of measure can depend on the shape of the distribution. The mean is affected by outliers, while the median is not.

在IB题目中，选择哪种度量取决于数据分布形态。均值受离群值影响，而中位数不受影响。

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4. Weighted Mean | 加权平均数

A weighted mean is used when some data values are more important than others. Each value is assigned a weight, and the mean is computed by multiplying each value by its weight.

当某些数据比其它数据更重要时，需要使用加权平均数。每个数值都被赋予一个权重，计算时用每个数值乘以其权重。

x̄ = Σ(wᵢxᵢ) / Σwᵢ

For example, if an IB internal assessment has three components with different percentage weights, you should use the weighted mean to find the final score.

例如，IB内部评估包含三个不同权重比例的组成部分时，应使用加权平均数来计算最终得分。

Weighted mean questions often appear in paper 1. Always check whether equal weights are implied before using the simple average.

加权平均在卷一考试中经常出现。在使用简单平均数之前，请务必确认是否隐含等权条件。

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5. Measures of Dispersion: Range and Interquartile Range | 离散程度：极差与四分位距

The range is the simplest measure of spread, calculated as the difference between the maximum and minimum values.

极差是最简单的离散程度度量，等于最大值与最小值之差。

Range = Max – Min

The interquartile range measures the spread of the middle 50% of the data. It is found by subtracting the lower quartile from the upper quartile.

四分位距度量数据中间50%的离散程度，等于上四分位数减去下四分位数。

IQR = Q₃ – Q₁

The IQR is more robust than the range because it is not affected by extreme values.

四分位距比极差更加稳健，因为它不受极端值影响。

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6. Variance and Standard Deviation | 方差与标准差

Variance and standard deviation measure how far data values are spread around the mean. A small value indicates that the data are close to the mean; a large value indicates more spread.

方差和标准差衡量数据围绕均值的离散程度。数值越小表示数据越接近均值；数值越大表示数据越分散。

For a population, the variance is the average of the squared deviations from the mean.

对于总体，方差是各数据与均值之差的平方的平均值。

σ² = Σ(x – μ)² / n

The population standard deviation is the square root of the variance.

总体标准差是方差的平方根。

σ = √( Σ(x – μ)² / n )

For a sample, the standard deviation formula uses n – 1 as the denominator.

对于样本，标准差公式使用 n – 1 作为分母。

s = √( Σ(x – x̄)² / (n – 1) )

In IB, standard deviation is often found using the GDC. However, you must be able to interpret its value and compare two data sets.

在IB考试中，标准差通常使用图形计算器求解。但你必须能解释其含义并比较两组数据。

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7. Quartiles, Percentiles and Box-and-Whisker Plots | 四分位数、百分位数与箱线图

Quartiles divide ordered data into four equal parts. Q₁ is the 25th percentile, Q₂ is the median or 50th percentile, and Q₃ is the 75th percentile.

四分位数将有序数据分为四等份。Q₁ 是第25百分位数，Q₂ 是中位数或第50百分位数，Q₃ 是第75百分位数。

A box-and-whisker plot displays the minimum, Q₁, median, Q₃, and maximum. It is useful for comparing distributions visually.

箱线图显示最小值、Q₁、中位数、Q₃ 和最大值，适合直观比较分布。

Outliers can be identified using the 1.5 × IQR rule. Any value below Q₁ – 1.5 × IQR or above Q₃ +

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