Variance and Standard Deviation | 方差与标准差

📚 Variance and Standard Deviation | 方差与标准差

In Edexcel A-Level Mathematics, variance and standard deviation are the primary measures of spread for a data set. They tell you how far values are typically from the mean, and they appear constantly in descriptive statistics, probability distributions, and hypothesis testing.

在 Edexcel A-Level 数学中，方差与标准差是描述数据离散程度的核心指标。它们衡量数据值通常偏离均值的程度，并且在描述统计、概率分布和假设检验中反复出现。

1. Mean as a starting point | 从均值出发

Before measuring spread, you must locate the centre of the data using the mean. For raw data x₁, x₂, …, xₙ, the mean is the sum of all values divided by the number of values.

在衡量离散程度之前，必须先使用均值确定数据中心。对于原始数据 x₁、x₂、…、xₙ，均值等于所有数据值之和除以数据个数。

x̄ = Σx / n

The mean is used as a reference point because every deviation x – x̄ measures how far a particular value lies from the centre.

均值被用作参照点，因为每个离差 x – x̄ 都衡量了某个特定数据值偏离中心的距离。

2. Defining variance | 方差的定义

Variance is the mean of the squared deviations from the mean. Squaring removes negative signs and gives greater weight to larger deviations.

方差是离差平方的均值。平方可以消除负号，并赋予较大离差更大的权重。

σ² = Σ(x – x̄)² / n

For a population, the symbol σ² is used. For a sample, many courses use s² with divisor n – 1, but Edexcel often uses the full data set variance with divisor n unless a sample is explicitly specified.

对于总体，通常使用符号 σ²。对于样本，许多课程使用除数为 n – 1 的 s²，但 Edexcel 通常使用除数为 n 的整体数据方差，除非题目明确说明是样本。

3. Standard deviation | 标准差

Standard deviation is the positive square root of the variance. It has the same units as the original data, which makes it easier to interpret in context.

标准差是方差的正平方根。它与原始数据具有相同的单位，因此在具体情境中更容易解释。

σ = √(Σ(x – x̄)² / n)

A small standard deviation means values are tightly clustered around the mean. A large standard deviation means values are widely spread.

标准差较小意味着数据值紧密聚集在均值周围。标准差较大意味着数据值分布较为分散。

4. Raw data calculation | 原始数据计算

For a small data set, you can calculate variance using a table of deviations. However, the alternative formula is usually faster and less error-prone in exams.

对于小型数据集，可以使用离差表计算方差。然而，在考试中替代公式通常更快且更不容易出错。

σ² = Σx² / n – x̄²

This formula is equivalent to the definition because Σ(x – x̄)² = Σx² – n x̄². You must calculate the mean first, then apply the formula.

该公式与定义等价，因为 Σ(x – x̄)² = Σx² – n x̄²。你必须先计算均值，然后再应用该公式。

5. Frequency distributions | 频数分布

When data are given in a frequency table, each value x has a frequency f. The mean becomes Σfx / Σf, and the variance is weighted by the frequencies.

当数据以频数表给出时，每个数值 x 都有对应的频数 f。均值变为 Σfx / Σf，方差则需要按频数加权。

σ² = Σf(x – x̄)² / Σf = Σfx² / Σf – x̄²

Here Σf is the total number of observations. Always use the same denominator Σf for both the mean and the variance.

这里的 Σf 是观测值的总数。在计算均值和方差时，必须始终使用相同的分母 Σf。

6. Grouped data | 分组数据

For grouped continuous data, you do not know the exact individual values. You use the midpoint of each class interval as the representative value x.

对于分组连续数据，你并不知道每个具体数据值。此时使用每个区间的中点作为代表值 x。

If a class is 10 ≤ x < 20, the midpoint is 15. With class width w, midpoint = (lower bound + upper bound) / 2.

如果某个组的区间为 10 ≤ x < 20，则中点为 15。若组距为 w，则中点 = （下界 + 上界）/ 2。

x = (lower + upper) / 2

The grouped variance is an estimate because using midpoints assumes values are evenly spread within each interval.

分组方差只是一个估计值，因为使用中点相当于假设每个区间内的数据值均匀分布。

7. Sample vs population | 样本与总体

When a data set is a sample from a larger population, the unbiased sample variance divides by n – 1 rather than n.

