Biased and Unbiased Estimates · 有偏与无偏点估计
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| point estimate/pɔɪnt ˈestɪmət/ | 点估计 | diǎn gū jì |
| unbiased/ʌnˈbaɪəst/ | 无偏 | wú piān |
Guessing the parameter
- A point estimate 点估计 is a single number from a sample used to estimate a parameter.
- $\bar{x}$ estimates $\mu$; $\hat{p}$ estimates $p$; $s$ estimates $\sigma$.
- It's our best single guess — but it's just one value from a varying statistic.
- The quality of an estimator depends on where it centers and how much it varies.
猜测参数
- 点估计是从样本得到的、用来估计参数的单个数。
- $\bar{x}$ 估计 $\mu$；$\hat{p}$ 估计 $p$；$s$ 估计 $\sigma$。
- 它是我们最好的单个猜测——但它只是一个变动统计量的一个取值。
- 一个估计量的好坏取决于它居中在哪里、以及它变动多少。
What "unbiased" means
- An estimator is unbiased 无偏 if its sampling distribution is centered on the true parameter.
- "Centered on" means the mean of the estimator equals the parameter.
- On average, over many samples, an unbiased estimator hits the target.
- $\bar{x}$ and $\hat{p}$ are unbiased estimators of $\mu$ and $p$.
“无偏”是什么意思
- 如果一个估计量的抽样分布以真实参数为中心，它就是无偏的。
- “以……为中心”意味着估计量的均值等于那个参数。
- 平均而言，在许多样本上，无偏估计量会命中目标。
- $\bar{x}$ 和 $\hat{p}$ 是 $\mu$ 和 $p$ 的无偏估计量。
Center = bias
- Bias is about the center of the sampling distribution, not any single estimate.
- If the estimator's mean sits off the parameter, it's biased — systematically too high or low.
- An unbiased estimator can still miss on any one sample (that's variability, not bias).
- Aim first for an estimator centered in the right place.
中心 = 偏倚
- 偏倚关乎抽样分布的中心，而非任何单个估计。
- 如果估计量的均值偏离参数，它就是有偏的——系统性地偏高或偏低。
- 无偏估计量在任何单个样本上仍可能偏离（那是变异性，不是偏倚）。
- 首先要争取一个居中在正确位置的估计量。
Variability shrinks with n
- The variability of an estimator (how much it bounces around) decreases as $n$ increases.
- Bigger samples → a narrower sampling distribution → more precise estimates.
- Bias and variability are separate goals: hit the center, and keep the spread small.
- The ideal estimator is unbiased with low variability.
变异性随 n 缩小
- 估计量的变异性（它跳动多少）随 $n$ 增大而减小。
- 更大的样本 → 更窄的抽样分布 → 更精确的估计。
- 偏倚和变异性是两个独立的目标：命中中心，并且保持分散小。
- 理想的估计量是无偏且变异性低的。
Bias and variability are different things. Bias is about the center of the sampling distribution (is it on target?); variability is about its spread (how much do estimates bounce?). A larger sample size shrinks variability but does not fix bias — a biased method stays off-center at any $n$ (echoing Unit 3).
偏倚和变异性是两回事。****偏倚关乎抽样分布的中心（对准目标了吗？）；变异性关乎它的分散（估计跳动多少？）。更大的样本量缩小变异性，但不能修复偏倚——有偏的方法在任何 $n$ 下都仍然偏离中心（呼应第 3 单元）。
Estimating a population mean $\mu = 100$.
- $\bar{x}$ is unbiased: the mean of its sampling distribution is exactly $100$.
- With $n = 25$ the estimates spread widely; with $n = 400$ they cluster tightly around $100$.
- Same center (unbiased), smaller variability — a better estimate.
估计一个总体均值 $\mu = 100$。
- $\bar{x}$ 是**无偏的：**它抽样分布的均值恰好是 $100$。
- 当 $n = 25$ 时估计分散很大；当 $n = 400$ 时它们紧紧聚集在 $100$ 附近。
- 相同的中心（无偏），更小的变异性——一个更好的估计。
A point estimate is a single-number guess of a parameter (e.g. $\bar{x}$ for $\mu$). An estimator is unbiased when the center of its sampling distribution equals the parameter. Bias concerns that center; variability concerns the spread and shrinks as $n$ grows — but a larger sample never removes bias.
点估计是对参数的单数猜测（如用 $\bar{x}$ 估 $\mu$）。当估计量抽样分布的中心等于参数时，它是无偏的。偏倚关乎那个中心；变异性关乎分散，且随 $n$ 增大而缩小——但更大的样本永远不能消除偏倚。
Center = bias, spread = variability · 中心 = 偏倚，分散 = 变异性
Unbiased means centered on the parameter; larger n narrows the spread. · 无偏意味着以参数为中心；n 越大分散越窄。
An estimator is unbiased when the center of its sampling distribution... · 当一个估计量抽样分布的中心……时，它是无偏的。
Unbiased = centered on the parameter. · 无偏 = 以参数为中心。
Increasing the sample size reduces an estimator's variability. · 增大样本量会减小估计量的变异性。
Bigger n → narrower sampling distribution. · n 越大 → 抽样分布越窄。
A larger sample size can fix the bias of a biased estimator. · 更大的样本量能修复有偏估计量的偏倚。
Size shrinks variability, not bias — a biased estimator stays off-center. · 规模缩小变异性，而非偏倚——有偏估计量仍偏离中心。
A single number from a sample used to estimate a parameter is a ___ estimate (one word). · 从样本得到、用来估计参数的单个数是 ___ 估计（填英文一词 point）。
A point estimate is the best single-number guess. · 点估计是最好的单数猜测。
Bias is a property of the ___ of the sampling distribution. · 偏倚是抽样分布的 ___ 的性质。
Bias = center off target; variability = spread. · 偏倚 = 中心偏离目标；变异性 = 分散。