📚 Year 9 CIE Statistics: Comparing UK University Entry Requirements | 英国大学申请要求对照
Every year, thousands of students use statistical thinking to choose the right university. By collecting data on entry requirements, such as A-level grades or UCAS tariff points, we can compare different institutions objectively. This article will guide you through the key steps: gathering data, organising it into tables and charts, calculating averages and measures of spread, and finally using box plots and cumulative frequency graphs to make informed decisions. Along the way, we will use a realistic dataset of entry offers from 20 UK universities for a popular course, helping you see how Year 9 statistics comes alive in university planning.
每年都有成千上万的学生运用统计思维选择适合自己的大学。通过收集入学要求数据（例如 A-level 成绩或 UCAS 学费积分），我们可以客观地比较不同院校。本文将带你经历关键步骤：收集数据、用表格和图表整理、计算平均值和离散程度，最后使用箱线图和累积频数图做出明智决策。在过程中，我们会使用一个包含 20 所英国大学对某热门专业录取要求的真实数据集，让你亲眼看到九年级统计学知识如何在大学规划中发挥作用。
1. Introduction to University Entry Requirements | 大学入学要求简介
When applying to UK universities, students typically need to meet specific academic conditions. These are often expressed as A-level grades, such as AAA or AAB, or as a total UCAS tariff score. Other qualifications like the International Baccalaureate (IB) or Scottish Highers are also used, but for simplicity we focus on A-level points. Understanding how these requirements vary between universities is the first step in making a data-driven comparison.
在申请英国大学时，学生通常需要满足特定的学术条件。这些条件通常表示为 A-level 成绩，例如 AAA 或 AAB，或者表示为 UCAS 学费积分总分。国际文凭（IB）或苏格兰高等证书等其他资格也会被使用，但为简单起见，我们专注于 A-level 绩点。了解这些要求在不同大学之间的差异，是进行数据驱动比较的第一步。
Imagine you are interested in studying Economics. By looking at 20 universities, you might find offers ranging from 128 tariff points (equivalent to ABB) up to 168 points (A*A*A). Instead of relying on subjective impressions, you can use statistical tools to summarise this spread. This turns a long list of numbers into clear, actionable information.
假设你对经济学感兴趣。通过查看 20 所大学，你可能会发现录取要求从 128 个学费积分（相当于 ABB）到 168 个积分（A*A*A）不等。与其依赖主观印象，你可以用统计工具来概括这种分布。这将一长串数字转变为清晰、可操作的信息。
2. Collecting Data on Entry Requirements | 收集入学要求数据
Our dataset consists of 20 UK universities offering Economics. For each, we record the typical A-level offer for the latest entry year and convert it into UCAS tariff points. The raw data, in points, is as follows: 144, 136, 152, 128, 144, 160, 136, 144, 152, 168, 136, 144, 128, 144, 152, 136, 144, 160, 144, 136. We obtained this information from university websites and UCAS course search, ensuring the data is reliable and up to date.
我们的数据集包含 20 所提供经济学课程的英国大学。我们为每所大学记录了最近一个入学年份的典型 A-level 录取要求，并将其转换为 UCAS 学费积分。原始数据（以积分计）如下：144, 136, 152, 128, 144, 160, 136, 144, 152, 168, 136, 144, 128, 144, 152, 136, 144, 160, 144, 136。我们从大学官网和 UCAS 课程搜索中获取了这些信息，确保数据可靠且最新。
Notice that the values range from 128 to 168. Already we can see that some universities set significantly higher bars. However, a simple list is hard to interpret. We need to organise the numbers to uncover patterns, such as the most common tariff point and how concentrated the requirements are.