当数据集是来自更大总体的样本时，无偏样本方差应除以 n – 1 而不是 n。

s² = Σ(x – x̄)² / (n – 1)

The n – 1 divisor corrects the bias that occurs because the sample mean is used in the calculation. In most Edexcel S1 and statistics questions, check the wording carefully before choosing n or n – 1.

除数 n – 1 可以修正由于在计算中使用样本均值而产生的偏差。在大多数 Edexcel S1 和统计题中，选择 n 还是 n – 1 之前必须仔细审题。

8. Linear transformations | 线性变换

If a variable y is obtained from x by a linear transformation y = ax + b, then the mean and standard deviation transform in predictable ways.

如果变量 y 由 x 经过线性变换 y = ax + b 得到，那么均值和标准差会以可预测的方式变化。

ȳ = ax̄ + b

σᵧ = |a|σₓ

Adding b shifts all values but does not change the spread. Multiplying by a scales the spread by |a|, so the variance is multiplied by a².

加上 b 会使所有数值平移，但不会改变离散程度。乘以 a 会使离散程度缩放 |a| 倍，因此方差会乘以 a²。

σᵧ² = a²σₓ²

9. Combining data sets | 合并数据集

If two data sets are combined, the overall mean is a weighted average of the two individual means.

如果将两个数据集合并，总均值是两个单独均值的加权平均值。

x̄ = (n₁x̄₁ + n₂x̄₂) / (n₁ + n₂)

The combined variance is not simply the average of the two variances. It must also include the separation between the two sub-group means.

合并方差并不是两个方差的简单平均值。它还必须包含两个子组均值之间的差异。

σ² = [n₁(σ₁² + (x̄₁ – x̄)²) + n₂(σ₂² + (x̄₂ – x̄)²)] / (n₁ + n₂)

This formula is particularly useful in examination questions that give summary statistics for two samples or groups.

这个公式在给出两个样本或组的汇总统计量的考试题中特别有用。

10. Interpreting spread | 离散程度的解释

Variance and standard deviation are only useful when compared with the context. A standard deviation of 5 cm has different meanings for human heights and for pencil lengths.

方差和标准差只有在结合具体情境时才有意义。标准差为 5 cm 对于人类身高和铅笔长度来说含义完全不同。

You can also compare spread between distributions only when the units are the same. Standard deviation is preferred over variance for interpretation because it uses the original units.

只有当单位相同时，才能比较不同分布的离散程度。解释数据时通常优先使用标准差而不是方差，因为标准差使用原始单位。

11. Common exam pitfalls | 常见考试易错点

Forgetting to square the mean in the alternative formula.

在替代公式中忘记将均值平方。
Using n instead of Σf in frequency table questions.

在频数表题目中误用 n 而不是 Σf。
Confusing population variance σ² with sample variance s².

混淆总体方差 σ² 与样本方差 s²。
Taking the square root before subtracting the squared mean.

在减去均值平方之前就取平方根。
Using class boundaries instead of midpoints for grouped data.

在分组数据中使用组界而不是中点。

A systematic approach, with clear intermediate steps, reduces these errors and helps secure method marks.

采用系统化步骤并写出清晰的中间过程，可以减少这些错误，并有助于获得方法分。

12. Worked example | 例题演示

The masses, in kg, of five packages are: 4, 6, 7, 9, 14. Find the mean, variance, and standard deviation.

五个包裹的质量（kg）为：4、6、7、9、14。求均值、方差和标准差。

First calculate the mean: x̄ = (4 + 6 + 7 + 9 + 14) / 5 = 40 / 5 = 8 kg.

首先计算均值：x̄ = （4 + 6 + 7 + 9 + 14）/ 5 = 40 / 5 = 8 kg。

Then compute Σx²: 4² + 6² + 7² + 9² + 14² = 16 + 36 + 49 + 81 + 196 = 378.

然后计算 Σx²：4² + 6² + 7² + 9² + 14² = 16 + 36 + 49 + 81 + 196 = 378。

σ² = Σx² / n – x̄² = 378 / 5 – 8² = 75.6 – 64 = 11.6

The variance is 11.6 kg². The standard deviation is √11.6 ≈ 3.41 kg.

方差为 11.6 kg²。标准差为 √11.6 ≈ 3.41 kg。

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Variance and Standard Deviation | 方差与标准差