Measures of Central Tendency | 集中趋势度量方法梳理

📚 Measures of Central Tendency | 集中趋势度量方法梳理

In statistics, central tendency refers to a single value that attempts to describe the centre of a data set. It is one of the first and most important summaries used when analysing data, and it is essential in A-Level Mathematics and Statistics.

在统计学中，集中趋势是指用一个数值来描述整个数据集的中心位置。它是数据分析中最基础、最重要的概括方式之一，在 A-Level 数学与统计学中十分关键。

1. What is Central Tendency? | 什么是集中趋势？

The three most common measures are the arithmetic mean, the median and the mode. The mean balances the total value of the data, the median marks the middle of the ordered list, and the mode identifies the most frequent or most typical value.

最常用的三种集中趋势度量是算术平均数（均值）、中位数与众数。均值平衡数据总和，中位数标记排序后数据的中间位置，众数则指出出现次数最多或最具代表性的数值。

In A-Level questions, you must also understand the weighted mean, the estimated mean for grouped data, the median interpolation formula and the modal class. These methods extend the same ideas to real and grouped data.

在 A-Level 题目中，你还需要理解加权平均数、分组数据的均值估计、中位数插值公式和众数所在组。这些方法将同样的思想推广到实际数据与分组数据中。

A measure of central tendency tries to give one representative value for the whole data set.

集中趋势度量试图为整个数据集给出一个有代表性的数值。
The choice of measure should always consider outliers, skewness and the type of variable.

选择哪种度量方式，必须要考虑异常值、数据偏态以及变量类型。

2. The Mean | 均值（算术平均数）

The arithmetic mean is found by adding all the data values and then dividing by the number of values. For a sample of n values x₁, x₂, …, xₙ, the sample mean is written as x̄.

算术平均数（均值）的计算方法是将所有数据值相加，再除以数据个数。对于包含 n 个数值 x₁, x₂, …, xₙ 的样本，样本均值记作 x̄。

x̄ = (x₁ + x₂ + … + xₙ) / n = Σx / n

The symbol Σ means ‘sum of’. If the data set is the whole population, we use μ instead of x̄. In most A-Level questions, the data are treated as a sample or as a set without population notation.

符号 Σ 表示“求和”。如果数据集是总体，我们用 μ 表示总体均值。在大多数 A-Level 题目中，数据通常按样本处理，因此使用 x̄。

Example: For the data 4, 8, 6, 10, 2, the sum is 30 and n = 5, so the mean is 30 ÷ 5 = 6.

示例：对于数据 4, 8, 6, 10, 2，总和为 30，n = 5，所以均值为 30 ÷ 5 = 6。

The mean uses every data value. It is easy to calculate algebraically and is the basis for variance and standard deviation. However, it is strongly affected by extreme outliers, so it may not describe skewed data well.

均值利用了每一个数据值，代数计算方便，也是方差和标准差的基础。但是，均值受极端异常值影响较大，因此在数据偏斜时不一定能很好代表整体。

3. The Median | 中位数

The median is the middle value when the data are arranged in ascending order. It is the 50th percentile and splits the data into two equal halves.

中位数是将数据按升序排列后位于正中间的值。它是第 50 百分位数，把数据分成相等的两半。

If n is odd, the median is the ((n + 1) / 2)-th ordered value. If n is even, the median is the average of the n/2-th and the (n/2 + 1)-th ordered values.

若 n 为奇数，中位数是排序后第 ((n + 1) / 2) 个值；若 n 为偶数，中位数是第 n/2 个值与第 (n/2 + 1) 个值的平均数。

Median position = (n + 1) / 2 (for an odd-sized ordered list)

Example 1: For 3, 5, 7, 9, 11, n = 5 and the median is the 3rd value, which is 7.

例 1：对于 3, 5, 7, 9, 11，n = 5，中位数是第 3 个值，即 7。

Example 2: For 3, 5, 7, 9, n = 4 and the median is (5 + 7) ÷ 2 = 6.

例 2：对于 3, 5, 7, 9，n = 4，中位数为 (5 + 7) ÷ 2 = 6。

The median is highly robust to extreme values. Changing one extreme value usually does not change the median, which makes it suitable for right-skewed or left-skewed data.

中位数对极端值具有很强的稳健性。改变一个极端值通常不会改变中位数，因此它适用于右偏或左偏的数据。

4. The Mode | 众数

The mode is the value that appears most frequently in a data set. A data set may have one mode, two modes, or more. If all values occur equally often, there is no mode.

众数是数据集中出现次数最多的数值。一组数据可能有一个众数、两个众数或多个众数。若所有数值出现次数相同，则该数据没有众数。

Example 1: For 2, 3, 3, 4, 5, the mode is 3 because it occurs twice.

