📚 Other Measures of Location | 其他位置度量
In statistics, a measure of location is a single value that describes the central or typical position of a data set. While the arithmetic mean is the most familiar measure, it is not always the most appropriate choice. Other measures of location, such as the median, mode, quartiles, percentiles, and weighted mean, provide alternative ways to summarise data, especially when outliers or skewed distributions are present.
在统计学中，位置度量是描述数据集中心或典型位置的单个数值。虽然算术平均值是最常见的度量，但它并不总是最合适的选择。其他位置度量，例如中位数、众数、四分位数、百分位数和加权平均数，提供了总结数据的替代方法，尤其是在存在异常值或偏态分布时。
1. Why ‘Other’ Measures? | 为什么需要其他位置度量？
In Edexcel A-Level Statistics, a measure of location summarises the centre of a data set. The arithmetic mean, often called the average, is widely used because it uses every data value in its calculation.
在 Edexcel A-Level 统计学中，位置度量用于概括数据集的中心。算术平均数（通常称为平均值）被广泛使用，因为它在计算中使用了每一个数据值。
However, the mean is sensitive to extreme values, known as outliers. A single very large or very small observation can pull the mean away from the typical value. For this reason, we need other measures such as the median and mode that are more robust to outliers and skewed data.
然而，平均数对极端值（即异常值）很敏感。一个非常大或非常小的观测值可能使平均数偏离典型值。因此，我们需要其他度量，例如中位数和众数，它们对异常值和偏态数据更具稳健性。
Other measures of location also allow us to summarise non-numeric or categorical data, where averaging is impossible or meaningless.
其他位置度量还允许我们总结非数值或分类数据，在这些数据中取平均是不可能的或没有意义的。
2. The Median | 中位数
The median is the middle value when data are arranged in ascending order. For an odd number of observations, the median is the (n + 1)/2-th value, where n is the sample size.
中位数是将数据按升序排列后位于中间的值。当观测值个数为奇数时，中位数是第 (n + 1)/2 个值，其中 n 是样本量。
For an even number of observations, the median is the midpoint of the two central values. In grouped frequency tables, linear interpolation is used to estimate the median within the median class interval.
当观测值个数为偶数时，中位数是两个中间值的中间点。在分组频数表中，使用线性插值在中位数所在组区间内估计中位数。
Example: For data 3, 7, 8, 11, 15, median = 8; for 3, 7, 8, 11, median = (7 + 8) / 2 = 7.5.
示例：对于数据 3、7、8、11、15，中位数 = 8；对于 3、7、8、11，中位数 = (7 + 8) / 2 = 7.5。
The median is not affected by extreme values because it depends only on the ordered position of the middle observation.
中位数不受极端值影响，因为它只依赖于中间观测值的有序位置。
3. The Mode | 众数
The mode is the value that occurs most frequently in a data set. A data set can have one mode (unimodal), two modes (bimodal), or no mode if all values occur equally often.
众数是数据集中出现频率最高的值。数据集可以有一个众数（单峰）、两个众数（双峰），或者如果所有值出现次数相同则没有众数。
In grouped data, the modal class is the class interval with the highest frequency, and the mode can be estimated using a formula involving the lower boundary, class width, and adjacent frequencies.
在分组数据中，众数所在组是频率最高的组区间，可以使用包含下限、组距和相邻频率的公式来估计众数。
The mode is particularly useful for categorical data where numerical averaging is impossible, such as the most common shoe size or preferred colour.
众数对于无法进行数值平均的分类数据特别有用，例如最常见的鞋码或最喜欢的颜色。
Unlike the mean and median, the mode can be used with non-numeric data, but it may not be unique and can be less informative for small data sets.
与平均数和中位数不同，众数可用于非数值数据，但它可能不唯一，并且对于小型数据集提供的信息可能较少。
4. Quartiles and the Interquartile Range | 四分位数与四分位距
Quartiles divide an ordered data set into four equal parts. The lower quartile Q₁ is the median of the lower half, the middle quartile Q₂ is the overall median, and the upper quartile Q₃ is the median of the upper half.
