A-Level Mathematics: Comprehensive Guide to Other Measures of Location | A-Level 数学：其他位置度量方法梳理

📚 A-Level Mathematics: Comprehensive Guide to Other Measures of Location | A-Level 数学：其他位置度量方法梳理

In descriptive statistics, a measure of location gives a single value that summarises a key positional feature of a data set. The mean and median are the most familiar measures, but many A-Level examination questions also require confidence with quartiles, percentiles, deciles, the mode and weighted averages. This guide offers a structured revision of these ‘other’ measures, including definitions, formulas, worked examples and common traps.

在描述统计学中，位置度量是用一个数值来概括数据集的某种位置特征。均数和中位数最为常见，但许多 A-Level 考试题还要求你熟练处理四分位数、百分位数、十分位数、众数和加权平均数。本指南系统梳理这些”其他”位置度量方法，涵盖定义、公式、例题与常见易错点，帮助你高效复习。

1. Introduction to Measures of Location | 位置度量方法导论

A measure of location summarises where the data are concentrated. The mean, median and mode are called measures of central tendency because they locate the centre of a data set. Quartiles, percentiles and deciles are also measures of location because they describe the position of an observation within the ordered data set.

位置度量用于概括数据集中所在的位置。均值、中位数和众数被称为集中趋势度量，因为它们定位数据集的中心。四分位数、百分位数和十分位数同样属于位置度量，因为它们描述某个观测值在有序数据集中的位置。

The mean can be pulled towards extreme values, while the median is resistant to outliers. Quartiles and percentiles give a fuller picture of the distribution, especially when the data are skewed or when we need thresholds such as the top 10% of candidates.

均值容易受极端值影响，而中位数则不受离群值干扰。四分位数和百分位数能更全面地反映分布形态，尤其适用于偏斜分布，或当我们需要划定排名阈值（如前 10% 的考生）时非常重要。

2. Quartiles and Percentiles: Definitions | 四分位数与百分位数的定义

Quartiles divide an ordered data set into four equal parts. The lower quartile Q₁ leaves approximately 25% of the data below it, the median Q₂ leaves 50% below it, and the upper quartile Q₃ leaves 75% below it.

四分位数将有序数据集分成四个相等的部分。下四分位数 Q₁ 约为 25% 的数据在其之下，中位数 Q₂ 约为 50% 的数据在其之下，上四分位数 Q₃ 约为 75% 的数据在其之下。

Percentiles divide data into 100 equal parts. The k-th percentile, written Pₖ, is the value below which k percent of the data lie. For example, P₉₀ is the value below which 90% of the data lie. Deciles divide data into 10 equal parts and are denoted D₁, D₂, …, D₉.

百分位数将数据分成 100 个相等的部分。第 k 百分位数记作 Pₖ，表示有 k% 的数据位于该值之下。例如，P₉₀ 就是 90% 的数据位于其下的数值。十分位数将数据分成 10 个相等的部分，记为 D₁, D₂, …, D₉。

The difference Q₃ − Q₁ is the interquartile range (IQR). Although IQR is a measure of spread, it is often calculated immediately after locating the quartiles, so you must be confident finding Q₁ and Q₃.

Q₃ − Q₁ 称为四分位距（IQR）。虽然四分位距是离散程度度量，但通常在求出四分位数后立刻计算，因此你必须熟练掌握 Q₁ 和 Q₃ 的求法。

3. Finding Quartiles from Discrete Data | 离散数据中的四分位数

Before finding quartiles, always sort the data in ascending order. For a discrete data set with n values, a common A-Level convention uses these positions:

在求四分位数之前，一定要先将数据按升序排列。对于含有 n 个观测值的离散数据集，一种常见的 A-Level 约定使用以下位置：

Q₁ position = (n + 1) / 4, Q₂ position = (n + 1) / 2, Q₃ position = 3(n + 1) / 4

If the position is an integer, take the corresponding data value. If the position is not an integer, many specifications suggest interpolating between the two nearest data values, while others instruct you to round up. Always check your exam board’s formula booklet and convention.

如果这个位置是整数，就直接取对应的数据值。如果位置不是整数，许多考试局建议在两个相邻数据值之间插值，而有些考试局则规定向上取整。请务必查阅你所在考试局的公式册和约定。

Example: For the ordered data set 4, 6, 8, 10, 12, 14, 16, we have n = 7.

