📚 Mode, Median, Percentiles and Skewness | 众数、中位数、百分位数与偏度
This unit covers the key measures of location and the concept of skewness in data distributions. You will learn how to calculate and interpret the mode, median, percentiles and quartiles, and how to use them to describe whether data are symmetric, positively skewed or negatively skewed.
本单元涵盖关键的位置度量以及数据分布中的偏度概念。你将学习如何计算和解释众数、中位数、百分位数和四分位数，并利用它们描述数据是对称、正偏还是负偏。
1. Why Measures of Location Matter | 为什么位置度量很重要
A measure of location summarises the centre of a data set with a single representative value. The three most common measures are the mode, the median and the mean, and each has advantages and disadvantages depending on the data type and distribution shape.
位置度量用一个具有代表性的数值概括数据集的中心。最常见的三种度量是众数、中位数和平均数，每种度量根据数据类型和分布形状各有优缺点。
In Edexcel A-Level Statistics, questions often ask you to choose the most appropriate measure and to justify your decision. For example, the median is preferred for skewed data because it is not pulled by extreme values, while the mode is the only measure that can be used for non-numeric categorical data.
在爱德思 A-Level 统计学中，题目经常要求你选择最合适的度量并说明理由。例如，对于偏斜数据，中位数更合适，因为它不受极端值影响；而众数是唯一可用于非数值分类数据的度量。
2. Mode: The Most Frequent Value | 众数：出现频率最高的值
The mode is the value or category that occurs most frequently in a data set. A data set may have one mode, more than one mode, or no mode at all if all values occur with equal frequency.
众数是数据集中出现频率最高的数值或类别。一个数据集可能有一个众数、多个众数，或者如果所有数值出现频率相同，则没有众数。
The mode is especially useful for categorical data, such as colour or type of transport, where calculating a mean or median would be impossible. However, for continuous numerical data, the mode can be less informative because it depends on how the data are grouped.
众数对于分类数据特别有用，例如颜色或交通方式，在这些情况下计算平均数或中位数是不可能的。然而，对于连续数值数据，众数的信息量可能较低，因为它取决于数据的分组方式。
3. Median: The Middle Value | 中位数：中间值
The median is the middle value when the data are arranged in ascending order. For raw data with n observations, the position of the median is given by (n + 1)/2. If n is odd, the median is the value in this position; if n is even, the median is the mean of the two middle values.
中位数是将数据按升序排列后位于中间的数值。对于有 n 个观测值的原始数据，中位数的位置由 (n + 1)/2 给出。如果 n 为奇数，中位数就是该位置的数值；如果 n 为偶数，中位数是两个中间数值的平均数。
Median position for raw data = (n + 1) / 2
原始数据的中位数位置 = (n + 1) / 2
For grouped data, the median is estimated by interpolation within the median class. The formula used in Edexcel is:
对于分组数据，中位数通过在中位数所在组内进行插值来估计。爱德思考试中使用的公式为：
Median = L + ((n/2 − F) / f) × w
中位数 = L + ((n/2 − F) / f) × w
Here L is the lower class boundary of the median class, F is the cumulative frequency before the median class, f is the frequency of the median class, and w is the class width. Always check whether the data are raw or grouped before choosing the method.
其中 L 是中位数所在组的下边界，F 是中位数所在组之前的累计频数，f 是中位数所在组的频数，w 是组距。在选择方法之前，务必检查数据是原始数据还是分组数据。
4. Mean vs Median: Key Differences | 平均数与中位数：关键区别
The mean uses every data value and is calculated as the sum of all values divided by the number of values. It is the most commonly used measure of location for symmetric distributions, but it is sensitive to outliers and extreme values.
平均数使用每一个数据值，计算方式为所有数值之和除以数值的个数。它是对称分布中最常用的位置度量，但对异常值和极端值很敏感。
The median is resistant to outliers because it depends only on the order of the data, not on the actual size of extreme values. This makes the median a better choice when the distribution is skewed or when outliers are present.
中位数对异常值具有抵抗力，因为它只取决于数据的顺序，而不取决于极端值的实际大小。因此，当分布偏斜或存在异常值时，中位数是更好的选择。
5. Quartiles and Percentiles | 四分位数与百分位数
Percentiles divide an ordered data set into 100 equal parts. The kth percentile, denoted Pₖ, is the value below which k% of the data lie. For raw data, the position of Pₖ is often estimated by k(n + 1)/100.
