📚 Other measures of location | 其他位置度量
In statistics, measures of location are used to describe the centre of a data set. While the mean, median and mode are the most commonly discussed, there are several other important measures that give a more detailed picture of the distribution. These include quartiles, percentiles, deciles and interpolated estimates for grouped data. This article focuses on those other measures of location that appear regularly in the Edexcel A-Level Mathematics syllabus, helping you understand how to find and interpret them with confidence.
在统计学中，位置度量用于描述数据集的中心。虽然均值、中位数和众数是最常讨论的，但还有其他重要的度量可以提供更细致的分布情况。这些度量包括四分位数、百分位数、十分位数以及分组数据的插值估计。本文重点介绍这些经常出现在 Edexcel A-Level 数学大纲中的其他位置度量，帮助你理解如何查找和解读它们，并增强信心。
1. Introduction to measures of location | 位置度量简介
A measure of location is a single value that summarises a central position within a data set. The three fundamental measures are the mean, the median and the mode, but they do not tell the whole story. When data is skewed or contains outliers, other measures such as quartiles and percentiles provide a more robust way to understand the spread and shape of the data. These additional measures are especially useful when constructing box plots, comparing distributions, and working with grouped frequency tables.
位置度量是概括数据集中某个中心位置的单一数值。三个基本的度量是均值、中位数和众数，但它们不能反映全部情况。当数据有偏态或包含异常值时，其他度量如四分位数和百分位数提供了更稳健的方式来理解数据的散布和形状。这些额外的度量在构建箱线图、比较分布以及处理分组频率表时特别有用。
In the Edexcel specification, you are expected to calculate quartiles for raw data, discrete frequency tables, and to estimate them for grouped continuous data using linear interpolation. You should also be familiar with percentiles and be able to interpret them in context. This article will break down each of these techniques with clear examples and explanations.
在 Edexcel 大纲中，你需要能够计算原始数据、离散频率表的四分位数，并能够使用线性插值法估计分组连续数据的四分位数。你还应该熟悉百分位数，并能在具体情境中加以解读。本文将借助清晰的示例和说明逐一分解这些技巧。
2. The mean vs. the median | 均值与中位数对比
Before diving into other measures, it is worth revisiting the difference between the mean and the median. The mean is computed by summing all data values and dividing by the number of observations. The median is the middle value when the data is arranged in order. The mean is sensitive to extreme values, whereas the median is resistant to them. Because of this, the median is often a better measure of location for skewed data.
在深入探讨其他度量之前，有必要重温一下均值和中位数之间的区别。均值是将所有数据值相加再除以观测值个数得出的。中位数是数据按顺序排列后的中间值。均值对极端值敏感，而中位数对极端值具有抵抗力。因此，对于偏态数据，中位数往往是更好的位置度量。
For example, consider the data set: 4, 5, 6, 7, 100. The mean is (4+5+6+7+100)/5 = 24.4, which is heavily pulled upwards by the outlier 100. The median is 6, which accurately reflects the centre of the bulk of the data. This limitation of the mean motivates the use of measures like quartiles, which are also resistant to outliers and provide a fuller summary.
例如，考虑数据集：4, 5, 6, 7, 100。均值为 (4+5+6+7+100)/5 = 24.4，被异常值 100 大幅拉高。中位数为 6，准确地反映了主要数据的中心。均值的这一局限性促使我们使用像四分位数这样的度量，它们同样对异常值稳健，并且能提供更全面的概括。
3. Quartiles: dividing data into four parts | 四分位数：将数据分成四部分
Quartiles split an ordered data set into four equal parts. The lower quartile (Q1) is the value below which 25% of the data falls. The median (Q2) splits the data into two halves, and the upper quartile (Q3) is the value below which 75% of the data falls. Together with the minimum and maximum, the quartiles form the five-number summary used in box plots.
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) 是数据中有 25% 的数值低于它的那个值。中位数 (Q2) 将数据分成两半，上四分位数 (Q3) 是数据中有 75% 的数值低于它的那个值。四分位数与最小值和最大值一起，构成了用于箱线图的五数概括。
There are several methods for finding quartiles, especially when the sample size is small. In Edexcel A-Level, when dealing with a small data set, you may use the convention of taking the median of the lower half for Q1 and the median of the upper half for Q3, discarding the median itself from both halves if the number of data points is odd. For larger discrete frequency tables, cumulative frequencies are used.
