Statistical Distributions Notes

Sampling Distribution

  • Definition: Probability distribution of a statistic from repeated random samples.

  • Key Concepts:

  • Population: Entire set of observations.

  • Sample: Subset of the population.

  • Statistic: Summary characteristic of a sample (e.g., sample mean).

  • Sampling Distribution: Distribution of a statistic over multiple samples.

Sampling Distribution of the Sample Mean (X)

  • Sample Mean: Average of a sample, represented as X.

  • Mean of Sampling Distribution: E(X) = Population Mean (µ).

  • Standard Error of Mean: SE(X) = σ/√n.

Central Limit Theorem (CLT)

  • Sampling distribution of sample mean approaches normal distribution as n increases (n ≥ 30).

  • Mean = Population Mean (µ). Standard deviation (SE) = σ/√n.

Importance of Sampling Distribution

  • Inferential Statistics: Estimates population parameters from sample statistics.

  • Hypothesis Testing: Determines the probability of sample means under null hypothesis.

  • Confidence Intervals: Provides range for population parameters.

Types of Sampling Distributions

  • Sampling Distribution of the Sample Mean (X).

  • Sampling Distribution of the Sample Proportion (p̂).

  • Sampling Distribution of the Sample Variance (s²).

Standard Normal Distribution

  • Definition: Special case of normal distribution (mean = 0, SD = 1).

  • PDF: f(x) = (1/√(2π)) e^(-x²/2).

  • Properties:

  • Symmetry around mean.

  • 68-95-99.7 Rule for standard deviations.

  • Application: Calculating Z-scores, confidence intervals, hypothesis testing.

Chi-Square Distribution

  • Definition: Distribution of sum of squares of standard normal variables.

  • PDF: f(x, ν) = (x^(ν/2 - 1) e^(-x/2)) / (2^(ν/2) Γ(ν/2)).

  • Properties: Non-negative, Mean = ν, Variance = 2ν.

  • Application: Goodness-of-fit tests, independence tests.

t-Distribution

  • Definition: Used when sample size is small and population SD is unknown.

  • PDF: f(t, ν) = [Gamma((ν + 1)/2)] / (√(νπ) Gamma(ν/2))(1 + t²/ν)^(-(ν + 1)/2).

  • Properties: Symmetric, approaches normal distribution as ν increases.

  • Application: Hypothesis testing, confidence intervals for means.

F-Distribution

  • Definition: Ratio of two independent chi-square variables.

  • PDF: Non-negative, positively skewed.

  • Properties: Mean = ν2 / (ν2 - 2) (ν2 > 2).

  • Application: Comparing variances in ANOVA, model comparisons.

Symbols in Statistics

  • µ (Mu): Population mean

  • x̄ (X-bar): Sample mean

  • σ (Sigma): Population standard deviation

  • s: Sample standard deviation

  • n: Sample size

  • E(X): Expected value of statistic

  • SE(X): Standard error of mean

  • p̂ (P-hat): Sample proportion

  • ν (Nu): Degrees of freedom

  • Γ (Gamma): Gamma function

  • Z: Z-score

  • χ² (Chi-square): Chi-square statistic

  • F: F-statistic

These symbols are fundamental in understanding and working with statistical concepts.