百分位数将有序数据集分成 100 等份。第 k 个百分位数记为 Pₖ，是使得 k% 的数据落在其下方的数值。对于原始数据，Pₖ 的位置通常用 k(n + 1)/100 来估计。
Pₖ position = k(n + 1) / 100
百分位数位置 = k(n + 1) / 100
Quartiles are special percentiles: Q₁ = P₂₅, Q₂ = P₅₀ = median, and Q₃ = P₇₅. For grouped data, quartiles and percentiles are found by interpolation, using the lower boundary of the appropriate class and the cumulative frequency before that class.
四分位数是特殊的百分位数：Q₁ = P₂₅，Q₂ = P₅₀ = 中位数，Q₃ = P₇₅。对于分组数据，四分位数和百分位数通过插值法求得，使用相应组的下边界和该组之前的累计频数。
Grouped percentile: Pₖ = L + ((kn/100 − F) / f) × w
分组百分位数：Pₖ = L + ((kn/100 − F) / f) × w
6. Interquartile Range and Percentile Range | 四分位距与百分位距
The interquartile range, or IQR, is the difference between the upper quartile and the lower quartile. It measures the spread of the middle 50% of the data and is not affected by outliers.
四分位距（IQR）是上四分位数与下四分位数之差。它衡量数据中间 50% 的分散程度，并且不受异常值影响。
IQR = Q₃ − Q₁
四分位距 = Q₃ − Q₁
A percentile range, such as the 10th to 90th percentile range, is sometimes used to describe the spread of the central 80% of data. Like the IQR, it reduces the influence of extreme values at both ends of the distribution.
百分位距，例如第 10 到第 90 百分位距，有时用于描述数据中间 80% 的分散程度。与四分位距一样，它减少了分布两端极端值的影响。
7. Introduction to Skewness | 偏度简介
Skewness describes the asymmetry of a distribution. A distribution is symmetric if the left and right sides are mirror images. A distribution is positively skewed if the right tail is longer, and negatively skewed if the left tail is longer.
偏度描述分布的不对称性。如果左右两侧互为镜像，则分布是对称的。如果右尾更长，则分布为正偏；如果左尾更长，则分布为负偏。
| Shape | 形状 | Tail direction | 尾部方向 | Mean vs median | 平均数与中位数 |
|---|---|---|
| Symmetric | 对称 | Both tails equal | 两尾相等 | Mean ≈ median | 平均数 ≈ 中位数 |
| Positive skew | 正偏 | Right tail longer | 右尾更长 | Mean > median | 平均数 > 中位数 |
| Negative skew | 负偏 | Left tail longer | 左尾更长 | Mean < median | 平均数 < 中位数 |
Skewness is important because it tells you which measure of location gives the best summary of the data. In a positively skewed distribution, the mean is pulled to the right by high values, so the median is often a more representative measure of centre.
偏度很重要，因为它能告诉你哪种位置度量最能概括数据。在正偏分布中，平均数被高值拉向右侧，因此中位数通常是更具代表性的中心度量。
8. Skewness and the Mean–Median–Mode Relationship | 偏度与平均数–中位数–众数关系
For a perfectly symmetric distribution, the mean, median and mode are approximately equal. For a positively skewed distribution, the mean is usually greater than the median, which is greater than the mode. For a negatively skewed distribution, the order is reversed.
对于完全对称的分布，平均数、中位数和众数大致相等。对于正偏分布，平均数通常大于中位数，中位数大于众数。对于负偏分布，顺序则相反。
Symmetric: mean ≈ median ≈ mode
对称：平均数 ≈ 中位数 ≈ 众数
Positive skew: mean > median > mode
正偏：平均数 > 中位数 > 众数
Negative skew: mean < median < mode
负偏：平均数 < 中位数 < 众数
This relationship is a useful quick check in exam questions. If you are given the mean, median and mode, you can identify the likely skewness without drawing the distribution.
这种关系是考试题目中一个有用的快速判断方法。如果给出平均数、中位数和众数，你就可以在不绘制分布的情况下判断可能的偏度。
9. Quartile-Based Skewness | 基于四分位数的偏度
Because quartiles are resistant to outliers, they provide a robust way to assess skewness. Compare the distance from Q₁ to Q₂ with the distance from Q₂ to Q₃. If Q₃ − Q₂ is greater than Q₂ − Q₁, the distribution is positively skewed; if it is smaller, the distribution is negatively skewed.
