Sampling Distribution Calculator

Sample Mean Distribution

x̄ ~ N(μ, σ/√n)

• Central Limit Theorem
• Normal approximation
• Sample size effects
• Distribution shape
• Sampling variability
• Mean convergence
• Probability density

Standard Error

SE = σ/√n

• Sampling precision
• Estimation accuracy
• Size relationship
• Variance reduction
• Confidence bounds
• Error magnitude
• Precision metrics

Confidence Intervals

CI = x̄ ± (z_α/2 × SE)

• Interval estimation
• Coverage probability
• Margin of error
• Level selection
• Critical values
• Precision control
• Interval width

Distribution Parameters

• Location measures
• Scale parameters
• Shape characteristics
• Symmetry properties
• Tail behavior
• Density features
• Moment structure

Sampling Theory

• Random sampling
• Independence assumption
• Sample representation
• Distribution theory
• Asymptotic behavior
• Limiting distributions
• Convergence rates

Inference Properties

• Unbiasedness
• Consistency
• Efficiency
• Sufficiency
• Robustness
• Power analysis
• Error control

Research Design

• Sample size planning
• Power calculations
• Effect detection
• Design efficiency
• Cost optimization
• Resource allocation
• Study planning

Quality Control

• Process monitoring
• Control limits
• Specification compliance
• Variation control
• Performance standards
• Acceptance sampling
• Risk assessment

Decision Making

• Hypothesis testing
• Risk evaluation
• Policy analysis
• Comparative studies
• Benchmark setting
• Performance assessment
• Strategic planning