📚 A-Level Further Mathematics: Bias and Standard Error of Estimators | A-Level 进阶数学：估计量的偏差与标准误差
In statistical inference, we use sample data to estimate unknown population parameters. An estimator is a rule or formula that tells us how to compute an estimate from a sample. Two fundamental properties of an estimator are its bias and its standard error.
在统计推断中，我们用样本数据来估计未知的总体参数。估计量是一种规则或公式，它告诉我们如何从样本中计算出一个估计值。估计量有两个基本性质：偏差和标准误差。
1. Definition of Bias | 偏差的定义
Let θ be a population parameter, and let θ̂ (theta-hat) be an estimator of θ based on a sample. The bias of θ̂ is defined as the difference between the expected value of the estimator and the true parameter value:
设 θ 是总体参数，θ̂（θ 帽）是基于样本的 θ 的估计量。θ̂ 的偏差定义为估计量的期望值与真实参数值之差：
Bias(θ̂) = E(θ̂) − θ
If the bias is zero, we say that θ̂ is an unbiased estimator. If the bias is positive, the estimator tends to overestimate the parameter; if negative, it tends to underestimate the parameter.
如果偏差为零，我们就说 θ̂ 是无偏估计量。如果偏差为正，估计量倾向于高估参数；如果偏差为负，则倾向于低估参数。
2. Unbiased Estimators | 无偏估计量
An estimator is unbiased if its expected value equals the true parameter, that is, E(θ̂) = θ. Unbiasedness is a desirable property because, on average over many random samples, the estimator hits the correct target.
如果估计量的期望值等于真实参数，即 E(θ̂) = θ，则该估计量是无偏的。无偏性是一个理想的属性，因为从许多随机样本平均来看，估计量能够命中正确的目标。
For example, the sample mean X̄ is an unbiased estimator of the population mean μ. This follows directly from the linearity of expectation:
例如，样本均值 X̄ 是总体均值 μ 的无偏估计量。这直接来自期望的线性性质：
E(X̄) = E( (1/n) ∑i=1n Xi ) = (1/n) ∑i=1n E(Xi) = μ
Here X₁, X₂, …, Xₙ are independent and identically distributed random variables with mean μ.
这里 X₁, X₂, …, Xₙ 是均值为 μ 的独立同分布随机变量。
3. Standard Error: Definition | 标准误差：定义
The standard error (SE) of an estimator is the standard deviation of its sampling distribution. It measures how much the estimator typically varies from sample to sample. A smaller standard error means the estimator is more precise.
估计量的标准误差是其抽样分布的标准差。它衡量估计量在不同样本之间的典型波动程度。标准误差越小，估计量就越精确。
SE(θ̂) = √Var(θ̂)
Do not confuse the standard error with the standard deviation of the population. The standard deviation describes spread in the population, while the standard error describes spread of the estimator.
不要把标准误差与总体标准差混淆。标准差描述总体中的离散程度，而标准误差描述估计量的离散程度。
4. Standard Error of the Sample Mean | 样本均值的标准误差
For a random sample of size n from a population with variance σ², the variance of the sample mean is Var(X̄) = σ²/n. Therefore the standard error of the sample mean is:
对于来自方差为 σ² 的总体、容量为 n 的随机样本，样本均值的方差为 Var(X̄) = σ²/n。因此样本均值的标准误差为：
SE(X̄) = σ / √n
This formula shows that quadrupling the sample size halves the standard error. Since the square root of n appears in the denominator, increasing n reduces the variability of X̄ only slowly.
这个公式表明，样本量增大到四倍时，标准误差减半。由于分母中出现的是 √n，增加 n 对 X̄ 变异的减小是缓慢的。
5. Estimating the Population Variance | 估计总体方差
In practice, the population variance σ² is usually unknown. We estimate it from the sample. A natural candidate is Sₙ² = (1/n) ∑(Xᵢ − X̄)². However, this estimator is biased for σ².
实际上，总体方差 σ² 通常是未知的。我们用样本估计它。一个自然的候选是 Sₙ² = (1/n) ∑(Xᵢ − X̄)²。然而，该估计量对于 σ² 是有偏的。
Because X̄ is itself a random variable that tends to be closer to the sample values than μ is, the average squared deviation ∑(Xᵢ − X̄)² is on average smaller than ∑(Xᵢ − μ)². This leads to a negative bias.
