Measures of Spread | 离散程度的度量

📚 Measures of Spread | 离散程度的度量

In statistics, measures of spread (also called measures of dispersion) describe how similar or varied a set of values are. While measures of central tendency such as the mean and median give a typical value, the spread tells us whether the data are tightly clustered or widely scattered. For Edexcel A-Level Mathematics, you need to understand the range, interquartile range, variance, and standard deviation, and know when to use each.

在统计学中，离散程度的度量（也称为变异程度的度量）描述一组数值之间的相似性或差异性。虽然均值和中位数等集中趋势度量给出了一个典型值，但离散程度告诉我们数据是紧密聚集还是广泛分散。对于爱德思A-Level数学，你需要理解极差、四分位距、方差和标准差，并知道何时使用它们。

1. The Range | 极差

The range is the simplest measure of spread. It is calculated by subtracting the smallest value from the largest value in a data set.

极差是最简单的离散程度度量。计算方法是用数据集中的最大值减去最小值。

Range = Largest value − Smallest value

For example, for the data set {3, 7, 9, 12, 15}, the range is 15 − 3 = 12.

例如，对于数据集 {3, 7, 9, 12, 15}，极差为 15 − 3 = 12。

The range is easy to calculate but is heavily affected by outliers. A single extreme value can make the range misleadingly large, so it is often used only for a quick, rough check.

极差容易计算，但受异常值影响很大。一个极端的值可能使极差产生误导性偏大，因此通常仅用于快速粗略的检查。

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

Quartiles divide a data set into four equal parts. The lower quartile (Q₁) is the median of the lower half of the data, and the upper quartile (Q₃) is the median of the upper half. The interquartile range (IQR) is the difference between Q₃ and Q₁.

四分位数将数据集分成四个相等的部分。下四分位数（Q₁）是数据下半部分的中位数，上四分位数（Q₃）是数据上半部分的中位数。四分位距（IQR）是 Q₃ 和 Q₁ 之差。

IQR = Q₃ − Q₁

Unlike the range, the IQR only considers the middle 50% of the data, making it resistant to outliers. It is particularly useful when comparing skewed distributions.

与极差不同，四分位距只考虑中间50%的数据，因此对异常值不敏感。在比较偏态分布时尤其有用。

3. Variance | 方差

Variance measures the average squared deviation of each data point from the mean. For a population, the variance is denoted by σ², while for a sample it is denoted by s². The formula for a population variance is:

方差衡量每个数据点与均值之间的平均平方偏差。对于总体，方差记作 σ²；对于样本，记作 s²。总体方差公式为：

σ² = Σ(x − μ)² / N

where μ is the population mean, N is the population size, and Σ denotes the sum over all data points.

其中 μ 是总体均值，N 是总体大小，Σ 表示对所有数据点求和。

In A-Level statistics, you will often use the alternative computational formula, which is easier for manual calculations:

在A-Level统计中，你经常会使用另一种更便于手工计算的公式：

σ² = (Σx² / N) − μ²

This formula shows that variance is the mean of the squares minus the square of the mean.

这个公式表明方差等于平方的均值减去均值的平方。

4. Sample Variance | 样本方差

When you have a sample rather than the whole population, you use a slightly different formula. The sample variance s² is given by:

当你处理的是样本而非整个总体时，使用的公式略有不同。样本方差 s² 为：

s² = Σ(x − x̄)² / (n − 1)

where x̄ is the sample mean and n is the sample size. Dividing by n − 1 instead of n is called Bessel’s correction, which provides an unbiased estimate of the population variance.

其中 x̄ 是样本均值，n 是样本大小。除以 n − 1 而不是 n 称为贝塞尔校正，它给出总体方差的无偏估计。

In Edexcel exams, you may be given raw data and asked to calculate the sample variance. Be careful whether the question says ‘population’ or ‘sample’ — the denominator changes.

在爱德思考式中，你可能会得到原始数据并要求计算样本方差。注意题目说的是“总体”还是“样本”——分母是不同的。

5. Standard Deviation | 标准差

The standard deviation is the square root of the variance. It is denoted by σ for a population and s for a sample. Because the variance is in squared units, the standard deviation brings the measure back to the original units of the data, making it easier to interpret.

标准差是方差的平方根。总体标准差记作 σ，样本标准差记作 s。由于方差的单位是平方单位，标准差将度量恢复到数据的原始单位，使其更易于解释。

σ = √(Σ(x − μ)² / N)

For a sample:

对于样本：

s = √(Σ(x − x̄)² / (n − 1))

The standard deviation is the most widely used measure of spread, especially when the data are approximately normally distributed.

标准差是最广泛使用的离散程度度量，尤其是当数据近似正态分布时。

6. Using Σx and Σx² | 使用 Σx 与 Σx²

In many exam questions, you will be given the summary statistics Σx and Σx² rather than the raw data. From these you can calculate the mean and variance directly.

在许多考试题目中，你会被给到汇总统计量 Σx 和 Σx²，而不是原始数据。据此你可以直接计算均值和方差。

The formulas become:

公式变为：

Mean = Σx / n
Variance (population) = Σx² / n − (Σx / n)²

For a sample, the variance is multiplied by n / (n − 1) to correct the bias.

对于样本，方差需要乘以 n / (n − 1) 以修正偏差。

Example: Given Σx = 150, Σx² = 5000, n = 10, the mean is 15, and the population variance is 5000/10 − (15)² = 500 − 225 = 275. The sample variance would be 275 × 10/9 ≈ 305.56.

