Mode, Median, Percentiles and Skewness | 众数、中位数、百分位数与偏度

📚 Mode, Median, Percentiles and Skewness | 众数、中位数、百分位数与偏度

In A-Level Further Mathematics, summarising data requires not only measuring the centre but also understanding the shape of a distribution. The mode, median and percentiles provide robust ways to describe location, while skewness tells us how symmetric, or asymmetric, the data really are.

在进阶数学中，描述数据集不仅需要衡量中心位置，更要理解数据的分布形态。众数、中位数与百分位数提供了稳健的位置度量，而偏度则揭示了数据是近似对称还是存在明显偏移。

1. The Mode | 众数

The mode is the value that occurs most frequently in a data set. For discrete data, we simply count frequencies and pick the value with the highest frequency. A data set may have one mode, two modes (bimodal), or more.

众数是一组数据中出现频率最高的数值。对于离散数据，我们只需统计频数并选取频数最大的值。一组数据可以有一个众数、两个众数（双峰）或更多。

For example, in the set {2, 3, 3, 5, 7, 7, 7, 9}, the mode is 7 because it appears three times.

例如，在集合 {2, 3, 3, 5, 7, 7, 7, 9} 中，众数为 7，因为它出现了三次。

For grouped continuous data, the modal class is the class interval with the highest frequency density, not simply the highest frequency if class widths differ.

对于分组连续数据，众数所在组是频率密度最大的组别。当组距不同时，不能仅凭频数大小判断。

2. The Median | 中位数

The median is the middle value when data are arranged in ascending order. It divides the data set into two equal halves. For n ordered values, the median is the (n + 1)/2-th value if n is odd, and the average of the (n/2)-th and (n/2 + 1)-th values if n is even.

中位数是将数据按升序排列后的中间值，它将数据集分为相等的两半。对于 n 个有序数值，若 n 为奇数，中位数是第 (n + 1)/2 个数值；若 n 为偶数，中位数是第 n/2 个与第 (n/2 + 1) 个数值的平均值。

For example, in {1, 4, 6, 8, 9}, the median is 6. In {1, 4, 6, 8}, the median is (4 + 6)/2 = 5.

例如，在 {1, 4, 6, 8, 9} 中，中位数为 6；在 {1, 4, 6, 8} 中，中位数为 (4 + 6)/2 = 5。

The median is resistant to extreme values, making it a preferred measure of centre for skewed distributions.

中位数不受极端值影响，因此在偏态分布中常被优先用作中心位置的度量。

3. Percentiles | 百分位数

The p-th percentile is the value below which p% of the data lie. For example, the 25th percentile is called the lower quartile (Q₁), the 50th percentile is the median (Q₂), and the 75th percentile is the upper quartile (Q₃).

第 p 百分位数是指有 p% 的数据小于或等于该值。例如，第 25 百分位数称为下四分位数（Q₁），第 50 百分位数即中位数（Q₂），第 75 百分位数称为上四分位数（Q₃）。

For a data set of size n, the position of the k-th percentile Pₖ is often estimated using the formula:

对于容量为 n 的数据集，第 k 百分位数的位置通常用以下公式估算：

Position of Pₖ = (k/100) × (n + 1)

If the resulting position is not an integer, linear interpolation between adjacent ordered values is used.

若计算出的位置不是整数，则在相邻有序数值之间进行线性插值。

4. Quartiles and the Interquartile Range | 四分位数与四分位距

Quartiles split the ordered data into four equal parts. The interquartile range (IQR) is defined as Q₃ − Q₁, which measures the spread of the middle 50% of the data.

四分位数将有序数据分成四个等份。四分位距（IQR）定义为 Q₃ − Q₁，它衡量了中间 50% 数据的离散程度。

For grouped continuous data, we use interpolation within the appropriate class interval. Given a cumulative frequency table, the formula for a quartile is:

对于分组连续数据，我们在相应组内使用插值。给定累积频数表，四分位数的公式为：

Q = L + [ ( (k/4) × n − F ) / f ] × w

where L is the lower class boundary of the quartile class, F is the cumulative frequency before that class, f is the frequency of that class, w is the class width, and k = 1, 2, 3 for Q₁, Q₂, Q₃ respectively.