由于四分位数对异常值具有抵抗力，它们提供了一种稳健的偏度评估方法。比较 Q₁ 到 Q₂ 的距离与 Q₂ 到 Q₃ 的距离。如果 Q₃ − Q₂ 大于 Q₂ − Q₁，则分布为正偏；如果较小，则分布为负偏。
Quartile skewness coefficient = (Q₃ − 2Q₂ + Q₁) / (Q₃ − Q₁)
四分位偏度系数 = (Q₃ − 2Q₂ + Q₁) / (Q₃ − Q₁)
The quartile skewness coefficient takes values between −1 and 1. A value of 0 indicates symmetry, a positive value indicates positive skew, and a negative value indicates negative skew. Edexcel questions may not require this formula, but you should understand the idea behind comparing quartile gaps.
四分位偏度系数的取值范围在 −1 到 1 之间。取值为 0 表示对称，正值表示正偏，负值表示负偏。爱德思考试可能不要求这个公式，但你应该理解比较四分位间距背后的思想。
10. Pearson’s Coefficient of Skewness | 皮尔逊偏度系数
Pearson’s coefficient of skewness uses the mean, median or mode together with the standard deviation to give a numerical measure of skewness. There are two common versions:
皮尔逊偏度系数使用平均数、中位数或众数以及标准差来给出偏度的数值度量。常见的两种形式为：
Pearson’s first coefficient = (mean − mode) / s
皮尔逊第一偏度系数 = (平均数 − 众数) / s
Pearson’s second coefficient = 3(mean − median) / s
皮尔逊第二偏度系数 = 3(平均数 − 中位数) / s
Here s represents the standard deviation. The second coefficient is often preferred because the median is more stable than the mode. A positive value means positive skew, a negative value means negative skew, and a value close to 0 indicates an approximately symmetric distribution.
其中 s 表示标准差。第二系数通常更受青睐，因为中位数比众数更稳定。正值表示正偏，负值表示负偏，接近 0 的值表示分布近似对称。
11. Comparing Distributions Using Box Plots | 使用箱线图比较分布
A box plot displays the minimum, Q₁, median, Q₃ and maximum, and sometimes outliers. The position of the median inside the box and the lengths of the whiskers give a clear visual indication of skewness.
箱线图显示最小值、Q₁、中位数、Q₃ 和最大值，有时还显示异常值。中位数在箱内的位置以及须的长度可以清楚地显示偏度。
- If the median is in the centre of the box and the whiskers are roughly equal, the distribution is symmetric.
- 如果中位数位于箱子的中心且两条须大致相等，则分布是对称的。
- If the median is closer to Q₁ and the upper whisker is longer, the distribution is positively skewed.
- 如果中位数更靠近 Q₁ 且上须更长，则分布为正偏。
- If the median is closer to Q₃ and the lower whisker is longer, the distribution is negatively skewed.
- 如果中位数更靠近 Q₃ 且下须更长，则分布为负偏。
When comparing two distributions using box plots, comment on the median, the interquartile range, the range and the skewness. Always put your comparison in the context of the data.
使用箱线图比较两个分布时，要评论中位数、四分位距、极差和偏度。始终将比较放在数据背景中进行。
12. Exam Tips and Common Errors | 考试技巧与常见错误
Always check whether the data are raw, grouped or already summarised in a cumulative frequency table. The median position formula differs for raw and grouped data, and using the wrong formula is a common mistake.
务必检查数据是原始数据、分组数据还是已经在累计频数表中汇总。原始数据和分组数据的中位数位置公式不同，使用错误公式是常见错误。
Do not confuse the median position with the median value. After calculating the position, you must return to the ordered data or the grouped frequency table to find the actual median value.
不要将中位数的位置与中位数本身混淆。计算位置后，必须回到有序数据或分组频数表中找出实际的中位数值。
When estimating quartiles from a cumulative frequency diagram, read the values carefully at 25%, 50% and 75% of the total frequency. For skewness, refer to the relative positions of the mean, median and mode, or use the shape of the box plot.
从累计频数图估计四分位数时，要在总频数的 25%、50% 和 75% 处仔细读取数值。对于偏度，要参考平均数、中位数和众数的相对位置，或使用箱线图的形状。
Finally, always state your answers in context. For example, instead of saying ‘the distribution is positively skewed’, say ‘the distribution of waiting times is
Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导