统计学

描述统计

Mean: x̄ = Σxᵢ / n—Average of all values

Median—Middle value when sorted

Mode—Most frequent value

Range = max - min—Spread of data

Variance: σ² = Σ(xᵢ-μ)²/N—Population variance

Sample var: s² = Σ(xᵢ-x̄)²/(n-1)—Bessel's correction

Std dev: σ = √(variance)—Measure of spread

IQR = Q3 - Q1—Interquartile range

概率

P(A) = favorable / total—0 ≤ P ≤ 1

P(A') = 1 - P(A)—Complement

P(A∪B) = P(A)+P(B)-P(A∩B)—Addition rule

P(A∩B) = P(A)·P(B|A)—Multiplication rule

P(A|B) = P(A∩B)/P(B)—Conditional probability

Independent: P(A∩B) = P(A)·P(B)—

Bayes: P(A|B) = P(B|A)P(A)/P(B)—Bayes' theorem

分布

n! = n×(n-1)×...×1—Factorial

nPr = n!/(n-r)!—Permutations

nCr = n!/(r!(n-r)!)—Combinations

With replacement: nʳ—

Multiplication principle—If A has m, B has n → m×n

检验

Normal: N(μ, σ²)—Bell curve, 68-95-99.7 rule

z = (x - μ) / σ—Standard score

Binomial: P(X=k) = nCk·pᵏ(1-p)ⁿ⁻ᵏ—n trials, prob p

Binomial: μ=np, σ²=np(1-p)—

Poisson: P(X=k) = e⁻λλᵏ/k!—Rate λ events

Poisson: μ=λ, σ²=λ—

Uniform: μ=(a+b)/2, σ²=(b-a)²/12—

Exponential: P(X≤x) = 1-e⁻λˣ—Waiting times

回归

68% within ±1σ—

95% within ±2σ—

99.7% within ±3σ—

z=1.645 → 90% CI—

z=1.960 → 95% CI—

z=2.576 → 99% CI—

Hypothesis Testing

H₀: null hypothesis—No effect / no difference

H₁: alternative hypothesis—Effect exists

p-value < α → reject H₀—α usually 0.05

Type I error (α)—Reject true H₀ (false positive)

Type II error (β)—Fail to reject false H₀

Power = 1 - β—Probability of detecting effect

t-test: compare means—t = (x̄-μ)/(s/√n)

Chi-squared: categorical data—χ² = Σ(O-E)²/E

Regression & Correlation

ŷ = a + bx—Simple linear regression

b = Σ(xᵢ-x̄)(yᵢ-ȳ)/Σ(xᵢ-x̄)²—Slope

a = ȳ - bx̄—Intercept

r = correlation coefficient—-1 ≤ r ≤ 1

R² = coefficient of determination—% of variance explained

Residual = yᵢ - ŷᵢ—Actual minus predicted

r² = R² for simple regression—

Confidence Intervals

CI = x̄ ± z*(σ/√n)—Known σ

CI = x̄ ± t*(s/√n)—Unknown σ, use t

Proportion: p̂ ± z*√(p̂(1-p̂)/n)—

Sample size: n = (z*σ/E)²—E = margin of error

df = n - 1—Degrees of freedom (1 sample)

Larger n → narrower interval—

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