请注意，这些数值的范围从 128 到 168。我们已经能看出有些大学的要求明显更高。但仅凭一列数字很难解读。我们需要整理这些数字，以便发现规律，比如最常见的积分值以及要求集中程度。
3. Types of Variables: Categorical and Numerical | 变量类型：分类与数值
In this investigation, the variable we are studying is 'university entry tariff points'. Since it is recorded as a number, it is a quantitative, discrete variable. Discrete means the values are distinct, countable amounts (usually whole numbers). If we had kept the original grades like AAA, AAB, that would have been categorical (ordinal) data. Transforming them into points allows us to perform more powerful numerical analysis.
在本调查中，我们研究的变量是“大学入学学费积分”。由于它被记录为数字，因此是一个定量离散变量。离散意味着数值是独立的、可数的量（通常是整数）。如果我们保留原始成绩等级（如 AAA、AAB），那将是分类（有序）数据。将其转换为积分使我们可以进行更强大的数值分析。
2, D = 24, E = 16. For example, AAA = 48 + 48 + 48 = 144 points, while AAB = 48 + 48 + 40 = 136 points. The conversion follows a consistent scale approved by UCAS.
使用数值数据意味着我们可以计算平均数、找出中位数并绘制直方图。这就是我们要转换 A-level 成绩的原因：A* = 56，A = 48，B = 40，C = 32，D = 24，E = 16。例如，AAA = 48 + 48 + 48 = 144 分，而 AAB = 48 + 48 + 40 = 136 分。这一转换遵循 UCAS 认可的统一定标。
4. Displaying Requirements: Frequency Tables | 展示要求：频数表
The first way to organise our 20 data points is a frequency table. We count how many universities set each tariff point requirement. The table below groups the data by distinct tariff values, showing frequency (f).
组织我们 20 个数据点的第一种方法是频数表。我们统计每个学费积分要求有多少所大学。下表按不同的积分值对数据进行分组，显示频数 (f)。
| Tariff points (x) | Frequency (f) |
|---|---|
| 128 | 2 |
| 136 | 6 |
| 144 | 7 |
| 152 | 3 |
| 160 | 1 |
| 168 | 1 |
| Total | 20 |
From this table, we can immediately see that 144 points is the most common requirement (7 universities), while the extremes 160 and 168 are rare. The mode of this dataset is 144 tariff points. A frequency table transforms an unordered list into a clear summary that highlights the central tendency.
从这张表我们可以立刻看出，144 分是最常见的录取要求（7 所大学），而极值 160 和 168 很少见。此数据集的众数是 144 学费积分。频数表将无序列表转变为清晰的摘要，突显了集中趋势。
5. Visualising Data: Bar Charts and Histograms | 数据可视化：条形图和直方图
Because our tariff points are discrete numerical values with gaps between them, a bar chart is the ideal graph. Each distinct x-value gets a separate bar whose height equals the frequency. If the data were continuous (e.g. time), we would use a histogram with no gaps, but here the gaps emphasise that you cannot have 135 tariff points under this scale.
由于我们的学费积分是离散数值且数值间有间隔，条形图是最理想的图表。每个不同的 x 值都有一个单独的条形，其高度等于频数。如果数据是连续的（例如时间），我们会使用没有间隔的直方图，但这里的间隔强调了在这个标尺下不存在 135 积分这样的值。
If you were to draw the bar chart, you would place 'Tariff points' on the horizontal axis and 'Frequency' on the vertical axis. The tallest bar would be at 144, reaching 7. This visual immediately communicates the popularity of that requirement. You can also quickly spot the skew: the distribution is slightly skewed to the right because of the high values 160 and 168.
如果要绘制条形图，你会把“学费积分”放在横轴上，把“频数”放在纵轴上。最高的条形会位于 144 处，达到 7。这种可视化立即传达了该要求的常见程度。你还可以迅速发现偏态：由于存在高值 160 和 168，该分布略微右偏。
The bar chart helps you compare the frequency of each tariff point at a glance. It is much easier to interpret than a table for many people. For example, it is obvious that more than half the universities set requirements between 136 and 144 points inclusive.