例如：已知 Σx = 150，Σx² = 5000，n = 10，则均值为 15，总体方差为 5000/10 − (15)² = 500 − 225 = 275。样本方差为 275 × 10/9 ≈ 305.56。

7. Effect of Linear Transformations | 线性变换的影响

If you add a constant to every data value, the mean and median increase by that constant, but the spread measures (range, IQR, variance, standard deviation) remain unchanged because the relative distances between values do not change.

如果你给每个数据值加上一个常数，均值和位数会相应增加该常数，但离散程度度量（极差、四分位距、方差、标准差）保持不变，因为值之间的相对距离没有改变。

If you multiply every data value by a constant k, the mean and standard deviation are multiplied by k, the variance is multiplied by k², and the range and IQR are multiplied by k.

如果你将每个数据值乘以常数 k，均值和标准差乘以 k，方差乘以 k²，极差和四分位距乘以 k。

These properties are frequently tested. You should be comfortable with the notation x’ = ax + b and its effect on summary statistics.

这些性质经常被考查。你应该熟悉记号 x’ = ax + b 以及它对汇总统计量的影响。

8. Coding and Data | 编码与数据

Coding is a technique used to simplify calculations, especially when the data are large or awkward. You can subtract a constant and divide by another constant to form coded data y = (x − a) / b, work out the mean and standard deviation of y, then transform back.

编码是一种用于简化计算的技术，特别是当数据很大或不便处理时。你可以减去一个常数并除以另一个常数，形成编码数据 y = (x − a) / b，求出 y 的均值和标准差，再转换回去。

The mean of x = a + b × mean of y. The standard deviation of x = |b| × standard deviation of y. Note that the variance of x = b² × variance of y.

x 的均值 = a + b × y 的均值。x 的标准差 = |b| × y 的标准差。注意 x 的方差 = b² × y 的方差。

In Edexcel past papers, coding is often used with large datasets or grouped frequency tables.

在爱德思历年真题中，编码常与大型数据集或分组频数表一起使用。

9. Measures of Spread for Grouped Data | 分组数据的离散程度度量

For a grouped frequency table with class midpoints x (after coding if necessary) and frequencies f, the mean is given by:

对于具有组中点 x（必要时编码后）和频数 f 的分组频数表，均值由下式给出：

x̄ = Σfx / Σf

The variance is given by:

方差为：

σ² = (Σfx² / Σf) − (x̄)²

For a sample variance, multiply by Σf / (Σf − 1).

对于样本方差，乘以 Σf / (Σf − 1)。

When calculating from a grouped table, remember that the actual values are not known; we approximate them using the midpoints. This introduces a small error, but it is acceptable for estimation.

当从分组表计算时，请记住实际值是未知的；我们使用组中点来近似。这引入了小误差，但用于估计是可以接受的。

10. Percentiles and Box Plots | 百分位数与箱线图

Percentiles divide data into 100 equal parts. The 25th percentile is the lower quartile Q₁, the 50th percentile is the median, and the 75th percentile is the upper quartile Q₃. A box plot (or box-and-whisker plot) visually displays the minimum, Q₁, median, Q₃, and maximum of a data set.

百分位数将数据分成100个相等的部分。第25百分位数是下四分位数 Q₁，第50百分位数是中位数，第75百分位数是上四分位数 Q₃。箱线图（也称为箱须图）直观地显示数据集的最小值、Q₁、中位数、Q₃ 和最大值。

Box plots are useful for comparing two distributions. If one box is longer than another, it indicates greater spread in the middle 50% of the data. Outliers may be shown as separate points beyond the whiskers.

箱线图用于比较两个分布。如果一个箱体比另一个更长，说明中间50%的数据离散程度更大。异常值可能显示为须线之外的点。

You should be able to read a box plot to find the median and interquartile range, and also sketch a box plot from summary statistics.

你应该能够从箱线图读取中位数和四分位距，也能根据汇总统计量绘制箱线图。

11. Choosing the Right Measure | 选择合适的度量

The choice of measure of spread depends on the data and the context.

离散程度度量的选择取决于数据和具体情境。

Measure	Advantage	Disadvantage
Range	Easy to calculate	Affected by outliers
IQR	Not affected by outliers	Uses only middle 50%
Variance / SD	Uses all data; mathematically useful	Affected by outliers; less intuitive

In general, if the data have extreme values or are skewed, the IQR is preferred. For roughly symmetric data, the standard deviation is the best choice because it uses every data point.

一般来说，如果数据存在极端值或偏态，则优先选择四分位距。对于近似对称的数据，标准差是最佳选择，因为它使用了每个数据点。

12. Common Exam Pitfalls | 常见考试陷阱

Students often make the following mistakes:

学生常犯以下错误：

Forgetting to square the deviations before summing them when calculating variance.
Using the population variance formula when the question asks for the sample variance (dividing by n rather than n − 1).
Confusing the median with quartiles when sorting data.
Neglecting to use the coded mean and variance when transforming back to original units.
Misreading a box plot, especially the whiskers and outliers.

计算方差时忘记先对偏差平方再求和。

当题目要求样本方差时却使用总体方差公式（除以 n 而不是 n − 1）。

排序数据时混淆中位数与四分位数。

转换回原始单位时忘记使用编码后的均值和方差。

误读箱线图，尤其是须线和异常值。

To avoid these errors, always write down the formula you are using, check whether you are dealing with a sample or a population, and verify your calculations with a quick estimate.

为了避免这些错误，始终写下你使用的公式，检查你处理的是样本还是总体，并通过快速估算来验证你的计算。

Understanding measures of spread is fundamental to statistical analysis. Mastering these formulas and their applications will help you tackle both mechanics and statistics questions in your Edexcel A-Level exam with confidence.

理解离散程度的度量是统计分析的基础。掌握这些公式及其应用将帮助你在爱德思A-Level考试中自信地处理统计问题。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Measures of Spread | 离散程度的度量