Uniform Distribution

Simplest continuous distribution with constant probability density across a bounded interval.

Distribution Overview & Constant-Density Behavior

The Uniform distribution represents complete uncertainty within specified bounds, where all values in the interval are equally likely. It serves as the foundation for random number generation and many statistical concepts.

Key Characteristics:

• • Shape: Rectangular (flat) between bounds
• • Support: Finite interval [a, b]
• • Parameters: Lower bound (a) and upper bound (b)
• • Density: Constant within the interval
• • Symmetry: Symmetric around midpoint

Standard Uniform U(0,1):

• • Mean: 0.5
• • Variance: 1/12 ≈ 0.0833
• • Standard Deviation: √(1/12) ≈ 0.289
• • Range: [0, 1]
• • PDF: f(x) = 1

Density Function & Moment Formulas

Probability density function

a,b

= lower and upper bounds of the interval

Cumulative distribution function

Mean and variance

Reading the Uniform Density Curve

Example Test & Uniformity Result Interpretation

Evidence of Uniformity

• • KS or Anderson-Darling p-value > 0.05
• • Histogram bars roughly equal in height
• • Sample mean near (a+b)/2
• • Chi-square test fails to reject equal frequencies

Evidence Against Uniformity

• • KS or Anderson-Darling p-value ≤ 0.05
• • Histogram shows clustering or skew
• • Sample variance far from (b-a)²/12
• • Chi-square test rejects equal frequencies

sixsigmacenter.com Implementation Guide

Example Data Layout for Uniform Distribution

Uniform distribution testing with Kolmogorov-Smirnov statistics and frequency analysis. Data should be scaled to [0,1] or specify custom bounds [a,b]. Sample Size Recommendation: Minimum 30-50 observations for reliable uniform distribution testing and parameter estimation in goodness-of-fit analysis.

This is how your spreadsheet or CSV file should be structured before you upload it. Match the column names, keep one row per observation, and leave the analysis to the platform.