Biased and Unbiased Estimates

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 $p$.

Bias is being off-centre; variability is spread. Larger samples reduce variability but not bias

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.

Write from an unbiased estimator

• The mean of a sampling distribution equals the population parameter.
• A different method centers somewhere else.