📚 Measures of Location and Spread | 位置与离散程度的度量
In Edexcel A-Level Mathematics, measures of location summarise the central position of a data set, while measures of spread describe how far the values are scattered around that centre. Both are essential for comparing distributions and for interpreting statistical information in real contexts.
在 Edexcel A-Level 数学中，位置度量用于概括数据集的中心位置，而离散程度度量则描述数值围绕中心分布的分散程度。两者对于比较数据分布以及在真实情境中解释统计信息都至关重要。
1. Key Measures of Location: Mean, Median and Mode | 主要位置度量：均值、中位数和众数
For a list of n observations x₁, x₂, ..., xₙ, the arithmetic mean is given by x̄ ata and uses every observation. The median is the value in the middle when the data are ordered; for n odd it is the (n+1)/2th value, and for n even it is the midpoint of the two central values. The mode is the value with the highest frequency, which may not be unique if several values tie for the highest frequency.
对于包含 n 个观测值的列表 x₁, x₂, ..., xₙ，算术均值由 x̄ = Σx/n 给出。它是数据的平衡点，并且使用了每一个观测值。中位数是将数据按大小排列后位于中间的值；当 n 为奇数时，它是第 (n+1)/2 个值，当 n 为偶数时，它是中间两个值的平均值。众数是出现频率最高的值，如果有多个值并列最高频率，则众数可能不唯一。
ta set 3, 5, 7, 7, 9, the mean is (3+5+7+7+9)/5 = 6.2, the median is 7, and the mode is 7.
例如：对于数据集 3, 5, 7, 7, 9，均值为 (3+5+7+7+9)/5 = 6.2，中位数为 7，众数为 7。
2. Choosing Between Mean, Median and Mode | 均值、中位数和众数的选择
The mean is normally preferred for roughly symmetric data because it uses all observations and supports further statistical inference. However, it is sensitive to extreme values and can give a misleading centre when the distribution is skewed. The median is more robust and is often used for house prices or incomes, where a small number of very large values could distort the mean.
对于大致对称的数据，通常优先使用均值，因为它使用了所有观测值，并且支持进一步的统计推断。然而，均值对极端值敏感，当分布偏斜时可能给出具有误导性的中心位置。中位数更为稳健，通常用于房价或收入等数据，因为少数非常大的值可能会扭曲均值。
The mode is mainly useful for qualitative data or for identifying the most common category. A data set can be unimodal, bimodal or multimodal, and in grouped data we refer to the modal class rather than an exact mode.
众数主要用于定性数据或识别最常见的类别。一个数据集可以是单峰的、双峰的或多峰的；在分组数据中，我们通常指众数所在组，而不是精确的众数值。
3. Measures of Spread: Range and Interquartile Range | 离散程度度量：极差和四分位距
late but depends only on the two extreme values, so a single outlier can make it very large.
极差是最大值与最小值之差：极差 = max(x) − min(x)。它计算简单，但仅取决于两个极端值，因此单个异常值就可能使其变得非常大。
The interquartile range (IQR) = Q₃ − Q₁ measures the spread of the middle 50% is the median of the upper half. The IQR is not affected by extreme values at either end.
四分位距 IQR = Q₃ − Q₁ 衡量数据中间 50% 的离散程度。为求四分位数，先将数据排序并找出中位数；然后 Q₁ 是下半部分的中位数，Q₃ 是上半部分的中位数。四分位距不受两端极端值的影响。
4. Percentiles and Box Plots | 百分位数和箱线图
A percentile Pₖ is the value below which k% of the data falls. The median is the 50th percentile, Q₁ is the 25th percentile and Q₃ is the 75th percentile. For large grouped data, linear interpolation estimates percentiles from a cumulative frequency graph or formula.
百分位数 Pₖ 是指有 k% 的数据落在其下方的值。中位数是第 50 百分位数，Q₁ 是第 25 百分位数，Q₃ 是第 75 百分位数。对于大型分组数据，可以使用累积频率图或公式通过线性插值来估计百分位数。
A box plot (box-and-whisker diagram) displays the minimum, Q₁, median, Q₃ and maximum. It is useful for comparing asymmetry and spread, but it does not show the mean or the detailed shape inside the box. When outliers are identified, whiskers usually extend to the most extreme non-outlier values, and outliers are plotted individually.
箱线图（盒须图）显示最小值、Q₁、中位数、Q₃ 和最大值。它有助于比较偏斜程度和离散程度，但不显示均值，也不显示箱体内部的详细形状。当识别出异常值时，须通常延伸到最极端的非异常值，异常值单独绘制。
5. Variance and Standard Deviation for Raw Data | 原始数据的方差和标准差
Variance measures the average squared distance from the mean: σ² = Σ(x − x̄)²/n. The standard deviation is its square root: σ = √[Σ(x − x̄ ance has squared units. For hand calculation, the computational formula is usually faster.
方差衡量数据与均值的平均平方距离：σ² = Σ(x − x̄)²/n。标准差是其平方根：σ = √[Σ(x − x̄)²/n]。标准差与原始数据的单位相同，而方差的单位是原始单位的平方。手算时，通常使用计算公式更快。
σ² = Σx²/n − (Σx/n)²
This formula gives the population variance for a complete data set. For a sample, some courses use n − 1 as the divisor; in Edexcel A-Level Mathematics, the divisor n is used unless the question states otherwise.
该公式给出完整数据集的总方差。对于样本，某些课程使用 n − 1 作为除数；在 Edexcel A-Level 数学中，除非题目另有说明，否则使用除数 n。
mean is 5, Σx² = 4 + 16 + 36 + 64 = 120, n = 4, so σ² = 120/4 − 5² = 30 − 25 = 5, and σ = √5 ≈ 2.24.
例如：对于数据 2, 4, 6, 8，均值为 5，Σx² = 4 + 16 + 36 + 64 = 120，n = 4，因此 σ² = 120/4 − 5² = 30 − 25 = 5，σ = √5 ≈ 2.24。
6. Using Frequency Tables for Mean and Standard Deviation | 使用频数表计算均值和标准差
When data are given in a frequency table with values x and frequencies f, the mean is x̄ = Σfx/Σf. The variance is calculated using Σfx and Σfx², where Σf is the total frequency.
当数据以频数表给出，包含数值 x 和频数 f 时，均值为 x̄ = Σfx/Σf。方差使用 Σfx 和 Σfx² 计算，其中 Σf 是总频数。
σ² = Σfx²/Σf − (Σfx/Σf)²
This avoids finding each individual deviation from the mean and is essential when data are summarised in tables. Always check that the mean is sensible before calculating the standard deviation.
这样可避免逐一计算每个值与均值的偏差，当数据以表格汇总时这种方法非常重要。在计算标准差之前，请先检查均值是否合理。
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