📚 Variance and Standard Deviation | 方差与标准差
In statistics, understanding how data values spread around a central point is just as important as knowing the average itself. Variance and standard deviation are the most widely used measures of dispersion, telling us whether data points cluster tightly around the mean or scatter widely. This article explains their definitions, calculations, interpretations and applications within the IB Mathematics curriculum.
在统计学中，理解数据值围绕中心点的分散程度与了解平均值本身同样重要。方差和标准差是最常用的离散度量，它们告诉我们数据点是紧密聚集在均值附近还是分散得很开。本文将解释它们的定义、计算、解读以及在IB数学课程中的应用。
1. What is Dispersion? | 什么是离散程度？
Dispersion refers to the variability or spread in a dataset. Two datasets may have the same mean but very different spreads. For example, the sets {48, 50, 52} and {10, 50, 90} both have a mean of 50, yet the second set is far more spread out. Variance and standard deviation quantify this spread by examining the average squared distance of each data point from the mean.
离散程度是指数据集中的变异或散布情况。两个数据集可能有相同的均值，但散布程度差异很大。例如，集合{48, 50, 52}和{10, 50, 90}的均值都是50，但第二个集合的分散程度要大得多。方差和标准差通过考察每个数据点与均值之间的平均平方距离来量化这种散布。
2. Variance: Definition and Formula (Population) | 总体方差的定义与公式
Population variance, denoted by σ², is the average of the squared deviations from the population mean μ. For a population of N values, the formula is:
总体方差用σ²表示，是各数据与总体均值μ的偏差平方的平均值。对于有N个值的总体，其公式为：
σ² = Σ(x − μ)² / N
Each data point x is subtracted from the mean, the difference is squared to avoid positive and negative deviations cancelling, and then the squared deviations are averaged.
将每个数据点x减去均值，对差值进行平方以避免正负偏差相互抵消，然后将平方后的偏差取平均值。
3. Standard Deviation: The Square Root of Variance | 标准差：方差的平方根
on, denoted by σ for a population, is simply the positive square root of the variance: σ = √σ². It brings the measure back to the original units of the data, making interpretation easier. If a dataset of heights has variance in cm², the standard deviation is in cm, directly comparable to individual heights.
总体标准差用σ表示，就是方差的正平方根：σ = √σ²。它将度量值恢复为数据的原始单位，使解读更为直观。如果一组身高数据的方差单位是cm²，那么标准差的单位就是cm，可以直接与各自身高比较。
4. Sample Variance and Bessel's Correction | 样本方差与贝塞尔校正
When we work with a sample of size n drawn from a larger population, we use the sample variance s² with Bessel's correction to obtain an unbiased estimate of the population variance:
当我们处理从较大总体中抽取的容量为n的样本时，需要使用带贝塞尔校正的样本方差s²来得到总体方差的无偏估计：
s² = Σ(x − x̄)² / (n − 1)
Here x̄ is the sample mean. Dividing by (n − 1) instead of n accounts for the fact that the sample mean itself shows less variability, and it prevents the variance from being underestimated.
其中x̄为样本均值。除以(n − 1)而不是n，是为了补偿样本均值本身变异较小的事实，从而避免方差被低估。
5. Computing Variance: Step-by-Step | 计算方差：步骤分解
Let us calculate the population variance for the dataset: 2, 4, 6, 8, 10.
让我们计算数据集 2, 4, 6, 8, 10 的总体方差。
Step 1: Find the mean μ = (2+4+6+8+10)/5 = 6.
Step 2: Compute deviations: −4, −2, 0, 2, 4.
Step 3: Square them: 16, 4, 0, 4, 16.
Step 4: Sum the squares: Σ(x − μ)² = 40.
Step 5: Divide by N = 5: σ² = 40/5 = 8.
第1步：求均值 μ = (2+4+6+8+10)/5 = 6。
第2步：计算偏差：−4, −2, 0, 2, 4。
第3步：平方偏差：16, 4, 0, 4, 16。
第4步：求和：Σ(x − μ)² = 40。
第5步：除以 N = 5：σ² = 40/5 = 8。
The standard deviation is σ = √8 ≈ 2.828.
标准差为 σ = √8 ≈ 2.828。
6. Alternative Computational Formula | 替代计算公式
An algebraically equivalent formula often simplifies hand calculations:
一个代数等价的公式常常能简化手工计算：
σ² = (Σx²)/N − μ² or σ² = (Σx² − (Σx)²/N) / N
100 = 220, (Σx)²/N = 30²/5 = 180, so σ² = (220 − 180)/5 = 8, matching the previous result.