查找四分位数有几种方法，尤其是在样本量较小的时候。在 Edexcel A-Level 中，处理小数据集时，通常采用的方法是取下半部分的中位数作为 Q1，取上半部分的中位数作为 Q3，如果数据点个数为奇数，则从两部分中都剔除整体的中位数。对于较大的离散频率表，则需要使用累积频率。
4. Finding quartiles for discrete data | 求离散数据的四分位数
When data is given as a simple list, sort it in ascending order. To find the position of the median and quartiles, use the formulas: position of Q2 = (n+1)/2, position of Q1 = (n+1)/4 and position of Q3 = 3(n+1)/4. If the position is a whole number, the quartile is the value at that position. If it ends in .5, take the mean of the two surrounding values.
当数据以简单的列表形式给出时，将其按升序排序。要找到中位数和四分位数的位置，可以使用公式：Q2 的位置 = (n+1)/2，Q1 的位置 = (n+1)/4，Q3 的位置 = 3(n+1)/4。如果位置是整数，四分位数就是该位置上的值。如果位置以 0.5 结尾，则取周围两个数值的均值。
For example, consider the ordered data: 3, 5, 7, 8, 12, 15, 18, 19, 21 (n=9). The median position is (9+1)/2 = 5, so Q2 = 12. The lower quartile position is (9+1)/4 = 2.5, so Q1 is the mean of the 2nd and 3rd values: (5+7)/2 = 6. The upper quartile position is 3(9+1)/4 = 7.5, so Q3 = (18+19)/2 = 18.5.
例如，考虑已排序的数据：3, 5, 7, 8, 12, 15, 18, 19, 21 (n=9)。中位数位置为 (9+1)/2 = 5，所以 Q2 = 12。下四分位数位置为 (9+1)/4 = 2.5，因此 Q1 是第 2 和第 3 个值的均值：(5+7)/2 = 6。上四分位数位置为 3(9+1)/4 = 7.5，因此 Q3 = (18+19)/2 = 18.5。
5. Percentiles and other fractiles | 百分位数及其他分位数
Quartiles are special cases of a broader family called fractiles. A fractile divides an ordered data set into equal groups. Percentiles split the data into 100 parts. The pth percentile is the value below which p% of the data falls. For instance, the 10th percentile (P10) leaves 10% of the data below it. The median is the 50th percentile, Q1 is the 25th percentile, and Q3 is the 75th percentile.
四分位数是更广泛的分位数家族的特例。分位数将有序数据集分成相等的组。百分位数将数据分成 100 个部分。第 p 百分位数是有 p% 数据低于它的数值。例如，第 10 百分位数 (P10) 使得 10% 的数据低于它。中位数就是第 50 百分位数，Q1 是第 25 百分位数，Q3 是第 75 百分位数。
Deciles are another common type of fractile; they divide the data into 10 parts. The first decile D1 is the 10th percentile, D2 is the 20th percentile, and so on. In Edexcel exams, you might be asked to estimate a specific percentile from a cumulative frequency graph, or to calculate it from grouped data using interpolation.
十分位数是另一种常见的分位数类型；它们将数据分成 10 个部分。第一个十分位数 D1 是第 10 百分位数，D2 是第 20 百分位数，依此类推。在 Edexcel 考试中，你可能会被要求从累积频率图中估计某个特定的百分位数，或者使用插值法根据分组数据计算百分位数。
6. Interpolation to estimate quartiles from grouped data | 用插值法估计分组数据的四分位数
When data is presented in a grouped frequency table without the original values, we cannot pinpoint the exact quartiles. Instead, we use linear interpolation to estimate them. This method assumes that the data points within each class interval are uniformly distributed. The formula for any fractile is: estimate = L + ((k – cfprev) / f) × w, where L is the lower class boundary, k is the target cumulative frequency, cfprev is the cumulative frequency before the class, f is the frequency of the class, and w is the class width.
