IGCSE CAIE Statistics: Case Study Practice | IGCSE CAIE 统计：案例分析实战演练

📚 IGCSE CAIE Statistics: Case Study Practice | IGCSE CAIE 统计：案例分析实战演练

Statistical case studies require you to move from raw data to interpretation. This article works through a realistic IGCSE CAIE Statistics scenario involving mobile phone use among 30 students. You will practise organising data, drawing diagrams, calculating averages and spread, estimating probability and solving conditional probability.

统计案例分析要求你从原始数据走向解释。本文通过一个真实的 IGCSE CAIE 统计情景，涉及 30 名学生的手机使用时间。你将练习整理数据、绘制图表、计算平均数和离散程度、估计概率以及求解条件概率。

1. Case Context and Data Set | 案例背景与数据集

The case study uses a sample of 30 students from a school statistics project. The variable is the time spent on mobile phones per day, measured in minutes. Understanding the context helps you choose the right diagram, average and spread. The raw data are shown below.

本案例使用某学校统计项目中 30 名学生的样本。变量是每天使用手机的时间，以分钟为单位。理解背景有助于选择正确的图表、平均数和离散程度指标。原始数据如下所示。

45, 48, 50, 53, 54, 55, 57, 58, 59, 60, 61, 62, 64, 66, 68, 69, 70, 71, 73, 75, 76, 77, 79, 80, 82, 84, 85, 88, 90, 92

这 30 个数值已经按升序排列。在考试中，排序后更容易找到中位数、四分位数和极值。请始终先检查数据是否有重复值或异常值，再开始计算。

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2. Data Organisation and Frequency Table | 数据整理与频数表

For continuous data, we usually group values into class intervals. In this case we can use intervals 40 ≤ x < 50, 50 ≤ x < 60, 60 ≤ x < 70, 70 ≤ x < 80 and 80 ≤ x < 100. A grouped frequency table helps summarise the distribution clearly.

对于连续数据，我们通常将数值分组。本例可以使用区间 40 ≤ x < 50、50 ≤ x < 60、60 ≤ x < 70、70 ≤ x < 80 和 80 ≤ x < 100。分组频数表有助于清晰地概括分布。

Class interval	Frequency	Relative frequency
40 ≤ x < 50	2	2/30 ≈ 0.067
50 ≤ x < 60	7	7/30 ≈ 0.233
60 ≤ x < 70	7	7/30 ≈ 0.233
70 ≤ x < 80	7	7/30 ≈ 0.233
80 ≤ x < 100	7	7/30 ≈ 0.233

Notice that the modal class is 50 ≤ x < 60, 60 ≤ x < 70, 70 ≤ x < 80 and 80 ≤ x < 100 because they all have frequency 7. Relative frequency is frequency divided by total frequency, which is essential for probability estimation.

注意众数所在组是 50 ≤ x < 60、60 ≤ x < 70、70 ≤ x < 80 和 80 ≤ x < 100，因为它们的频数都是 7。相对频率等于频数除以总频数，这是估计概率的重要基础。

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3. Histograms and Bar Charts | 直方图与条形图

For continuous data, use a histogram with no gaps between bars. In this case all class intervals have equal width, so the vertical axis can show frequency directly. If the class widths were unequal, you would need to calculate frequency density.

对于连续数据，应使用无间隙的直方图。本例所有组距相等，因此纵轴可以直接表示频数。如果组距不相等，则需要计算频率密度。

Always label the horizontal axis with the variable and unit, and the vertical axis with frequency or frequency density. A bar chart is only for categorical or discrete data, so it would be inappropriate for this continuous variable.

务必给横轴标上变量和单位，给纵轴标上频数或频率密度。条形图仅适用于分类数据或离散数据，因此不适合这个连续变量。

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4. Measures of Central Tendency | 集中趋势的度量

The mean is the total of all values divided by the number of values. The sum of these 30 values is 2051. The median is the middle value after sorting. Because n is even, the median is the average of the 15th and 16th values, which are 68 and 69. There is no mode because no value repeats in the raw data.

均值是所有数值之和除以数值个数。这 30 个数值的总和为 2051。中位数是排序后位于中间的值。因为样本量为偶数，中位数是第 15 个和第 16 个值的平均数，即 68 和 69 的平均数。原始数据中没有重复值，因此没有众数。

Mean = Σx / n = 2051 / 30 ≈ 68.4 min

Median = (68 + 69) / 2 = 68.5 min

In grouped data, the modal class is the interval with the highest frequency. Here four intervals share the highest frequency, so the modal class is not unique. You should report all intervals with frequency 7 or describe the modal class as the 50-99 range.

在分组数据中，众数所在组是频数最高的区间。本例有四个区间频数并列最高，因此众数所在组不唯一。你应该报告所有频数为 7 的区间，或将众数所在组描述为 50 到 99 的范围。

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

Range is the simplest measure of spread. It is the maximum minus the minimum. Here the range is 92 − 45 = 47 minutes. However, the range is easily affected by extreme values.

极差是最简单的离散程度指标，等于最大值减最小值。本例极差为 92 − 45 = 47 分钟。但极差容易受极端值影响。

The interquartile range, IQR, uses the middle 50 percent of the data. The lower quartile Q1 is the median of the lower half, and the upper quartile Q3 is the median of the upper half. For this data set, Q1 = 58 and Q3 = 79, so IQR = 79 − 58 = 21 minutes.

四分位距 IQR 使用数据中间 50% 的范围。下四分位数 Q1 是下半部分数据的中位数，上四分位数 Q3 是上半部分数据的中位数。对于本数据集，Q1 = 58，Q3 = 79，因此 IQR = 79 − 58 = 21 分钟。

Range = Max − Min = 92 − 45 = 47 min

IQR = Q3 − Q1 = 79 − 58 = 21 min

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6. Box-and-Whisker Plot | 箱线图

The five-number summary consists of minimum, Q1, median, Q3 and maximum. For this data set the summary is 45, 58, 68.5, 79, 92. A box plot displays these values on a horizontal or vertical scale.

五数概括由最小值、下四分位数、中位数、上四分位数和最大值组成。本数据集的五数概括为 45、58、68.5、79、92。箱线图在水平或垂直数轴上展示这些数值。

The box runs from Q1 to Q3, with a vertical line at the median. Whiskers extend from the box to the minimum and maximum if there are no outliers. To check

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