以数据 2,4,6,8,10 为例：Σx² = 4+16+36+64+100 = 220，(Σx)²/N = 30²/5 = 180，因此 σ² = (220 − 180)/5 = 8，与之前结果一致。
7. Why Square the Deviations? | 为什么要将偏差平方？
Squaring eliminates the sign of deviations so that negative and positive differences do not cancel each other out. Moreover, squaring gives more weight to values farther from the mean, highlighting outliers. The variance therefore responds sensitively to extreme values, which can be both an advantage and a limitation when analysing skewed data.
平方可以消除偏差的符号，使正负差值不会相互抵消。此外，平方会给远离均值的数值赋予更大的权重，从而突出异常值。因此方差对极端值反应灵敏，这在分析偏态数据时既是优点也是局限。
8. Properties of Variance and Standard Deviation | 方差和标准差的性质
Variance and standard deviation have useful algebraic properties:
- he variance is always non-negative: σ² ≥ 0; it is zero only when all values are identical.
- Adding a constant c to all data shifts the mean but does not change the variance or standard deviation.
- Multiplying all data by a constant k multiplies the standard deviation by |k| and the variance by k².
方差和标准差具有实用的代数性质：
- 方差始终非负：σ² ≥ 0；仅当所有数值都相同时才为零。
- 将所有数据加上一个常数c会平移均值，但不改变方差或标准差。
- 将所有数据乘以常数k，会使标准差乘以|k|，方差乘以k²。
9. Variance and Standard Deviation for Grouped Data | 分组数据的方差与标准差
When data are presented in a frequency table, we use the midpoints of class intervals and incorporate frequencies f. The population variance formula for grouped data becomes:
当数据以频数表呈现时，我们使用组中值并纳入频数f。分组数据的总体方差公式变为：
σ² = Σf(x − μ)² / Σf or σ² = (Σfx²)/Σf − μ²
Σfx / Σf the estimated mean. The square root of the result gives the grouped standard deviation.
这里x代表组中值，f代表该组的频数，μ = Σfx / Σf 是估计均值。结果的平方根即为分组数据的标准差。
10. Worked Example and Interpretation | 实例演示与解读
An IB student surveys the weekly hours of homework for 10 classmates: 3, 5, 5, 6, 7, 8, 8, 9, 10, 14. Treat the data as a sample. Compute the sample variance and standard deviation.
一位IB学生调查了10位同学每周的家庭作业小时数：3, 5, 5, 6, 7, 8, 8, 9, 10, 14。将数据视为样本，计算样本方差和标准差。
Step 1: sample mean x̄ = (3+5+5+6+7+8+8+9+10+14)/10 = 75/10 = 7.5.
.25, (7-7.5)²=0.25, (8-7.5)²=0.25 (twice), (9-7.5)²=2.25, (10-7.5)²=6.25, (14-7.5)²=42.25.
ep 3: sum of squares = 20.25 + 6.25×2 + 2.25 + 0.25 + 0.25×2 + 2.25 + 6.25 + 42.25 = 87.5.
Step 4: s² = 87.5 / (10−1) = 87.5/9 ≈ 9.72.
Step 5: s = √9.72 ≈ 3.12 hours.
第1步：样本均值 x̄ = (3+5+5+6+7+8+8+9+10+14)/10 = 75/10 = 7.5。
第2步：偏差平方：(3-7.5)²=20.25, (5-7.5)²=6.25（两次）, (6-7.5)²=2.25, (7-7.5)²=0.25, (8-7.5)²=0.25（两次）, (9-7.5)²=2.25, (10-7.5)²=6.25, (14-7.5)²=42.25。
第3步：平方和 = 20.25 + 6.25×2 + 2.25 + 0.25 + 0.25×2 + 2.25 + 6.25 + 42.25 = 87.5。
第4步：s² = 87.5 / (10−1) = 87.5/9 ≈ 9.72。
第5步：s = √9.72 ≈ 3.12 小时。
The standard deviation of about 3.12 hours indicates that typical homework hours vary by roughly 3 hours above or below the mean of 7.5 hours. A low standard deviation would imply most students study a similar amount, while a high one suggests greater disparity in study habits.
标准差约3.12小时表明，典型的作业时间在均值7.5小时上下约3小时的范围内波动。低标准差意味着大多数学生学习量相近，而高标准差则反映学习习惯的差异较大。
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