IB Mathematics: Variance and Standard Deviation | IB数学：方差与标准差

📚 IB Mathematics: Variance and Standard Deviation | IB数学：方差与标准差

When summarising a data set, the mean alone is not enough. Two sets of data can have the same average yet look very different. Variance and standard deviation measure how spread out the values are around the mean.

在概括一组数据时，仅有平均数是不够的。两组数据的平均值可能相同，但分布形态却大不相同。方差与标准差正是用来度量数据在平均值附近离散程度的工具。

1. Why Dispersion Matters | 为什么离散程度重要

Measures of central tendency, such as mean and median, describe the “centre” of the data. Measures of dispersion, such as variance and standard deviation, describe the “spread”. A small spread means most values are close to the centre; a large spread means the values are widely scattered.

均值、中位数等集中趋势度量描述的是数据的“中心”，而方差、标准差等离散程度度量描述的是数据的“散布程度”。离散程度小，说明大部分数值靠近中心；离散程度大，说明数值分布得很散。

In quality control, a small standard deviation indicates consistent products.

在质量控制中，较小的标准差意味着产品质量稳定。
In finance, a larger standard deviation means higher risk.

在金融领域，较大的标准差意味着较高的风险。

2. Population Variance | 总体方差

For a finite population with N elements, let μ be the population mean. The population variance, denoted by σ², is the average of the squared deviations from the mean.

对于包含 N 个元素的有限总体，设 μ 为总体均值。总体方差记作 σ²，是各数据离均差的平方的平均值。

σ² = Σ (xᵢ − μ)² / N

The unit of variance is the square of the original unit. The Greek letter σ is called sigma.

方差的单位是原数据单位的平方。希腊字母 σ 读作“西格玛”。

An alternative computational form is σ² = Σ xᵢ² / N − μ², which is often useful for hand calculations.

另一种等价的计算形式是 σ² = Σ xᵢ² / N − μ²，在手工计算时往往很方便。

3. Sample Variance | 样本方差

When the population is too large, we use a sample. For a sample of size n with sample mean x̄, the sample variance is denoted by s². The divisor is n − 1, not n.

当总体过大时，我们使用样本。对于容量为 n、样本均值为 x̄ 的样本，样本方差记作 s²。除数不是 n，而是 n − 1。

s² = Σ (xᵢ − x̄)² / (n − 1)

The n − 1 is called Bessel’s correction. It makes s² an unbiased estimate of the population variance. This is expected in IB statistics questions.

这个 n − 1 称为贝塞尔校正，它使 s² 成为总体方差的无偏估计。IB 统计学题目中通常要求使用这个公式。

4. Standard Deviation | 标准差

The standard deviation is the positive square root of the variance. It restores the original units, so it is easier to interpret.

标准差是方差的算术平方根。它恢复了原来的单位，因此更易于解释。

σ = √(σ²) , s = √(s²)

Population standard deviation is written as σ; sample standard deviation is written as s.

总体标准差记作 σ，样本标准差记作 s。

5. Worked Example: Ungrouped Data | 实例：未分组数据

Consider the data set 4, 6, 8. The mean is x̄ = (4 + 6 + 8)/3 = 6. The deviations are −2, 0, and 2.

考虑数据集 4, 6, 8。均值 x̄ = (4 + 6 + 8)/3 = 6。离差分别是 −2、0 和 2。

Σ (xᵢ − x̄)² = (−2)² + 0² + 2² = 8

For a population, σ² = 8/3 ≈ 2.67 and σ = √(8/3) ≈ 1.63. For a sample, s² = 8/(3 − 1) = 4 and s = 2.

若视为总体，σ² = 8/3 ≈ 2.67，σ = √(8/3) ≈ 1.63。若视为样本，s² = 8/(3 − 1) = 4，s = 2。

6. Grouped Data and Frequency Tables | 分组数据与频数表

When data are grouped into intervals, we lose the original values. We use the midpoint of each class as a representative value. The variance is then calculated with frequency weights.

当数据按区间分组时，我们无法得知原始数值，因此用每一组的组中值作为代表值。此时方差的计算需要以频数为权重。

σ² = Σ fᵢ (xᵢ − μ)² / Σ fᵢ

Here fᵢ is the frequency of the i-th class and xᵢ is its midpoint.

其中 fᵢ 是第 i 组的频数，xᵢ 是该组的组中值。

7. Units and Interpretation | 单位与解释

If the original data are measured in metres, the variance is in square metres, while the standard deviation is in metres. This makes the standard deviation more meaningful when describing real-world data.

如果原始数据以米为单位，那么方差的单位是平方米，而标准差的单位仍是米。这使得标准差在描述实际数据时更有意义。

For many symmetric, mound-shaped distributions, roughly 68% of the data lie within 1 standard deviation of the mean, and 95% within 2 standard deviations.

对于许多对称的钟形分布，大约 68% 的数据落在均值左右 1 个标准差之内，95% 落在 2 个标准差之内。

8. Properties: Translations and Scaling | 性质：平移与缩放

If every value in the data set is increased by a constant c, the mean increases by c but the variance and standard deviation do not change, because the spread is unchanged.

若数据集中的每个数值都加上常数 c，均值增加 c，但方差与标准差不变，因为离散程度没有改变。

Var(X + c) = Var(X) , SD(X + c) = SD(X)

If every value is multiplied by a constant a, the variance is multiplied by a² and the standard deviation is multiplied by |a|.

若每个数值都乘以常数 a，则方差变为原来的 a² 倍，标准差变为原来的 |a| 倍。

Var(aX) = a² Var(X) , SD(aX) = |a| SD(X)

9. Comparing Variance and Standard Deviation | 方差与标准差的比较

Both quantities measure spread, but they are used in different ways. Variance is mathematically convenient: it is additive for independent variables and appears in the formula for the normal distribution. Standard deviation is easier to communicate because it has the same unit as the data.

两者都度量离散程度，但在使用上各有侧重。方差在数学上更方便：对于独立变量具有可加性，且出现在正态分布的公式中。标准差更便于交流，因为它与数据单位相同。

Use standard deviation when reporting results to a general audience.

向普通读者报告结果时使用标准差。
Use variance in further statistical derivations.

在进一步的统计推导中使用方差。

10. Common Mistakes and Exam Tips | 常见错误与考试技巧

Confusing population variance and sample variance: check whether the data represent the whole population.

混淆总体方差与样本方差：判断数据是来自全部总体还是样本。
Forgetting to take the square root when the question asks for standard deviation.

题目要求标准差时忘记开平方根。
Using n instead of n − 1 for sample variance.

计算样本方差时用了 n 而不是 n − 1。
For grouped data, using class boundaries instead of midpoints.

在分组数据中，使用组界而不是组中值。
Always show the formula and substitute values clearly; examiners award method marks for your working.

始终清楚写出公式并代入数值；阅卷时步骤分来自你的求解过程。

11. Conclusion | 总结

Variance and standard deviation are essential measures of spread in IB mathematics. Know the symbols, the n − 1 correction, and how to apply the ideas to grouped data. Practise with your GDC, but also learn to do short calculations by hand.

方差与标准差是 IB 数学中描述离散程度的重要工具。必须牢记符号、n − 1 校正，以及如何将概念应用于分组数据。既要练习计算器操作

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IB Mathematics: Variance and Standard Deviation | IB数学：方差与标准差