📚 Variance and Standard Deviation Formulas Explained | 方差与标准差公式详解
In statistics, variance and standard deviation are the two most important measures of spread. They tell us how far the data values are scattered around the mean. Understanding these measures is essential for IB Mathematics, especially in the Statistics and Probability topic.
在统计学中，方差和标准差是衡量离散程度最重要的两个指标。它们告诉我们数据值围绕平均值分布的分散程度。理解这些概念对于IB数学尤为重要，尤其是在统计与概率专题中。
1. What Are Variance and Standard Deviation? | 什么是方差与标准差？
Variance measures the average squared deviation of each data point from the mean. A large variance means the data points are spread out widely, while a small variance means they are clustered closely around the mean.
方差度量每个数据点与平均值之间的平均平方偏差。方差越大，说明数据点分布越分散；方差越小，说明数据点紧密聚集在平均值附近。
Standard deviation is simply the square root of the variance. It is expressed in the same units as the original data, which makes it easier to interpret than variance.
标准差就是方差的平方根。它与原始数据具有相同的单位，因此比方差更易于解释。
2. Population Variance Formula | 总体方差公式
When we have data for an entire population, we use the population variance, written as σ² (sigma squared).
当我们拥有整个总体的数据时，我们使用总体方差，记为σ²（西格玛平方）。
σ² = Σ(xᵢ – μ)² / N
Here, xᵢ represents each data value, μ is the population mean, and N is the total number of data points in the population.
其中，xᵢ表示每个数据值，μ是总体平均值，N是总体中数据点的总数。
To calculate it, subtract the mean from each value, square the result, add all squares together, and then divide by N.
计算方法是：将每个值减去平均值，对结果平方，将所有平方和相加，再除以N。
3. Sample Variance Formula | 样本方差公式
In IB Mathematics, we often work with a sample, which is a subset of the population. For sample data, we use s² to denote the sample variance, and the formula has a slightly different denominator.
在IB数学中，我们经常处理样本，即总体中的一个子集。对于样本数据，我们用s²表示样本方差，其公式的分母略有不同。
s² = Σ(xᵢ – x̄)² / (n – 1)
Here, x̄ is the sample mean, and n is the sample size. The denominator is n – 1 instead of n. This correction is called Bessel’s correction.
其中，x̄是样本平均值，n是样本容量。分母是n – 1而不是n。这一修正称为贝塞尔修正。
4. Standard Deviation | 标准差
The standard deviation is the square root of the variance. For a population, it is denoted by σ; for a sample, it is denoted by s.
标准差是方差的平方根。对于总体，用σ表示；对于样本，用s表示。
σ = √(Σ(xᵢ – μ)² / N)
s = √(Σ(xᵢ – x̄)² / (n – 1))
Because variance squares the deviations, its units are the square of the original units. Taking the square root gives us a measure of spread that is directly comparable to the mean.
由于方差对偏差进行了平方，其单位是原始单位的平方。取平方根后，我们得到一个可以直接与平均值比较的离散程度度量。
5. Why Divide by n – 1? | 为什么除以 n – 1？
When we calculate the sample variance, if we divided by n instead of n – 1, the sample variance would tend to underestimate the true population variance. This is because the sample mean is itself estimated from the data and is closer to the sample values than the true population mean is.
当我们计算样本方差时，如果除以n而不是n – 1，样本方差往往会低估真实的总体方差。这是因为样本平均值本身是从数据中估计的，它比真实的总体平均值更接近样本数据值。
Using n – 1 increases the value of s² slightly, making it an unbiased estimator of the population variance. This is a key concept in IB Mathematics SL and HL.
使用n – 1会略微增大s²的值，使其成为总体方差的无偏估计量。这是IB数学SL和HL中的一个关键概念。
6. Calculating from a Frequency Table | 频数表计算
When data is presented in a frequency table, we need to adjust our calculations. If we have values x₁, x₂, …, xₖ with frequencies f₁, f₂, …, fₖ, then:
当数据以频数表形式呈现时，我们需要调整计算方法。如果我们有数值x₁, x₂, …, xₖ，对应频数f₁, f₂, …, fₖ，则：
σ² = Σ fᵢ(xᵢ – μ)² / N
where N = Σ fᵢ is the total frequency. This formula applies to grouped or ungrouped frequency tables.
其中N = Σ fᵢ是总频数。这个公式适用于分组或未分组的频数表。
For grouped data with class midpoints, we treat the midpoint of each class as the representative xᵢ value for that class.
对于分组数据，我们使用每组的中点作为该组xᵢ的代表值。
7. Properties of Variance and Standard Deviation | 方差与标准差的性质
Variance and standard deviation have several useful properties that help simplify calculations.
方差和标准差具有几个有用的性质，可以简化计算。
- Variance is always non-negative: σ² ≥ 0 and s² ≥ 0. It is zero only when all data values are identical.
- 方差永远非负：σ² ≥ 0 且 s² ≥ 0。只有当所有数据值都相同时才等于零。
- Standard deviation has the same unit as the data: This makes it easier to interpret in real-world contexts.
