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Central Limit Theorem: Summary of Formulas

Module by: Dr. Barbara Illowsky, Susan Dean

Formula 1: Central Limit Theorem for Sample Means (Averages)

X¯ X ~ N ( μ X , σ X n ) N( μ X , σ X n ) Mean for Averages ( X¯ ) ( X ): μ X μ X

Formula 2: Central Limit Theorem for Sample Means (Averages) Z-Score and Standard Error of the Mean

z = x¯ - μ X ( σ X n ) z= x - μ X ( σ X n ) Standard Error of the Mean (Standard Deviation for Averages (X¯)( X )): σ X n σ X n

Formula 3: Central Limit Theorem for Sums

ΣXΣX ~ N [ ( n ) ⋅ μ X , n ⋅ σ X ] N[(n)⋅ μ X , n ⋅ σ X ] Mean for Sums ( ΣX ) (ΣX): n ⋅ μ X n⋅ μ X

Formula 4: Central Limit Theorem for Sums Z-Score and Standard Deviation for Sums

z = Σx − n ⋅ μ X n ⋅ σ X z= Σx − n ⋅ μ X n ⋅ σ X Standard Deviation for Sums ( ΣX ) (ΣX): n ⋅ σ X n ⋅ σ X

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