Biased and Unbiased Estimates
Introduced| English |
|---|
| point estimate/pɔɪnt ˈestɪmət/ |
| unbiased/ʌnˈbaɪəst/ |
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.
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 $p$.
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.
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.
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).
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.
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.
Center = bias, spread = variability
Unbiased means centered on the parameter; larger n narrows the spread.
An estimator is unbiased when the center of its sampling distribution...
Unbiased = centered on the parameter.
Increasing the sample size reduces an estimator's variability.
Bigger n → narrower sampling distribution.
A larger sample size can fix the bias of a biased estimator.
Size shrinks variability, not bias — a biased estimator stays off-center.
A single number from a sample used to estimate a parameter is a ___ estimate (one word).
A point estimate is the best single-number guess.
Bias is a property of the ___ of the sampling distribution.
Bias = center off target; variability = spread.