条形图有助于你一眼比较每个积分的频数。对许多人来说，它比表格更容易解读。例如，很明显超过一半的大学将入学要求设定在 136 至 144 分（含）之间。
6. UCAS Tariff Points: A Common Scale | UCAS 学费积分：一个通用标尺
different types of credentials fairly. The scale was updated in 2017, so we are using the current values. For A-levels, the points per grade are: A* = 56, A = 48, B = 40, C = 32, D = 24, E = 16. You can add up the best three A-levels or other accepted qualifications.
UCAS 学费积分是一个将各种资格证书和成绩转换为单一数字的积分系统。它使大学能够公平地比较持有不同类型证书的申请者。该标尺在 2017 年进行了更新，因此我们使用的是当前数值。对于 A-level，每个等级的积分是：A* = 56，A = 48，B = 40，C = 32，D = 24，E = 16。你可以将最好的三个 A-level 成绩或其它认可的资格证书累加。
Using tariff points simplifies statistical analysis. Instead of juggling letter grades like A*AB, we can work with a single number. For instance, our university dataset includes offers like 144 (AAA), 152 (AAB with an A in an EPQ, or A*AB, depending on context), and 168 (A*A*A). This common metric allows us to calculate averages and spreads meaningfully.
使用学费积分简化了统计分析。我们无需纠结于像 A*AB 这样的字母等级，而可以用一个单一数字来处理。例如，我们的大学数据集包括 144（AAA）、152（例如 A*AB 或含有 EPQ 加分）和 168（A*A*A）等录取要求。这一通用度量标准使我们能够有意义地计算平均值和离散程度。
7. Measures of Central Tendency: Mean and Median | 集中趋势测量：均数和中位数
2×3) + (160×1) + (168×1) = 256 + 816 + 1008 + 456 + 160 + 168 = 2864. Then divide by 20: mean = 2864 / 20 = 143.2 tariff points. The formula can be written as:
为了概括分布的中心，我们计算均数（算术平均值）。我们将所有积分相加： 136×6) + (144×7) + (152×3) + (160×1) + (168×1) = 256 + 816 + 1008 + 456 + 160 + 168 = 2864。然后除以 20：均数 = 2864 / 20 = 143.2 学费积分。公式可以写为：
x̄ = Σfx / Σf = 2864 / 20 = 143.2
144, so median = 144 points. The mode, as we saw, is also 144. The mean is slightly lower than the median, which hints at a small negative skew in the sample because the low end includes 128s, but overall the distribution is fairly symmetric.
中位数是数据排序后位于中间的值。因为有 20 个值，中位数位于第 10 和第 11 个值之间。排序后的列表显示这两个值都是 144，因此中位数 = 144 分。正如我们所见，众数也是 144。均数略低于中位数，这暗示样本有轻微的负偏态，因为低端包含 128，但总体分布相当对称。
Knowing that the typical entry requirement is around 143–144 points gives a realistic target. Most universities in this group expect the equivalent of AAA or slightly higher. This is actionable information for a student planning A-level subject choices.
了解到典型的入学要求约为 143-144 分，可以设定一个现实的目标。该组中大多数大学预期成绩相当于 AAA 或略高。这对于正在规划 A-level 科目选择的学生来说是可操作的信息。
8. Measures of Spread: Range and Interquartile Range | 离散程度测量：极差和四分位距
uirements are. The range is the simplest measure: maximum minus minimum. Here, range = 168 – = 40 points. This shows a 40-point gap between the easiest and hardest offers in our set.
集中趋势只告诉了我们部分情况。我们还需要知道入学要求的离散程度。极差是最简单的衡量方式：最大值减去最小值。这里，极差 = 168 – 128 = 40 分。这表明在我们的数据集中，最容易和最难录取标准之间有 40 分的差距。
r quartile Q3 (median of the upper half). For our 20 ordered values: Q1 is the average of the 5th and 6th values, both 136, so Q1 = 136. Q3 is the average of the 15th and 16th values, both 152, so Q3 = 152. Therefore, IQR = Q3 – Q1 = 152 – 136 = 16 points.
