Binomial Sample Size Calculator

Calculate minimum sample sizes for reliability testing and determine demonstrated reliability from test results using non-parametric and parametric (Weibull) one-sided binomial distribution methods

Binomial Sample Size Calculator

Calculate the minimum sample size required to demonstrate a reliability level with a given confidence using one-sided binomial distribution.

Formula:

n = ln(1 - C) / ln(R)

Where: R = Reliability, C = Confidence, k = Allowed Failures

Target reliability level (e.g., 95 for 95%)

Statistical confidence (e.g., 90 for 90%)

Maximum number of failures allowed in test (max 20)

Minimum Sample Size:

N/A

Test N/A with 0 or fewer failures to demonstrate 95.00% reliability with 90.0% confidence

Sample Size vs. Desired Reliability

Required sample sizes across different reliability targets with multiple confidence levels.

Allowed Failures: 0
%

Demonstrated Reliability Calculator (Non-Parametric)

Calculate the demonstrated reliability based on test results (reverse one-sided binomial calculation)

Use this calculator to:

  • Determine reliability demonstrated by your test results
  • Validate if your test meets reliability requirements
  • Plan follow-up testing if needed

(%)

Demonstrated Reliability:

N/A

With 30 units tested, 0 failure(s) observed, and 90.0% confidence, the demonstrated reliability is at least N/A

Reliability vs. Sample Size

Lower and higher sample sizes with multiple confidence levels to visualize demonstrated reliability.

Allowed Failures: 0
%

Parametric Binomial Test Calculator

Calculate test time and sample size using Weibull distribution combined with binomial test planning. This calculator matches Method 2A (Calculate Samples) and Method 2B (Calculate Test Time) from the Reliability Analytics Toolkit. Given a reliability requirement at mission time and Weibull shape parameter, the calculator determines the optimal test parameters to demonstrate that requirement.

Core Equations:

Weibull Distribution:

R_test = exp[-(T_test / η)^β]

Binomial Confidence:

1 - C = Σ(i=0 to f) [C(n,i) × (1-R_test)^i × R_test^(n-i)]

Method:

  1. Given R_mission at T_mission and β, solve for characteristic life (η)
  2. Use η and β to calculate R_test at T_test (Method 2A) or find T_test (Method 2B)
  3. Use R_test in binomial equation to calculate sample size n

Where:

  • T_mission = Mission time (required reliability time)
  • R_mission = Required reliability at mission time
  • C = Confidence level
  • f = Maximum allowable failures
  • n = Sample size
  • R_test = Reliability at test time T_test
  • η (eta) = Characteristic life (calculated from mission requirements)
  • β (beta) = Shape parameter

Time at which reliability requirement must be met (T_mission)

Required reliability at mission time (R_mission)

Weibull shape parameter (beta). Characteristic life (η) is calculated from mission requirements.

Available test time (T_test)

Statistical confidence (C)

Maximum allowable failures (f)

About Binomial Sample Size Calculation

The binomial sample size calculator is used in reliability engineering to determine the minimum number of units that must be tested to demonstrate a required reliability level with a specified confidence. This is a one-sided test, meaning it demonstrates that reliability is at least the target value.

  • Zero-failure test: The simplest case where all units must pass. Formula: n = ln(1 - C) / ln(R)
  • Allowing failures: When some failures are acceptable, the calculation uses the cumulative binomial distribution
  • One-sided test: Demonstrates that reliability is at least the target value, not exactly equal to it
  • Common applications: Reliability demonstration tests, qualification testing, acceptance testing