- 标准差与数据单位相同：这使得它在实际情境中更容易解释。
- Variance is not robust to outliers: Because it squares each deviation, extreme values have a large influence on the variance.
- 方差对异常值不稳健：因为它对每个偏差进行平方，极端值对方差有很大影响。
8. Effect of Adding or Multiplying by a Constant | 常数加减乘的影响
If we add a constant c to every data value, the variance and standard deviation do not change because the spread remains the same. The mean shifts, but the distances between values stay identical.
如果我们在每个数据值上加上常数c，方差和标准差都不变，因为离散程度没有改变。平均值会移动，但数据值之间的距离保持不变。
If we multiply every data value by a constant c, the new variance becomes c² times the original variance, and the new standard deviation becomes |c| times the original standard deviation.
如果我们将每个数据值乘以常数c，新的方差变为原来方差的c²倍，新的标准差变为原来标准差的|c|倍。
Var(a + bX) = b² · Var(X)
SD(a + bX) = |b| · SD(X)
These properties are frequently tested in IB exam questions involving linear transformations of data.
这些性质在IB考试中经常涉及数据线性变换的题目中考查。
9. Using Calculator (GDC) | 使用计算器（GDC）
In IB Mathematics, you are expected to use a graphing display calculator (GDC) to compute variance and standard deviation quickly.
在IB数学中，你需要学会使用图形计算器（GDC）快速计算方差和标准差。
- Enter the data into a list (e.g., L1).
- 进入一个列表（如L1）输入数据。
- Use the 1-variable statistics function (usually found under STAT or Calc menu).
- 使用单变量统计功能（通常在STAT或Calc菜单下）。
- Read the values: the GDC gives both the sample standard deviation (usually labeled s or σₓ without n-1) and the population standard deviation (σ).
- 读取数值：GDC会给出样本标准差（通常标记为s或σₓ，使用n-1）和总体标准差（σ）。
Be careful to select the correct one based on whether you have a sample or a population. Many IB questions specify “consider the data as a sample” or “as a population.”
要根据数据是样本还是总体来准确选择。许多IB题目会明确说明“将数据视为样本”或“视为总体”。
10. Common Mistakes | 常见错误
Students often make the following mistakes when working with variance and standard deviation.
学生在处理方差和标准差时常犯以下错误。
- Forgetting to square the deviations: Variance requires the sum of squared deviations, not the sum of absolute deviations.
- 忘记对偏差平方：方差需要偏差平方和，而不是绝对偏差之和。
- Using wrong denominator: Use n – 1 for sample variance, N for population variance. Confusing them is a common error.
- 使用错误的分母：样本方差用n – 1，总体方差用N。混淆二者是常见错误。
- Interpreting standard deviation as the “average distance”: It is actually the root mean square deviation, not the average absolute deviation.
- 将标准差解释为“平均距离”：它实际上是均方根偏差，而不是平均绝对偏差。
- Rounding too early: Keep at least three significant figures during intermediate steps to avoid inaccuracies.
- 过早四舍五入：在中间步骤中至少保留三位有效数字，以免影响精确度。
11. Worked Example | 例题详解
Consider the sample data: 4, 6, 8, 10, 12. Find the sample mean, sample variance and sample standard deviation.
考虑样本数据：4, 6, 8, 10, 12。求样本平均值、样本方差和样本标准差。
Step 1: Calculate the sample mean.
步骤1：计算样本平均值。
x̄ = (4 + 6 + 8 + 10 + 12) / 5 = 40 / 5 = 8
Step 2: Compute each deviation from the mean, square it, and add them.
步骤2：计算每个偏差的平方并求和。
Σ(xᵢ – x̄)² = (4-8)² + (6-8)² + (8-8)² + (10-8)² + (12-8)²
= 16 + 4 + 0 + 4 + 16 = 40
Step 3: Divide by n – 1 = 4 to get the sample variance.
步骤3：除以n – 1 = 4得到样本方差。
s² = 40 / 4 = 10
Step 4: Take the square root to get the standard deviation.
步骤4：取平方根得到标准差。
s = √10 ≈ 3.16
Notice that if this data were a population, the variance would be 40 / 5 = 8, and the standard deviation would be √8 ≈ 2.83. The choice of sample versus population matters.
注意，如果这组数据是总体，方差会是40 / 5 = 8，标准差会是√8 ≈ 2.83。样本与总体的选择很重要。
12. Summary | 总结
Variance and standard deviation are fundamental measures of dispersion in IB Mathematics. Remember that variance is the average squared deviation from the mean, and standard deviation is its square root.
方差和标准差是IB数学中衡量离散程度的基本指标。记住：方差是离均差平方的平均值，标准差是它的平方根。
Always distinguish between population (σ², σ) and sample (s², s) formulas. The sample formula uses n – 1 to provide an unbiased estimate of the population variance. Frequency tables, linear transformations, and GDC calculations are essential skills for exam success.
务必区分总体（σ², σ）和样本（s², s）公式。样本公式使用n – 1以获得总体方差的无偏估计。频数表、线性变换和GDC计算是考试成功的关键技能。
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