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 是下半部分的中位数，中四分位数 Q₂ 是整体中位数，上四分位数 Q₃ 是上半部分的中位数。
The interquartile range (IQR) is Q₃ – Q₁. It measures the spread of the middle 50% of the data and is not affected by extreme values, making it a robust measure of spread.
四分位距 (IQR) 为 Q₃ – Q₁。它衡量中间 50% 数据的离散程度，不受极端值影响，因此是一种稳健的离散度量。
Edexcel often asks for quartiles from a cumulative frequency graph or by interpolation from grouped data. Always use the exact method required by the question.
Edexcel 常要求根据累积频率图或通过分组数据插值求四分位数。始终使用题目要求的精确方法。
Quartiles are used alongside the median to describe the centre and spread of a data set without relying on the mean or standard deviation.
四分位数与中位数一起使用，用于描述数据集的中心和离散程度，而不依赖于平均数或标准差。
5. Percentiles | 百分位数
A percentile indicates the value below which a given percentage of observations fall. For example, the 90th percentile P₉₀ is the value such that 90% of the data are less than or equal to it.
百分位数表示给定百分比的数据低于该值的界限。例如，第 90 百分位数 P₉₀ 是使得 90% 的数据小于或等于该值的值。
Percentiles are used extensively in standardised testing and growth charts. In Edexcel exams, you may need to estimate a percentile from a cumulative frequency diagram by reading across from the required percentage on the cumulative frequency axis.
百分位数广泛用于标准化考试和生长曲线图。在 Edexcel 考试中，可能需要通过从累积频率轴的所需百分比处水平读取，从累积频率图中估计百分位数。
Note that Q₁ = P₂₅, Q₂ = P₅₀ = median, and Q₃ = P₇₅. This connection helps link quartiles and percentiles.
注意 Q₁ = P₂₅、Q₂ = P₅₀ = 中位数、Q₃ = P₇₅。这种联系有助于将四分位数和百分位数联系起来。
When data are grouped, calculating a percentile usually requires interpolation within the appropriate class interval, just like estimating the median.
当数据分组时，计算百分位数通常需要在适当的组区间内进行插值，就像估计中位数一样。
6. Weighted Mean | 加权平均数
When different data values have different levels of importance, we use a weighted mean. Each value xᵢ is multiplied by its weight wᵢ, and the sum is divided by the total weight: weighted mean = Σwᵢxᵢ / Σwᵢ.
当不同数据值具有不同的重要性水平时，我们使用加权平均数。每个值 xᵢ 乘以其权重 wᵢ，总和除以总权重：加权平均数 = Σwᵢxᵢ / Σwᵢ。
For example, if coursework counts for 40% and an exam for 60%, the final score is 0.4 × coursework + 0.6 × exam. This is a weighted mean with weights 0.4 and 0.6.
例如，如果课程作业占 40%，考试占 60%，最终成绩为 0.4 × 课程作业 + 0.6 × 考试。这是权重为 0.4 和 0.6 的加权平均数。
Edexcel questions often require calculating a weighted mean from a frequency table where the frequencies act as weights, or from percentages that sum to 100%.
Edexcel 题目常要求根据频率表（其中频率充当权重）或总和为 100% 的百分比计算加权平均数。
The weighted mean gives a more accurate summary when some observations should contribute more heavily to the central value than others.
当某些观测值对中心值的贡献应大于其他观测值时，加权平均数能给出更准确的概括。
7. Geometric Mean and Its Uses | 几何平均数及其用途
The geometric mean of n positive values is the n-th root of their product: GM = (x₁ × x₂ × … × xₙ)^(1/n). It is appropriate for averaging ratios, growth rates, and index numbers.
n 个正值的几何平均数是它们乘积的 n 次方根：GM = (x₁ × x₂ × … × xₙ)^
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