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第 21 週 · 離差度量與統計 · 第 5 節

標準分(z-score)與正態分佈

不同科目、不同試卷的分數不能直接比!標準分 z 把任何分數翻譯成「離自己那組均值幾個標準差」,於是能跨組比較。再配上鐘形的正態分佈和 68–95–99.7 法則,估比例一眼看穿。

▶ 第一關: 標準分 z = (x − μ) / σ
raw scores (mean=70, sigma=10)5060708085z scale (translated)-2-10+1+1.5
mean (z=0)z < 0below meanz > 0above meansign tells which side of the mean
▶ 第二關: 用 z 跨組比較與反推
different scales, cannot compare rawMath: mean=70, sigma=10A=80Eng: mean=60, sigma=5B=75same z scale: now we can compare-10123A z=1B z=3
rank by z (bigger z = more outstanding)A z=11B z=33
▶ 第三關: 正態分佈與 68–95–99.7
mean (mu)peak at the meantailtailbell curve, symmetric about mu
68%95%99.7%0-11-22-3368% / 95% / 99.7% inside +-1,+-2,+-3 sigma
z = 0 (mu)-1+1-2+2z axis under the bellmu maps to 0, +-sigma maps to +-1
⚔ BOSS: 標準分綜合
z-score cheat sheetz = (x - mean) / sigmax = mean + z * sigmabigger z =more outstanding68 / 95 / 99.7+-1 +-2 +-3 sigmacompare z, not raw scores