当数据以分组频率表的形式呈现且没有原始数值时，我们无法精确确定四分位数。此时我们使用线性插值法进行估计。该方法假设每个组距内的数据点是均匀分布的。任何分位数的公式为：估计值 = L + ((k – cfprev) / f) × w，其中 L 是组下限，k 是目标累积频率，cfprev 是该组之前的累积频率，f 是该组的频率，w 是组距宽度。
To find the median, k = n/2. For the lower quartile, k = n/4, and for the upper quartile, k = 3n/4. The process involves constructing a cumulative frequency column, identifying the class in which k falls, and then applying the formula. This is a key skill tested in the Edexcel statistics papers.
要找到中位数，k = n/2。对于下四分位数，k = n/4；对于上四分位数，k = 3n/4。这个过程包括构建累积频率列，确定 k 落在哪个组中，然后应用公式。这是 Edexcel 统计试卷中考查的关键技能。
7. Step-by-step interpolation for the median | 逐步插值求中位数
Let’s work through an example. Suppose a frequency table shows the heights of 80 plants: 0–10 cm: 5 plants; 10–20 cm: 12; 20–30 cm: 28; 30–40 cm: 20; 40–50 cm: 15. First, calculate cumulative frequencies: 5, 17, 45, 65, 80. The median position is n/2 = 40. The 40th value falls in the class 20–30 cm, because the cumulative frequency before this class is 17, and 40 is between 17 and 45. The lower boundary L = 20, f = 28, cfprev = 17, w = 10.
让我们通过一个例子来详细说明。假设一个频率表显示了 80 株植物的高度：0–10 cm：5 株；10–20 cm：12 株；20–30 cm：28 株；30–40 cm：20 株；40–50 cm：15 株。首先计算累积频率：5, 17, 45, 65, 80。中位数的位置是 n/2 = 40。第 40 个值落在 20–30 cm 组内，因为该组之前的累积频率为 17，而 40 处于 17 和 45 之间。下限 L = 20，f = 28，cfprev = 17，w = 10。
Plug these into the formula: Median = 20 + ((40 – 17) / 28) × 10 = 20 + (23/28) × 10 = 20 + 8.214 = 28.2 cm (to 1 d.p.). This means we estimate that half of the plants are shorter than 28.2 cm. Notice how the interpolation spreads the 28 plants evenly across the 10 cm interval.
将这些数值代入公式：中位数 = 20 + ((40 – 17) / 28) × 10 = 20 + (23/28) × 10 = 20 + 8.214 = 28.2 cm（保留 1 位小数）。这意味着我们估计一半的植物高度低于 28.2 cm。注意插值是如何将 28 株植物均匀分布在这 10 cm 的区间内的。
8. Interpolation for the lower and upper quartiles | 下四分位数和上四分位数的插值
Using the same data, estimate Q1. The position is n/4 = 80/4 = 20. The cumulative frequency before the 20–30 cm class is 17, and its cumulative reaches 45. Thus Q1 also lies in 20–30 cm, with L = 20, f = 28, cfprev = 17. Q1 = 20 + ((20 – 17) / 28) × 10 = 20 + (3/28) × 10 ≈ 20 + 1.071 = 21.1 cm. For Q3, k = 3n/4 = 60. The 60th value is in the 30–40 cm class (cf before = 45, cf after = 65). Here L = 30, f = 20, cfprev = 45, w = 10. Q3 = 30 + ((60 – 45) / 20) × 10 = 30 + (15/20) × 10 = 30 + 7.5 = 37.5 cm.
使用同样的数据，估计 Q1。位置为 n/4 = 80/4 = 20。20–30 cm 组之前的累积频率为 17，该组的累积达到 45。因此 Q1 也落在 20–30 cm 组内，L = 20，f = 28，cfprev = 17。Q1 = 20 + ((20 – 17) / 28) × 10 = 20 + (3/28) × 10 ≈ 20 + 1.071 = 21.1 cm。对于 Q3，k = 3n/4 = 60。第 60 个值在 30–40 cm 组内（该组前 cf = 45，该组后 cf = 65）。这里 L = 30，f = 20，cfprev = 45，w = 10。Q3 = 30 + ((60 – 45) / 20) × 10 = 30 + (15/20) × 10 = 30 + 7.5 = 37.5 cm。
The interquartile range (IQR) is Q3 – Q1 = 37.5 – 21.1 = 16.4 cm. This gives a measure of spread that is unaffected by extreme values. Checking the five-number summary: min = 0, Q1 ≈ 21.1, median ≈ 28.2, Q3 = 37.5, max = 50. This summary can be used to draw a box plot.
