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第 21 週 · 離差度量與統計 · 第 3 節

標準差與方差

最重要的離差度量。標準差就是「每個數離平均大約多遠」的典型距離;平方再開方,是為了把單位還原回去。這節把離差、方差、標準差一步步算給你看,配滿圖例。

▶ 第一關: 離差 deviation
mean μ = 5x−μ < 0x−μ > 0deviation = each point minus the mean
mean μ=74 (−3)6 (−1)11 (+4)sum of deviations: −3 −1 +4 = 0positives and negatives cancel out
1²=12²=44²=16squaring magnifies big deviations
▶ 第二關: 方差與標準差
x − μ(x−μ)²average= variancesqrt = σsquare: drop the signsqrt: restore unitdeviation → square → average → sqrtvariance has squared unit, σ restores it
data: 2 4 4 4 5 5 7 9 , N=8xx−μ(x−μ)²2 4 4 4-3 -1 -1 -19 1 1 15 5 7 90 0 2 40 0 4 16μ = 40/8 = 5Σ(x−μ)² = 32variance = 32/8 = 4σ = sqrt(4) = 2
▶ 第三關: 標準差的意義與應用
small σ: tightlarge σ: spreadsame mean, different σ (spread)
mean μtypical distance = σσ ≈ how far a point sits from the mean
quick recall: 4 steps1) mean μ2) deviation x−μ , square it3) variance = Σ(x−μ)² / N4) std σ = sqrt(variance)bigger σ = more spread ; σ ≥ 0
⚔ BOSS: 方差與標準差綜合