例 1：对于 2, 3, 3, 4, 5，众数为 3，因为它出现了两次。

Example 2: For 1, 2, 3, 4, 5, there is no mode because every value appears once.

例 2：对于 1, 2, 3, 4, 5，没有众数，因为每个值都只出现一次。

Example 3: For 1, 1, 2, 2, 3, the data are bimodal with modes 1 and 2.

例 3：对于 1, 1, 2, 2, 3，数据是双峰的，众数为 1 和 2。

The mode is the only measure of central tendency that can be used with nominal categorical data, such as colours or transport types. However, it may not exist and it does not use all the information in the data.

众数是唯一可用于名义分类变量的集中趋势度量，例如颜色或交通工具类型。但众数可能不存在，而且它没有利用到数据的全部信息。

5. Weighted Mean | 加权平均数

Sometimes some values are more important than others. In that case, we assign each value a weight wᵢ and calculate the weighted mean rather than the simple mean.

有时某些数值比其他数值更重要。此时我们给每个数值赋予权重 wᵢ，并计算加权平均数，而不是简单平均数。

x̄_w = (w₁x₁ + w₂x₂ + … + wₙxₙ) / (w₁ + w₂ + … + wₙ) = Σwᵢxᵢ / Σwᵢ

Weighted means are common when combining coursework marks and final exam marks. If a coursework component has weight 20% and an exam component has weight 80%, the exam must influence the final mark much more strongly.

加权平均数常见于将平时作业成绩和考试成绩合成为总成绩时。如果平时作业权重为 20%，考试权重为 80%，那么考试对总成绩的影响必然大得多。

Example: Coursework mark = 70 with weight 20, exam mark = 60 with weight 80. The weighted mean is (20 × 70 + 80 × 60) ÷ (20 + 80) = (1400 + 4800) ÷ 100 = 62.

示例：平时作业 70 分，权重 20；考试 60 分，权重 80。加权平均数为 (20 × 70 + 80 × 60) ÷ (20 + 80) = (1400 + 4800) ÷ 100 = 62。

Remember that the simple mean would have been (70 + 60) ÷ 2 = 65, which overstates the final mark because it ignores the higher weight of the exam.

注意：简单平均数会是 (70 + 60) ÷ 2 = 65，这会高估总成绩，因为它忽略了考试权重更大的事实。

6. Mean for Grouped Data | 分组数据的均值估计

When raw data are grouped into classes, we no longer know the individual values. To estimate the mean, we assume every value in a class is equal to the class midpoint.

当原始数据被分成若干组后，我们不再知道每个具体数值。为了估计均值，我们假设每一组内的所有值都等于该组的组中值。

x̄ ≈ Σfᵢxᵢ / Σfᵢ

Here xᵢ is the midpoint of class i and fᵢ is the frequency of class i.

其中 xᵢ 是第 i 组的组中值，fᵢ 是第 i 组的频数。

Class \| 组别	Midpoint x \| 组中值 x	Frequency f \| 频数 f	fx
0 ≤ x < 10	5	5	25
10 ≤ x < 20	15	8	120
20 ≤ x < 30	25	12	300
30 ≤ x < 40	35	5	175

From the table, Σf = 30 and Σfx = 25 + 120 + 300 + 175 = 620. Therefore the estimated mean is 620 ÷ 30 ≈ 20.67.

由上表可知，Σf = 30，Σfx = 25 + 120 + 300 + 175 = 620。因此估计均值为 620 ÷ 30 ≈ 20.67。

This result is an estimate, not the exact mean, because the original individual values are unknown. The midpoint assumption works best when the data are roughly evenly spread within each class.

这个结果是估计值而不是精确均值，因为原始个体数值未知。当每类内的数据近似均匀分布时，组中值假设效果最好。

7. Median for Grouped Data | 分组数据的中位数估计

For grouped data, the median is found by interpolation. First construct a cumulative frequency column, then locate the class that contains the n/2-th value.

对于分组数据，中位数通过线性插值求得。首先建立累计频数列，然后找到包含第 n/2 个值的组。

Median = L + ((n/2 – F) / f) × c

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 c is the class width.

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，c 是组距。

Using the same grouped data, n = 30, so n/2 = 15. The cumulative frequencies are 5, 13, 25, 30. The median class is 20 ≤ x < 30 because its cumulative frequency first reaches 15.

使用同一组分组数据，n = 30，因此 n/2 = 15。累计频数为 5, 13, 25, 30。中位数所在组为 20 ≤ x < 30，因为该组的累计频数首次达到 15。

Here L = 20, F = 13, f = 12 and c = 10, so the median is 20

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Measures of Central Tendency | 集中趋势度量方法梳理