因为 X̄ 本身是随机变量，它比 μ 更接近样本值，所以平均而言 ∑(Xᵢ − X̄)² 小于 ∑(Xᵢ − μ)²。这导致了负偏差。
It can be shown that E(Sₙ²) = ((n−1)/n) σ². Thus the bias is −σ²/n. To correct this, we divide by n−1 instead of n.
可以证明 E(Sₙ²) = ((n−1)/n) σ²。因此偏差为 −σ²/n。为了修正这一点，我们用 n−1 而不是 n 来除。
6. Unbiased Sample Variance | 无偏样本方差
The unbiased estimator of the population variance is defined as:
总体方差的无偏估计量定义为：
S² = (1/(n−1)) ∑i=1n (Xᵢ − X̄)²
This estimator satisfies E(S²) = σ². The factor 1/(n−1) is called Bessel’s correction. It removes the negative bias from the natural estimator.
该估计量满足 E(S²) = σ²。因子 1/(n−1) 称为贝塞尔校正。它消除了自然估计量中的负偏差。
Note that S² is unbiased for σ², but S itself (the sample standard deviation) is not an unbiased estimator of σ because the square root is a nonlinear transformation. For A-Level purposes, we usually focus on unbiasedness of variances.
注意，S² 是 σ² 的无偏估计，但 S 本身（样本标准差）不是 σ 的无偏估计，因为平方根是非线性变换。在 A-Level 中，我们通常关注方差的无偏性。
7. Estimating the Standard Error from Data | 由数据估计标准误差
To estimate SE(X̄) when σ² is unknown, replace σ by the sample standard deviation s, where s = √S². This gives the estimated standard error:
当 σ² 未知时，要用样本标准差 s（s = √S²）代替 σ 来估计 SE(X̄)。由此得到标准误差的估计值：
estimated SE(X̄) = s / √n
This value is often called the standard error of the mean in statistical software. It is a sample statistic and itself varies from sample to sample.
该值在统计软件中通常称为均值的标准误差。它是一个样本统计量，本身也会随样本不同而变化。
Example: A sample of n = 25 observations has sample standard deviation s = 4.0. Then the estimated standard error of the mean is 4.0/√25 = 0.8.
例：n = 25 的样本，样本标准差 s = 4.0。则均值的标准误差估计值为 4.0/√25 = 0.8。
8. The Central Limit Theorem and Standard Error | 中心极限定理与标准误差
The central limit theorem (CLT) states that if X₁, X₂, …, Xₙ is a large random sample from any distribution with mean μ and variance σ², then the sample mean X̄ is approximately normally distributed with mean μ and variance σ²/n.
中心极限定理指出：如果 X₁, X₂, …, Xₙ 是来自均值为 μ、方差为 σ² 的任何分布的大容量随机样本，那么样本均值 X̄ 近似服从均值为 μ、方差为 σ²/n 的正态分布。
X̄ ≈ N(μ, σ²/n) for large n
The standard error therefore plays a key role in constructing confidence intervals and carrying out hypothesis tests. For example, a 95% confidence interval for μ is approximately X̄ ± 1.96 × SE(X̄), when the normal approximation is valid.
因此，标准误差在构造置信区间和进行假设检验中起关键作用。例如，在正态近似有效时，μ 的 95% 置信区间近似为 X̄ ± 1.96 × SE(X̄)。
9. Mean Squared Error and the Bias–Variance Trade-off | 均方误差与偏差-方差权衡
The mean squared error (MSE) of an estimator combines its bias and variance:
估计量的均方误差组合了其偏差和方差：
MSE(θ̂) = E[(θ̂ − θ)²] = [Bias(θ̂)]² + Var(θ̂)
This decomposition shows that an estimator with some bias can sometimes have a smaller MSE than an unbiased one, if its variance is much smaller. However, in most A-Level contexts, we prefer unbiased estimators among those with comparable variance.
