Σ學習通 · W21D4 · 127 / 129
0 / 15
第 21 週 · 離差度量與統計 · 第 4 節

比較離差與數據變換

上一節學會了標準差。這節學兩件 DSE 必考的事:怎樣用標準差「比較」兩組數誰更穩定;以及對每個數據做「加一個數 / 乘一個數」之後,平均和標準差會怎麼變 —— 配滿圖例。

▶ 第一關: 用標準差比較兩組
Class A: small std = 3tight, stableClass B: large std = 8spread, unstablesame mean 70, smaller std = more stable
smaller std ← more consistentσ=2.5tighterσ=6.1wider spreadbar length = how spread out the data is
▶ 第二關: 數據變換 ① 每項 + 常數
add k to every value: whole row shifts rightoriginalgap 30after + kgap still 30same spacing ⇒ std unchanged
deviation x − mean stays the samex − meanmean(x+k) − (mean+k)mean + kboth arrows same length
why + k cancels outnew dev = (x + k) − (mean + k)= x − mean + k − kk − k = 0= x − mean (unchanged ⇒ σ same)
▶ 第三關: 數據變換 ② 每項 × 常數
multiply by a: whole row stretchesoriginalgap 20after × 3gap 60 = 3 × 20spacing × 3 ⇒ std × 3
std scales by |a|old σ = 4new σ = |a| × 4 = 12×3 makes the spread 3 times wider
transform ax + b = stretch then shiftx× a (stretch: σ × |a|)+ b (shift: σ unchanged)only a changes σ; b does not
⚔ BOSS: 變換綜合
effect on mean / std / variancetransformmeanstdvar+ kμ + kσσ²× a|a|σa²σ²ax + baμ + b|a|σa²σ²b shifts mean only; a scales the spread