例：对于有序数据 4, 6, 8, 10, 12, 14, 16，n = 7。

Q₁ position = (7 + 1) / 4 = 2, so Q₁ = 6.

Q₁ 的位置 = (7 + 1) / 4 = 2，因此 Q₁ = 6。
Q₂ position = (7 + 1) / 2 = 4, so Q₂ = 10.

Q₂ 的位置 = (7 + 1) / 2 = 4，因此 Q₂ = 10。
Q₃ position = 3(7 + 1) / 4 = 6, so Q₃ = 14.

Q₃ 的位置 = 3(7 + 1) / 4 = 6，因此 Q₃ = 14。

Notice that Q₂ is exactly the median, so Q₁ and Q₃ describe how the lower and upper halves of the data are positioned.

注意 Q₂ 正好是中位数，所以 Q₁ 和 Q₃ 描述的是数据的下半部分和上半部分所处的位置。

4. Interpolation for Grouped Data | 分组数据的插值法

When data are grouped into intervals, individual values are lost, so we estimate quartiles using linear interpolation on cumulative frequency. Let L be the lower class boundary of the quartile class, F the cumulative frequency before that class, f the frequency of that class, and c the class width.

当数据按区间分组后，原始数值已经丢失，因此我们使用累计频率进行线性插值来估计四分位数。设 L 为四分位数所在组的下边界，F 为该组之前的累计频数，f 为该组的频数，c 为组宽。

Q₁ = L + ( (n/4 − F) / f ) × c

Q₂ = L + ( (n/2 − F) / f ) × c

Q₃ = L + ( (3n/4 − F) / f ) × c

Consider the following frequency table for 50 student heights.

考虑以下 50 名学生身高的频数表。

Height (cm)	Frequency	Cumulative frequency
140 – 149	5	5
150 – 159	12	17
160 – 169	20	37
170 – 179	8	45
180 – 189	5	50

For Q₁, position = 50/4 = 12.5. The first cumulative frequency that reaches at least 12.5 is 17, so the Q₁ class is 150 – 159, with boundaries 149.5 and 159.5.

对于 Q₁，位置 = 50/4 = 12.5。第一个达到至少 12.5 的累计频数是 17，因此 Q₁ 所在组为 150 – 159，其边界为 149.5 和 159.5。

Q₁ = 149.5 + ( (12.5 − 5) / 12 ) × 10 = 149.5 + 6.25 = 155.75 cm

For Q₃, position = 3 × 50 / 4 = 37.5. The cumulative frequency before the 170 – 179 class is 37, so the Q₃ class is 170 – 179.

对于 Q₃，位置 = 3 × 50 / 4 = 37.5。170 – 179 组之前的累计频数是 37，因此 Q₃ 所在组为 170 – 179。

Q₃ = 169.5 + ( (37.5 − 37) / 8 ) × 10 = 169.5 + 0.625 = 170.125 cm

5. Percentiles and Deciles | 百分位数与十分位数

For grouped data, the k-th percentile can be estimated by the same interpolation logic:

对于分组数据，第 k 百分位数的估计仍然使用相同的插值逻辑：

Pₖ = L + ( (kn/100 − F) / f ) × c

Here L, F, f and c are defined for the class that contains the kn/100-th value. For deciles, replace kn/100 by kn/10.

这里的 L、F、f 和 c 是针对包含第 kn/100 个数据的组而言。对于十分位数，将 kn/100 换成 kn/10 即可。

In the height example, to estimate P₉₀, position = 90 × 50 / 100 = 45. The cumulative frequency before the 180 – 189 class is 45, so P₉₀ lies at the lower boundary of that class, 179.5 cm. This makes sense because the 45th value is the first value in the final class.

在身高例子中，要估计 P₉₀，位置 = 90 × 50 / 100 = 45。180 – 189 组之前的累计频数是 45，所以 P₉₀ 位于该组的下边界处，即 179.5 cm。这是合理的，因为第 45 个数据恰好是最后一组的第一个数据。

Percentiles are especially useful for reporting rank-based results, such as “the top 10% of students scored at least 82 marks”.