其中 L 为四分位数所在组的下边界，F 为该组之前的累积频数，f 为该组频数，w 为组距，k 分别取 1、2、3 对应 Q₁、Q₂、Q₃。

5. Calculating Percentiles from Grouped Data | 分组数据中的百分位数计算

To find any percentile, first determine the cumulative frequency. Then locate the class interval containing the desired rank ( p/100 × n ). Use the same interpolation structure as above, replacing (k/4) × n with (p/100) × n.

要计算任意百分位数，首先建立累积频数，然后确定包含目标秩次 (p/100 × n) 的组别，并使用与上述相同的插值结构，将 (k/4) × n 替换为 (p/100) × n。

Example: The weights (kg) of 50 students are summarised as follows:

示例：50 名学生的体重（kg）汇总如下：

Weight (kg)	Frequency	Cumulative Frequency
40 – 49	5	5
50 – 59	12	17
60 – 69	20	37
70 – 79	9	46
80 – 89	4	50

For Q₁ (k = 1), rank = (1/4) × 50 = 12.5. The quartile class is 50 – 59 because the cumulative frequency reaches 17 there. Using L = 50, F = 5, f = 12, w = 10:

对于 Q₁（k = 1），秩次 = (1/4) × 50 = 12.5。由于累积频数在该组达到 17，四分位数所在组为 50 – 59。取 L = 50、F = 5、f = 12、w = 10：

Q₁ = 50 + ((12.5 − 5) / 12) × 10 = 50 + 6.25 = 56.25 kg

6. Interpreting Cumulative Frequency Graphs | 累积频率图的解读

Cumulative frequency graphs (ogives) are used to estimate the median, quartiles and percentiles. Plot the upper class boundaries against cumulative frequency, draw a smooth curve, and read values horizontally.

累积频率图（拱形图）常用于估算中位数、四分位数和百分位数。将上组界与累积频数描点并绘制平滑曲线，然后水平读取相应数值。

To find the median, locate n/2 on the vertical axis, draw a horizontal line to the curve, then drop a vertical line to the x-axis. Similar procedures apply for Q₁ at n/4 and Q₃ at 3n/4.

求中位数时，在纵轴上找到 n/2，画水平线与曲线相交，再向横轴作垂线。求 Q₁ 和 Q₃ 时，分别在 n/4 和 3n/4 处重复相同步骤。

7. Skewness: Definition and Visual Meaning | 偏度：定义与直观意义

Skewness describes the asymmetry of a frequency distribution. A symmetric distribution has zero skewness; the left and right tails are balanced. Positive skewness (right skew) means the right tail is longer and the bulk of data lies on the left. Negative skewness (left skew) means the left tail is longer.

偏度描述的是频率分布的不对称程度。对称分布的偏度为零，左右尾部平衡。正偏（右偏）表示右尾更长，数据主体位于左侧；负偏（左偏）则表示左尾更长。

For a right-skewed distribution, the mean is typically greater than the median, which is greater than the mode. The mean is pulled toward the long tail by extreme high values.

在右偏分布中，通常有均值大于中位数、中位数大于众数的关系。均值受到极端大值的影响而被拉向长尾方向。

8. Measuring Skewness | 偏度的度量

Pearson’s first coefficient of skewness uses the mode:

皮尔逊第一偏度系数使用众数：

Skewness = (mean − mode) / standard deviation

Pearson’s second coefficient uses the median and is more robust:

皮尔逊第二偏度系数使用中位数，因此更具稳健性：

Skewness = 3(mean − median) / standard deviation

For the quartile measure (Bowley’s skewness):

四分位偏度（鲍利偏度）为：

Skewness = (Q₃ + Q₁ − 2×Q₂) / (Q₃ − Q₁)

These coefficients are dimensionless. Positive values indicate positive skew, negative values indicate negative skew, and zero indicates symmetry.

这些系数均为无量纲量。正值表示正偏，负值表示负偏，零表示对称。

9. Relationship Between Mean, Median and Mode | 均值、中位数与众数的关系

In a symmetric distribution, mean = median = mode. In a positively skewed distribution, mode < median < mean. In a negatively skewed distribution, mean < median < mode.

