Variance and Standard Deviation | 方差与标准差

📚 Variance and Standard Deviation | 方差与标准差

Variance and standard deviation are fundamental measures of dispersion in statistics, quantifying how spread out a data set is around its mean. Understanding these concepts is essential for A-Level Edexcel Mathematics, as they underpin hypothesis testing, regression analysis and probability distributions.

方差和标准差是统计学中度量离散程度的基本指标，用来衡量数据集相对于其均值的分散程度。掌握这些概念对于爱德思A-Level数学至关重要，因为它们是假设检验、回归分析和概率分布的基础。

1. Introduction to Variance and Standard Deviation | 方差与标准差简介

In statistics, simply knowing the average of a data set is not enough to describe its distribution. Two data sets can have the same mean but very different spreads. Variance measures the average squared deviation from the mean, while standard deviation is its square root, providing a measure in the original units of the data.

在统计学中，仅仅知道数据集的平均值不足以描述其分布。两个数据集可以有相同的均值，但分散程度可能截然不同。方差衡量的是各数据点与均值之差的平方的平均值，而标准差是方差的平方根，提供了以原始数据单位表示的离散度量。

Standard deviation is particularly useful because it expresses variability in the same units as the data, making it easier to interpret. A low standard deviation indicates that data points tend to be close to the mean, whereas a high standard deviation indicates that data points are spread out over a wider range.

标准差的实用性在于它用与数据相同的单位来表示变异性，因此更容易理解。标准差小表示数据点倾向于聚集在均值附近，而标准差大则表示数据点分布在较宽的范围内。

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2. Population Variance: Definition and Formula | 总体方差：定义与公式

For a population of size N with values x₁, x₂, …, xₙ, the population mean is denoted by μ. The population variance, σ², is the mean of the squared deviations from μ. The formula is:

对于一个大小为N的总体，其数值为x₁、x₂、…、xₙ，总体均值记为μ。总体方差σ²是各数据与μ的偏差平方的平均值。公式如下：

σ² = Σ(xᵢ − μ)² / N

Here Σ represents the sum over all i from 1 to N. Each deviation (xᵢ − μ) is squared to ensure negative differences do not cancel out positive ones, and then the squared deviations are averaged.

其中Σ表示对所有i从1到N求和。每个偏差(xᵢ − μ)被平方，以确保负偏差不会与正偏差相互抵消，然后对平方后的偏差求平均值。

It is essential to note that σ² is always non-negative. A variance of zero means all data values are identical. The larger the variance, the more spread out the data.

必须注意σ²始终是非负的。方差为零意味着所有数据值都相同。方差越大，数据越分散。

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3. Population Standard Deviation | 总体标准差

The standard deviation is simply the positive square root of the variance. For a population, it is denoted by σ. By taking the square root, we return to the original units of measurement, which aids interpretation.

标准差就是方差的正平方根。对于总体，用σ表示。通过开平方根，我们回到了原始的计量单位，这有助于解释。

σ = √σ² = √[ Σ(xᵢ − μ)² / N ]

While variance is in squared units (e.g., cm² if the data are in cm), the standard deviation is in the same units as the data (cm). This makes standard deviation the preferred measure for reporting spread in most real-world applications.

方差使用的是平方单位（例如，如果数据以厘米为单位，则方差单位为平方厘米），而标准差使用与数据相同的单位（厘米）。这使得标准差成为大多数实际应用中报告离散程度的优选度量。

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4. Sample Variance: Why n-1? | 样本方差：为何除以n-1？

When dealing with a sample rather than a whole population, we estimate the population variance using the sample variance s². The formula divides by (n − 1) instead of n, where n is the sample size. This correction, known as Bessel’s correction, ensures that s² is an unbiased estimator of the population variance σ².

当处理样本而非整个总体时，我们使用样本方差s²来估计总体方差。公式除以(n − 1)而不是n，其中n为样本大小。这一修正称为贝塞尔校正，它确保s²是总体方差σ²的无偏估计量。

s² = Σ(xᵢ − x̄)² / (n − 1)

Here x̄ is the sample mean. Using n − 1 accounts for the fact that the sample mean is used in place of the unknown population mean, which consumes one degree of freedom. Without this adjustment, the sample variance would systematically underestimate the true variance.

其中x̄为样本均值。使用n − 1是因为用样本均值代替了未知的总体均值，这消耗了一个自由度。如果不进行这种调整，样本方差将系统性地低估真实方差。

In A-Level examinations, you must identify whether a data set represents a population or a sample before choosing the correct denominator.

在A-Level考试中，你必须先判断数据集代表的是总体还是样本，然后再选择正确的分母。

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5. Sample Standard Deviation | 样本标准差

The sample standard deviation s is the square root of the sample variance. It is used as an estimator of the population standard deviation σ. Although s² is an unbiased estimator, s itself is slightly biased, but it is still widely used in practice.

样本标准差s是样本方差的平方根，用作总体标准差σ的估计量。虽然s²是无偏估计量，s本身略有偏倚，但在实践中仍广泛使用。

s = √[ Σ(xᵢ − x̄)² / (n − 1) ]

When using a calculator, there are often two keys: σₙ for population standard deviation and σₙ₋₁ for sample standard deviation. Ensure you select the correct one according to the context.

使用计算器时，通常有两个键：σₙ用于总体标准差，σₙ₋₁用于样本标准差。务必根据上下文选择正确的键。

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6. Calculation with Ungrouped Data | 未分组数据的计算

To calculate variance for a small ungrouped data set, follow these steps: compute the mean, find each deviation from the mean, square these deviations, sum them, then divide by N (population) or n−1 (sample). A step-by-step approach reduces errors.

计算一个小型未分组数据集的方差，可遵循以下步骤：计算均值，找出每个数据点与均值的偏差，对这些偏差平方，求和，然后除以N（总体）或n−1（样本）。逐步进行可以减少错误。

Consider the data set: 2, 4, 6, 8, 10. The mean is (2+4+6+8+10)/5 = 6. The deviations are −4, −2, 0, 2, 4. Squaring gives 16, 4, 0, 4, 16. Their sum is 40. For a population, variance = 40/5 = 8; for a sample, variance = 40/4 = 10.

考虑数据集：2, 4, 6, 8, 10。均值为(2+4+6+8+10)/5 = 6。偏差分别为−4、−2、0、2、4。平方后为16、4、0、4、16。总和为40。对于总体，方差 = 40/5 = 8；对于样本，方差 = 40/4 = 10。

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7. Worked Example 1: Small Data Set | 例题1：小数据集

A sample of five students’ test scores: 12, 15, 14, 18, 16. Calculate the sample variance and sample standard deviation. First, find the sample mean: x̄ = (12+15+14+18+16)/5 = 75/5 = 15. Deviations: −3, 0, −1, 3, 1. Squares: 9, 0, 1, 9, 1. Sum of squares = 20. Sample variance s² =

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