📚 Mastering Handling Data 5 | 数据处理进阶篇
Welcome to the fifth installment of our Handling Data series! In this article, we consolidate the core statistical and probability skills you need for the Edexcel IGCSE Mathematics examination. We move beyond simple averages and charts into grouped data, cumulative frequency, box plots, scatter graphs, and probability tree diagrams — the topics that frequently appear in the higher-tier papers.
欢迎来到数据处理系列第五篇！在这篇文章中，我们将巩固 Edexcel IGCSE 数学考试所需的核心统计与概率技能。我们将超越简单的平均数和图表，深入学习分组数据、累积频率、箱线图、散点图和概率树形图——这些内容在高级别试卷中频繁出现。
1. Types of Data | 数据类型
Before any calculation begins, you must recognise the nature of your data. Data can be classified as qualitative (describing categories, such as favourite colour) or quantitative (numerical). Quantitative data is further divided into discrete data (countable values, such as the number of students in a class) and continuous data (measurable values, such as height or time).
在进行任何计算之前，你必须先认清数据的性质。数据可分为定性数据（描述类别，如最喜欢的颜色）和定量数据（数值型）。定量数据又进一步分为离散数据（可计数的数值，如班级学生人数）和连续数据（可测量的数值，如身高或时间）。
In the Edexcel exams, questions may ask you to state whether a variable is discrete or continuous, or to give an example of each. Remember: shoe size is discrete; shoe length is continuous. Exam questions often test this distinction directly, so have a clear definition ready.
在 Edexcel 考试中，题目可能要求你判断某个变量是离散的还是连续的，或者各举一个例子。记住：鞋码是离散数据；鞋长是连续数据。考试题经常直接考查这一区分，所以请准备好清晰的定义。
2. Frequency Tables | 频数表
A frequency table records how often each value appears. For grouped data, values are collected into class intervals. You must be able to read, complete and interpret these tables quickly and correctly.
频数表记录了每个数值出现的次数。对于分组数据，数值被归入若干个组距。你必须能够快速且准确地读取、补充和解读这些表格。
every data value belongs to exactly one class. For continuous data, use boundaries such as '150 ≤ h < 160 cm'. Notice that the lower boundary is included and the upper boundary is excluded.
在构造组距时，确保组与组之间不重叠，每个数据值恰好属于一个组。对于连续数据，使用如“150 ≤ h < 160 cm”这样的边界。注意下边界包含在内，上边界不包含在内。
| Height (cm) | Tally | Frequency |
| 150 ≤ h < 155 | |||| | 4 |
| 155 ≤ h < 160 | |||| ||| | 8 |
| 160 ≤ h < 165 | |||| | | 6 |
| 165 ≤ h < 170 | || | 2 |
The modal class is the class with the highest frequency — here it is '155 ≤ h < 160' lasses share the highest frequency, the data is bimodal and you should state both classes.
众数类（模态组）是频率最高的组——在这里是“155 ≤ h < 160”。如果有两个组频率并列最高，则数据呈双峰分布，你应该把两个组都写出来。
3. Measures of Central Tendency | 集中趋势的度量
The mean, median and mode summarise where the centre of the data lies. Each has its own strengths and weaknesses, and the exam will ask you to choose the most appropriate average for a given situation.
平均数、中位数和众数概括了数据的中心位置。它们各有优缺点，考试会要求你为给定情境选择最合适的平均数。
For ungrouped data, the mean is found by adding all values and dividing by how many there are:
对于未分组数据，平均数通过将所有值相加除以数值个数得出：
mean = (sum of all values) ÷ (number of values)
The median is the middle value when data is arranged in ascending order. If there are n values, the median is at position (n + 1) ÷ 2. When n is even, take the average of the two middle values. The mode is the most frequently occurring value.
中位数是数据按升序排列后的中间值。如果有 n 个数值，中位数位于第 (n + 1) ÷ 2 位。当 n 为偶数时，取中间两个数值的平均值。众数是出现次数最多的数值。
For grouped data, we cannot find the exact mean; we estimate it using the mid-point of each class interval. Multiply each mid-point by its frequency, add these products, then divide by the total frequency:
对于分组数据，我们无法求出精确的平均数；需要使用每个组距的中点值来估算。将每个中点值乘以对应频数，将乘积相加，再除以总频数：
estimated mean = Σ(mid-point × frequency) ÷ Σ(frequency)
When presenting your answer, write the method clearly: show the mid-points, the products, and the final division. Many examiners award method marks even if the arithmetic goes slightly wrong.
在作答时，请清晰地写出方法：列出中点值、乘积以及最后的除法。很多阅卷官即使算数稍有错误也会给方法分。
4. Measures of Spread | 离散程度的度量
range = largest value − smallest value. However, the range is easily distorted by outliers — a single extreme value can make it misleadingly large.
离散程度的度量告诉我们数据的分散程度。最简单的是极差 =
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