四分位距 (IQR) 为 Q3 – Q1 = 37.5 – 21.1 = 16.4 cm。这提供了一个不受极端值影响的散布度量。检查五数概括：最小值 = 0，Q1 ≈ 21.1，中位数 ≈ 28.2，Q3 = 37.5，最大值 = 50。这个概括可用于绘制箱线图。
9. Deciles and percentiles from grouped data | 从分组数据求十分位数和百分位数
The interpolation method extends naturally to any fractile. To estimate the pth percentile, first compute k = (p/100) × n. For example, to find the 90th percentile in the plant data, k = (90/100) × 80 = 72. The 72nd value falls in the 40–50 cm class, because cumulative frequency before is 65, and k = 72 is between 65 and 80. Then use L = 40, f = 15, cfprev = 65, w = 10. P90 = 40 + ((72 – 65) / 15) × 10 = 40 + (7/15) × 10 ≈ 40 + 4.67 = 44.7 cm.
插值方法可以自然地扩展到任何分位数。要估计第 p 百分位数，首先计算 k = (p/100) × n。例如，要在植物数据中求第 90 百分位数，k = (90/100) × 80 = 72。第 72 个值落在 40–50 cm 组内，因为该组之前的累积频率为 65，且 k = 72 处于 65 和 80 之间。然后使用 L = 40，f = 15，cfprev = 65，w = 10。P90 = 40 + ((72 – 65) / 15) × 10 = 40 + (7/15) × 10 ≈ 40 + 4.67 = 44.7 cm。
Deciles are handled in a similar way; for D4, k = (4/10) × n = 0.4 × 80 = 32. This falls in the 20–30 cm class. D4 = 20 + ((32 – 17) / 28) × 10 ≈ 20 + 5.357 = 25.4 cm. These measures are helpful for creating cumulative frequency curves and for interpreting position within a distribution without assuming symmetry.
十分位数以类似的方式处理；对于 D4，k = (4/10) × n = 0.4 × 80 = 32。这落在 20–30 cm 组内。D4 = 20 + ((32 – 17) / 28) × 10 ≈ 20 + 5.357 = 25.4 cm。这些度量有助于绘制累积频率曲线，并在不假设对称性的情况下解读分布中的位置。
10. Comparing measures of location | 位置度量的比较
Choosing the most appropriate measure of location depends on the data’s distribution and the presence of outliers. The mean is useful when the data is symmetric and free from extreme values. The median is preferred in skewed distributions or when outliers exist. Percentiles and quartiles give a more detailed picture and are essential for constructing box plots and understanding variability.
选择最合适的位置度量取决于数据的分布以及是否存在异常值。当数据对称且没有极端值时，均值十分有用。在偏态分布或存在异常值时，中位数更为可取。百分位数和四分位数提供了更详细的描述，对于构建箱线图和理解变异性必不可少。
In real-world contexts, such as house prices or income data, the median and percentiles are commonly reported because they are not distorted by a few very high values. For instance, the 10th and 90th percentiles of income give a better sense of inequality than the mean alone. The Edexcel exam may ask you to justify your choice of measure, so be ready to explain these properties.
在现实世界中，例如房价或收入数据，通常报告中位数和百分位数，因为它们不会受到少数极高值的扭曲。例如，收入的第 10 和第 90 百分位数比单一的均值更能反映不平等程度。Edexcel 考试可能会要求你论证所选度量的合理性，因此请准备好解释这些性质。
11. Worked example: continuous data | 实例：连续数据
A second worked example ties everything together. The table shows the time (in minutes) spent on homework by 50 students: 0–15 min: 6 students; 15–30 min: 10; 30–45 min: 18; 45–60 min: 12; 60–90 min: 4. Construct cumulative frequencies: 6, 16, 34, 46, 50. Estimate the median, interquartile range, and the 80th percentile.