这个分解表明，如果一个估计量有偏，但其方差小得多，其 MSE 有时可能小于无偏估计量。然而，在大多数 A-Level 情境中，在方差相当的估计量中我们优先选择无偏估计量。
For the sample variance, the biased estimator Sₙ² has MSE that may be smaller than S² in some cases, but S² is preferred for its unbiasedness.
对于样本方差，有偏估计量 Sₙ² 的 MSE 在某些情况下可能比 S² 小，但 S² 因其无偏性而被优先使用。
10. Worked Example: Bias and Standard Error | 典型例题：偏差与标准误差
Question: A random sample of n observations is taken from a normal distribution with mean μ and variance σ². The estimator T = (1/n) ∑(c Xᵢ) is proposed for μ. Find the bias and variance of T.
题：从正态分布 N(μ, σ²) 中抽取 n 个随机观测值。提出估计量 T = (1/n) ∑(c Xᵢ) 来估计 μ。求 T 的偏差和方差。
Solution: Since E(Xᵢ) = μ, we have E(T) = (1/n) ∑ c μ = c μ. Thus Bias(T) = E(T) − μ = (c − 1) μ.
解：因为 E(Xᵢ) = μ，所以 E(T) = (1/n) ∑ c μ = c μ。因此 Bias(T) = E(T) − μ = (c − 1) μ。
For the variance, because the Xᵢ are independent, Var(T) = Var( (1/n) ∑ c Xᵢ ) = (c²/n²) ∑ Var(Xᵢ) = (c²/n²) × n σ² = c² σ² / n.
对于方差，由于 Xᵢ 相互独立，Var(T) = Var( (1/n) ∑ c Xᵢ ) = (c²/n²) ∑ Var(Xᵢ) = (c²/n²) × n σ² = c² σ² / n。
If c = 1, T is unbiased and its standard error is σ/√n, matching the sample mean. If c ≠ 1, T is biased, but its variance may change in the opposite direction.
若 c = 1，则 T 无偏，其标准误差为 σ/√n，与样本均值一致。若 c ≠ 1，则 T 有偏，但其方差可能朝相反方向变化。
11. Common Exam Pitfalls | 常见考试误区
One common mistake is using n in the denominator for the sample variance. Remember that the unbiased variance uses n−1. Another mistake is writing SE(X̄) = σ/√n when σ is unknown, without replacing it with s/√n.
一个常见错误是在样本方差中使用 n 作分母。记住无偏方差使用 n−1。另一个错误是当 σ 未知时仍写 SE(X̄) = σ/√n，而没有用 s/√n 代替。
Also, be careful to distinguish between unbiasedness and consistency. A consistent estimator is one whose distribution collapses to the true value as n→∞; an unbiased estimator is not necessarily consistent, and vice versa.
另外，注意区分无偏性和一致性。一致估计量是当 n→∞ 时其分布收敛到真实值；无偏估计量不一定一致，反之亦然。
Finally, the standard error is a theoretical quantity based on the true σ, while the estimated standard error is a data-based approximation. In exam questions, read carefully whether σ is given or must be estimated.
最后，标准误差是基于真实 σ 的理论量，而标准误差的估计值是基于数据的近似。在考试题目中，要仔细阅读 σ 是已知还是必须从数据估计。
12. Summary | 总结
Bias measures systematic error of an estimator: Bias(θ̂) = E(θ̂) − θ. Standard error measures random sampling variability: SE(θ̂) = √Var(θ̂). The sample mean X̄ is unbiased for μ with standard error σ/√n. The sample variance S² = 1/(n−1) ∑(Xᵢ − X̄)² is unbiased for σ². When σ is unknown, use s/√n for the estimated standard error.
偏差衡量估计量的系统误差：Bias(θ̂) = E(θ̂) − θ。标准误差衡量随机抽样变异性：SE(θ̂) = √Var(θ̂)。样本均值 X̄ 是 μ 的无偏估计，其标准误差为 σ/√n。样本方差 S² = 1/(n−1) ∑(Xᵢ − X̄)² 是 σ² 的无偏估计。当 σ 未知时，使用 s/√n 作为标准误差的估计值。
Published by TutorHao | Further Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导Cancel reply