百分位数特别适合报告基于排名的结果，例如”排名前 10% 的学生至少获得了 82 分”。

6. The Mode and Modal Class | 众数与模态组

The mode is the value that occurs most frequently in a data set. A data set may have one mode, more than one mode, or no mode if all values appear with the same frequency.

众数是一组数据中出现次数最多的数值。一组数据可以有一个众数、多个众数，如果所有数值出现的次数相同，也可能没有众数。

For grouped data, the modal class is the class interval with the highest frequency. We usually quote the modal class rather than a single estimated value.

对于分组数据，模态组就是频数最高的组。通常我们直接给出模态组，而不是估计一个单一数值。

In the height table above, the class 160 – 169 has the largest frequency, 20, so the modal class is 160 – 169 cm.

在上面的身高表中，160 – 169 组频数最高，为 20，因此模态组为 160 – 169 cm。

The mode is simple to interpret, but it is not always a good positional measure. If the data are roughly symmetric and unimodal, the mean, median and mode are close to each other; if the data are skewed, they can differ greatly.

众数易于理解，但并非总是好的位置度量。如果数据近似对称且只有一个峰值，那么均值、中位数和众数十分接近；如果数据偏斜，它们可能相差很大。

7. Weighted Mean | 加权平均数

A weighted mean is used when different data values have different levels of importance. If the values are x₁, x₂, …, xₙ with corresponding weights w₁, w₂, …, wₙ, then the weighted mean is:

当不同数据值的重要程度不同时，需要使用加权平均数。若数据值为 x₁, x₂, …, xₙ，对应权重为 w₁, w₂, …, wₙ，则加权平均数为：

Weighted mean = (Σ wᵢxᵢ) / (Σ wᵢ) = (w₁x₁ + w₂x₂ + … + wₙxₙ) / (w₁ + w₂ + … + wₙ)

Example: A coursework score of 70% has weight 30%, and an examination score of 80% has weight 70%. The weighted mean is:

例：某课程作业成绩为 70%，权重为 30%；考试成绩为 80%，权重为 70%。加权平均数为：

(70 × 0.3 + 80 × 0.7) / (0.3 + 0.7) = (21 + 56) / 1 = 77%

A common mistake is to average 70 and 80 to get 75%. Because the exam has greater weight, the weighted mean is 77%, which is closer to the examination mark.

一个常见错误是直接取 70 和 80 的平均值得 75%。由于考试成绩权重更大，加权平均数为 77%，更接近考试成绩。

8. Choosing the Right Measure | 如何选择合适的度量

The mean should be used when the data are reasonably symmetric and contain no extreme outliers. It uses all data values and is algebraically convenient.

当数据比较对称且没有极端离群值时，应使用均值。均值利用了所有数据值，并且代数性质良好。

The median is preferred when the data are skewed or contain outliers, because it is based on position rather than on every individual value.

当数据偏斜或存在离群值时，应优先使用中位数，因为它基于位置而非所有个体值。

The mode is useful when we need the most typical category, especially for qualitative data. However, it may be unstable if a data set has several similar frequencies.

当我们需要最具代表性的类别时，尤其是定类数据，众数很有用。不过，如果多个频数接近，众数可能不稳定。

Quartiles and percentiles are helpful when reporting thresholds: bottom 25%, median benchmark, top 10%, etc. They are also used to compute the interquartile range and to detect outliers.

四分位数和百分位数适合报告阈值：最低 25%、中位数基准、前 10% 等。它们也用于计算四分位距、识别离群值。

9. Worked Example: Combined Measures | 综合例题

Consider the ordered discrete data set: 15, 18, 20, 22, 25, 28, 30, 33, 36. We have n = 9.

考虑有序离散数据集：15, 18, 20, 22, 25, 28, 30, 33, 36。这里 n = 9。

First calculate the mean:

首先计算均值：

Mean = (15 + 18 + 20 + 22 + 25 + 28 + 30 + 33 + 36) / 9 = 227 / 9 = 25.22

Median position = (9 + 1) / 2 = 5, so the median is the 5th value:

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A-Level Mathematics: Comprehensive Guide to Other Measures of Location | A-Level 数学：其他位置度量方法梳理