第二个实例将所有内容整合在一起。下表显示了 50 名学生花在家庭作业上的时间（分钟）：0–15 分钟：6 名学生；15–30 分钟：10 名；30–45 分钟：18 名；45–60 分钟：12 名；60–90 分钟：4 名。构建累积频率：6, 16, 34, 46, 50。估计中位数、四分位距和第 80 百分位数。
Median: k = 50/2 = 25. The 25th value is in the 30–45 min class. L = 30, cfprev = 16, f = 18, w = 15. Median = 30 + ((25 – 16) / 18) × 15 = 30 + (9/18) × 15 = 30 + 7.5 = 37.5 minutes. Q1: k = 50/4 = 12.5. This is in the 15–30 min class, L = 15, cfprev = 6, f = 10. Q1 = 15 + ((12.5 – 6) / 10) × 15 = 15 + (6.5/10) × 15 = 15 + 9.75 = 24.75 minutes. Q3: k = 37.5, falls in the 45–60 min class. L = 45, cfprev = 34, f = 12. Q3 = 45 + ((37.5 – 34) / 12) × 15 = 45 + (3.5/12) × 15 ≈ 45 + 4.375 = 49.4 minutes (1 d.p.). IQR = 49.4 – 24.75 = 24.65 minutes.
中位数：k = 50/2 = 25。第 25 个值位于 30–45 分钟组内。L = 30，cfprev = 16，f = 18，w = 15。中位数 = 30 + ((25 – 16) / 18) × 15 = 30 + (9/18) × 15 = 30 + 7.5 = 37.5 分钟。Q1：k = 50/4 = 12.5。这落在 15–30 分钟组内，L = 15，cfprev = 6，f = 10。Q1 = 15 + ((12.5 – 6) / 10) × 15 = 15 + (6.5/10) × 15 = 15 + 9.75 = 24.75 分钟。Q3：k = 37.5，落在 45–60 分钟组内。L = 45，cfprev = 34，f = 12。Q3 = 45 + ((37.5 – 34) / 12) × 15 = 45 + (3.5/12) × 15 ≈ 45 + 4.375 = 49.4 分钟（保留 1 位小数）。IQR = 49.4 – 24.75 = 24.65 分钟。
The 80th percentile: k = 80% × 50 = 40. This is in the 45–60 min class with L = 45, cfprev = 34, f = 12. P80 = 45 + ((40 – 34) / 12) × 15 = 45 + (6/12) × 15 = 45 + 7.5 = 52.5 minutes. Interpreting these results, we can say that 50% of students spent between about 24.8 and 49.4 minutes, and 80% spent less than 52.5 minutes on homework.
第 80 百分位数：k = 80% × 50 = 40。这落在 45–60 分钟组内，L = 45，cfprev = 34，f = 12。P80 = 45 + ((40 – 34) / 12) × 15 = 45 + (6/12) × 15 = 45 + 7.5 = 52.5 分钟。解读这些结果，我们可以说 50% 的学生花费的时间大约在 24.8 到 49.4 分钟之间，并且 80% 的学生花费的时间少于 52.5 分钟。
12. Summary and exam tips | 总结与考试技巧
Other measures of location, such as quartiles, percentiles and deciles, enrich our understanding of a data set beyond the mean and median. Mastering linear interpolation for grouped data is essential for Edexcel A-Level Mathematics. Always set out your working clearly: construct the cumulative frequency table, identify the correct class, and substitute carefully into the interpolation formula.
其他位置度量，如四分位数、百分位数和十分位数，丰富了我们对数据集的理解，超越了均值和中位数。掌握分组数据的线性插值对 Edexcel A-Level 数学至关重要。始终清晰地展示你的解题过程：构建累积频率表，确定正确的组，并仔细代入插值公式。
In the exam, double-check your values for k. Use n/2 for the median, n/4 for Q1 and 3n/4 for Q3. For percentiles, multiply the decimal fraction by n. Remember that linear interpolation is an estimation method; your answers should be given to a sensible degree of accuracy. Practise with different class widths and boundary conventions, as these can appear in real exam questions.
在考试中，仔细检查你的 k 值。中位数用 n/2，下四分位数用 n/4，上四分位数用 3n/4。对于百分位数，将小数分数乘以 n。请记住，线性插值是一种估计方法；你的答案应给出合理的精度。练习不同的组距以及组限的惯例，因为它们可能出现在真实考试题中。
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