Central Limit Theorem (CLT) 中央極限定理

When
population mean = μ
population variance = σ²

random sample of size n
random samples: X₁, X₂, ..., X_n,

z-score/z-value/standard score z分數/標準化值 (Wikipedia/維基百科)

sum of samples/sample sum 樣本和 S_n = (X₁ + X₂ + ... + X_n) = ΣX_i where i = 1 to n
sample mean 樣本平均數 x̄ = (X₁ + X₂ + ... + X_n)/n = S_n/n
sample variance 樣本變異數 s²
sample standard deviation 樣本標準差 s
sample size n

if σ is known:

Z = (x̄ - μ)/(σ/√ n) = (Sn - n)/σ√n

When n → ∞ (n >= 30),

Z ~ N(0, 1)

One special case of gamma distribution:
chi-square distribution 卡方分布 (χ²) α = v/2 or n/2, β = 2 for gamma distribution.

χ² = (n - 1)s²/σ² with n-1 degrees of freedom.

if σ is unknown and n is large:

s ≈ σ

Student t-distribution/Student's t-distribution 學生t分布
t-distribution t分布

T = (x̄ - μ)/(s/√ n)  with n-1 degrees of freedom.

t-table:
row: probability α at t_α
column - degree of freedom

參考資料

Essentials of Probability & Statistics for Engineers & Scientists (Walpole at el.)，機率與統計 (繆紹綱譯，滄海2013)，p201, 242-248

機率與統計 (陳鍾誠網站)

Mathematical statistics with applications in R, Ramachandran & Tsokos, 2nd edition (2015), p162-164, 191-194

Student's t-distribution (Wikipedia)