在对称分布中，均值 = 中位数 = 众数。在正偏分布中，众数 < 中位数 < 均值。在负偏分布中，均值 < 中位数 < 众数。

This relationship is an easy and powerful way to determine the direction of skewness quickly. However, it is only a rough rule and does not give the magnitude of skewness.

这个关系是快速判断偏度方向的简单而有效的方法。但它只是一个粗略规则，不能给出偏度的大小。

In Edexcel problems, you may be given two of the three measures and asked to deduce skewness. For instance, if mean = 25, median = 30, then mean < median, so the data are negatively skewed.

在 Edexcel 考题中，有时会给出三个度量中的两个并要求判断偏度。例如，若均值 = 25，中位数 = 30，则均值 < 中位数，数据负偏。

10. Box Plots and Skewness | 箱线图与偏度

A box plot (box-and-whisker diagram) visually displays the minimum, Q₁, median, Q₃ and maximum. The positions of the median within the box and the lengths of the whiskers indicate skewness.

箱线图（箱须图）直观展示最小值、Q₁、中位数、Q₃ 和最大值。中位数在箱中的位置以及须的长度可指示偏度。

If the median is closer to Q₁, the distribution is positively skewed. If it is closer to Q₃, the distribution is negatively skewed. A symmetric distribution has the median near the centre of the box and equal whisker lengths.

若中位数更靠近 Q₁，则分布正偏；若更靠近 Q₃，则分布负偏。对称分布的中位数位于箱体中心附近，且左右须长度相等。

11. Histograms, Frequency Density and the Mode | 直方图、频率密度与众数

For continuous data with unequal class widths, frequency density = frequency / class width. The mode is estimated from the histogram by constructing diagonal lines within the tallest rectangle to find the modal value.

对于组距不等的连续数据，频率密度 = 频数 / 组距。众数的估计方法是在直方图最高的矩形内作交叉对角线，交点对应的横坐标即为众数。

The formula for the mode of grouped data using the modal class is:

分组数据众数的计算公式为：

Mode = L + [ d₁ / (d₁ + d₂) ] × w

where L is the lower boundary of the modal class, d₁ is the difference between the modal class frequency and the previous class frequency, d₂ is the difference between the modal class frequency and the next class frequency, and w is the modal class width.

其中 L 为众数组的下边界，d₁ 为众数组频数与前一组的频数之差，d₂ 为众数组频数与后一组的频数之差，w 为众数组的组距。

12. Common Mistakes and Exam Tips | 常见错误与考试提示

One common error is using the midpoint of a class as the mode for continuous data without considering frequency density. Another is misplacing the quartile position: for grouped data, use (k/4) × n, not (n+1)/4.

常见错误之一是在连续数据中直接使用组中值作为众数，而未考虑频率密度；另一个错误是四分位数位置：对于分组数据，应使用 (k/4) × n，而非 (n+1)/4。

When using interpolation, always use class boundaries, not class midpoints, and verify your results with a cumulative frequency graph.

使用插值公式时，务必使用组界而非组中值，并利用累积频率图检验结果。

Always order discrete data before finding the median.
For skewed data, prefer the median and IQR over the mean and standard deviation.
Check whether a question asks for percentiles (P₅₀) or quartiles (Q₂) — they are the same value but written differently.

求离散数据中位数前务必先排序。
面对偏斜数据，优先使用中位数和 IQR，而非均值和标准差。
注意题目要求写百分位数（P₅₀）还是四分位数（Q₂）——两者数值相同但写法不同。

Pearson’s coefficients can exceed ±1; they are not bounded by ±1. Do not confuse the sign with the magnitude of data values.

皮尔逊偏度系数可以超过 ±1，它并不受限于 ±1 区间。不要将系数的符号与数据值的大小混淆。

In summary, the mode, median and percentiles give a complete picture of a distribution’s central tendency and spread, while skewness summarises its shape. Mastering these measures is essential for interpreting data in Edexcel A-Level Further Mathematics and for tackling exam questions efficiently.

总之，众数、中位数和百分位数提供了分布中心位置与离散程度的完整图像，而偏度概括了其形态。掌握这些度量是 Edexcel A-Level 进阶数学中解读数据、高效解答考题的关键。

Published by TutorHao | Further Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Mode, Median, Percentiles and Skewness | 众数、中位数、百分位数与偏度