對稱分布之期望值若存在則必與中位數相等. 當期望值存在時, 統計學家常用樣本 平均數建立對稱分布之期望值的信賴區間. 但當期望值不存在時通常用樣本中位 數來建立對稱分布之中位數的信賴區間 (參考 Casella and Berger(2002)). 以常態 分布及柯西分布為例, 因此二分布之機率密度函數非常接近, 常常誤判. 可能將樣 本平均數用於柯西分布, 將樣本中位數用於常態分布, 此時統計學家希望知道此誤 判所導致信賴區間覆蓋機率之變化. 此外, 就固定分布而言, 樣本平均數及樣本中 位數在建立信賴區間之表現的比較, 亦為統計之重要問題. 本文將討論樣本平均數 及樣本中位數在一維及二維常態及柯西分布之位置參數 (location parameters) 的 區間估計 (interval estimation), 藉以比較此二統計量之優劣.;The mean (if it exists) of a symmetric distribution must be equal to the median. Statistician usually construct the confidence interval from sample mean when the mean exist, and use sample median to construct confidence interval when the mean does not exist. Using Normal distribution and Cauchy distribution as examples, we misjudge often since the P.D.F. of these two distribution are similar. We may use sample mean on Cauchy or use sample median on Gaussian. Statistician want to observe the variation for this misjudgement and compare these two statistics. In this paper, we compare the performance of sample mean and sample median on interval estimations of Gaussian and Cauchy location parameters.