四分位距 (IQR) 更加稳健，因为它忽略了极值。我们首先找出下四分位数 Q1（下半部分的中位数）和上四分位数 Q3（上半部分的中位数）。对于我们的 20 个排序值：Q1 是第 5 和第 6 个值的平均数，两者都是 136，因此 Q1 = 136。Q3 是第 15 和第 16 个值的平均数，两者都是 152，因此 Q3 = 152。所以，IQR = Q3 – Q1 = 152 – 136 = 16 分。
An IQR of 16 means the middle 50% of universities set requirements within a relatively narrow band of 16 points. This suggests that for many Economics courses, the expected tariff is tightly clustered around the 136–152 range. Only a few universities demand significantly more or less.
IQR 为 16 意味着中间 50% 的大学将入学要求设定在相对较窄的 16 分区间内。这表明对于许多经济学课程，预期的学费积分紧密聚集在 136-152 范围内。只有少数大学的要求明显更高或更低。
9. Box Plots for Comparing Universities | 箱线图用于大学比较
36, median = 144, Q3 = 152, maximum = 168. Draw a box from Q1 to Q3 with a line at the median, and whiskers to the min and max.
箱线图使用五数概括：最小值、Q1、中位数、Q3 和最大值。对于我们的完整数据集：最小值 = 128，Q1 = 136，中位数 = 144，Q3 = 152，最大值 = 168。绘制一个从 Q1 到 Q3 的盒子，并在中位数处画一条线，然后画须延伸至最小值和最大值。
To truly compare two institutions, let us extract a subset. Imagine University X (a Russell Group university) gave offers to 10 of the students in our data: 144, 152, 144, 160, 136, 144, 152, 144, 168, 136. Meanwhile, University Y (a modern university) offered: 128, 136, 128, 144, 136, 136, 144, 136, 144, 136. Construct two box plots side by side.
为了真正比较两所院校，我们提取一个子集。假设大学 X（一所罗素集团大学）向我们的数据中的 10 名学生发放了录取：144, 152, 144, 160, 136, 144, 152, 144, 168, 136。同时，大学 Y（一所现代大学）发放了：128, 136, 128, 144, 136, 136, 144, 136, 144, 136。并排构建两个箱线图。
ry high offer. For University Y: min=128, Q1=136, median=136, Q3=140 (average of 136 and 144? 6,144,144,144 -> 144. So min=128, Q1=136, median=136 (average of 5th and 6th:136 and 136), Q3=144, max=144. The box plot for Y is symmetric, centred lower, and more compact.
对于大学 X：最小值=136，Q1=144，中位数=144，Q3=152，最大值=168。盒子较短，但上须较长，表明存在右偏且有一个非常高的录取要求。对于大学 Y：最小值=128，Q1=136，中位数=136，Q3=144，最大值=144。Y 的箱线图对称，中心较低，更加紧凑。
Comparing box plots instantly reveals that University X generally demands higher tariff points (median 144 vs 136) and has a wider spread. This visual tool helps students quickly assess which university might be a 'reach' or a 'safety' choice.
并排比较箱线图可以立即揭示，大学 X 普遍要求更高的学费积分（中位数 144 对比 136），且分布更广。这一可视化工具帮助学生快速判断哪所大学可能是“冲刺”或“保底”选择。
10. Interpreting Cumulative Frequency Graphs | 解读累积频数图
A cumulative frequency graph allows you to estimate the number of universities below a certain tariff point. First, we construct a cumulative frequency table. For the full 20-university dataset, we add a cumulative frequency column.
累积频数图让你可以估算低于某个学费积分的大学数量。首先，我们构建累积频数表。对于完整的 20 所大学数据集，我们添加累积频数列。
| Tariff points (≤ x) | Cumulative frequency |
|---|---|
| 128 | 2 |
| 136 | 8 |
| 144 | 15 |
|
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