📚 Measures of Central Tendency | 数据集中趋势的度量
In statistics, a measure of central tendency is a single value that attempts to describe a set of data by identifying the central position within that set. These measures are sometimes called measures of central location. They summarise a whole dataset with one representative number, which helps us compare different datasets easily.
在统计学中，集中趋势度量是试图通过识别数据中的中心位置来描述一组数据的单一数值。这些度量有时也被称为中心位置度量。它们用一个有代表性的数字概括整个数据集，帮助我们轻松比较不同的数据集。
This article covers the three main measures: mean, median and mode, along with weighted mean and grouped data methods, tailored for IB Mathematics students. We will explain the concepts, show the relevant formulas, and provide worked examples so that you can apply them with confidence in exams.
本文涵盖三大主要度量：均值、中位数和众数，以及加权平均数和分组数据处理方法，专为IB数学学生设计。我们将解释概念、展示相关公式并提供实际例题，帮助你自信地在考试中运用它们。
1. What is Central Tendency? | 什么是集中趋势？
Central tendency refers to the tendency of data to cluster around a central value. The three classic measures are the mean, median and mode. Each gives a different perspective on the “centre” of the data, and the most appropriate measure depends on the type of data and its distribution.
集中趋势指的是数据围绕某个中心值聚集的趋势。三种经典度量是均值、中位数和众数。每种度量都从不同角度反映数据的”中心”，而最合适的度量取决于数据类型及其分布。
Central tendency is important because it condenses a large set of data into a single number. This allows quick comparisons between datasets and serves as the foundation for more advanced statistical methods, such as variance and standard deviation. In IB Mathematics, this topic appears in both Analysis and Approaches (AA) and Applications and Interpretation (AI) courses, usually in the statistics section.
集中趋势之所以重要，是因为它将大量数据浓缩为一个数字。这样可以快速比较不同数据集，并为更高级的统计方法（如方差和标准差）奠定基础。在IB数学中，该主题出现在分析与方法（AA）和应用与解释（AI）两门课程中，通常位于统计部分。
Understanding central tendency is not just about memorising formulas. It requires knowing which measure best summarises a given situation, how to handle outliers, and how to interpret the result in context.
理解集中趋势不仅仅是记忆公式。它需要知道哪种度量最能概括特定情况、如何处理异常值，以及如何在具体情境中解释结果。
2. The Mean | 均值
The mean is the arithmetic average of all data values. It is found by summing all values and dividing by the number of values. The sample mean is denoted by x̄, while the population mean is denoted by μ.
均值是所有数据值的算术平均值。它是将所有数值相加后除以数值个数得到的。样本均值用x̄表示，总体均值用μ表示。
For a sample of n values x₁, x₂, …, xₙ, the mean is given by:
对于包含n个数值x₁, x₂, …, xₙ的样本，均值公式为：
x̄ = (x₁ + x₂ + … + xₙ)/n = (∑xᵢ)/n
Example: For the data set 4, 8, 6, the mean is (4+8+6)/3 = 6.
例如：对于数据集4, 8, 6，均值是(4+8+6)/3 = 6。
The mean uses every value in the dataset, so it is the most informative measure when the data are symmetric and contain no extreme values. However, it is sensitive to outliers, which can pull the mean towards one tail of the distribution.
均值使用了数据集中的每一个数值，因此当数据对称且没有极端值时，它是最有信息量的度量。然而，均值对异常值敏感，异常值会将均值拉向分布的一端。
When using a GDC (graphical display calculator) in IB exams, you can enter the data list and use the statistics mode to find the mean quickly. Always check whether the question asks for the mean of a sample or a population, and round your final answer to the required degree of accuracy.
在IB考试中使用图形计算器（GDC）时，你可以输入数据列表并使用统计模式快速求出均值。务必检查题目要求的是样本均值还是总体均值，并按要求的精度四舍五入最终答案。
3. The Median | 中位数
The median is the middle value when the data are arranged in ascending order. If the number of values is odd, the median is the central value; if it is even, the median is the average of the two central values.
中位数是数据按升序排列后的中间值。如果数值个数为奇数，中位数是正中间的值；如果为偶数，中位数是中间两个值的平均数。
Steps to find the median:
求中位数的步骤：
- Arrange the data in ascending order.
- 将数据按升序排列。
- Find the position: (n+1)/2 for odd n; for even n, the two middle positions are n/2 and n/2+1.
- 确定位置：n为奇数时，位置为(n+1)/2；n为偶数时，两个中间位置为n/2和n/2+1。
For example, in the data 1, 3, 7, the median is 3. In 1, 3, 7, 9, the median is (3+7)/2 = 5.
例如，在数据1, 3, 7中，中位数是3。在1, 3, 7, 9中，中位数是(3+7)/2 = 5。
The median is not affected by outliers, making it a robust measure of central tendency. This is especially useful for skewed distributions, such as income or house price data, where a few extreme values could otherwise distort the average.
中位数不受异常值影响，因此是一种稳健的集中趋势度量。这对于偏态分布尤其有用，例如收入或房价数据，其中少数极端值可能会扭曲平均值。
4. The Mode | 众数
The mode is the value that appears most frequently in a dataset. A dataset may have one mode (unimodal), two modes (bimodal), more than two modes (multimodal), or no mode if all values are unique.
众数是数据集中出现频率最高的数值。一个数据集可能有一个众数（单峰）、两个众数（双峰）、多个众数（多峰），或者如果所有数值都唯一则没有众数。
Example: In the data 2, 3, 3, 5, the mode is 3.
例如：在数据2, 3, 3, 5中，众数是3。
The mode is the only measure of central tendency that can be used for categorical or nominal data. For example, in a survey asking students their favourite subject, the mode is the subject chosen by the largest number of students.
众数是唯一可以用于类别数据或名义数据的集中趋势度量。例如，在一项询问学生最喜欢科目的调查中，众数
Published by TutorHao | IB Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导Cancel reply