Sample Size Calculator (Cochran Finite Population Correction)

Find required respondents for a target margin of error and population size.

Required Finite Sample Size (n)
370 respondents
Infinite Population Sample (n₀) 385 respondents
Sampling Fraction (n / N) 3.7 %
Margin of Error 5 %
Inputs
Z-score for confidence level (1.96 = 95% confidence).
%
Estimated population proportion p (%) — 50% maximizes variance.
%
Acceptable margin of error e (%).
Total target population size N.
Results & Graph
Required Finite Sample Size (n)
370 respondents
Infinite Population Sample (n₀)
385 respondents
Sampling Fraction (n / N)
3.7 %
Margin of Error
5 %

Required Finite Sample Size (n)

Graph 1 • Z Critical Sensitivity Recalculates the domain equation across 50% to 150% of the primary input while holding other parameters constant.
0 respondents 429.8 respondents 859.7 respondents 50%: 96 respondents 96 50% 75%: 212 respondents 212 75% 100%: 370 respondents 370 100% 125%: 567 respondents 567 125% 150%: 796 respondents 796 150%
Graph 2 • Primary & Supporting Metric Comparison Compares the primary output alongside supporting domain benchmarks.
Required Finite Sample Size (n) 370 respondents Infinite Population Sample (n₀) 385 respondents Sampling Fraction (n / N) 3.7 % Margin of Error 5 %
Graph 3 • Exact Output Table & Units Displays each calculated output in its exact domain unit.
Required Finite Sample Size (n)370 respondentsInfinite Population Sample (n₀)385 respondentsSampling Fraction (n / N)3.7 %Margin of Error5 %
Summary: Cochran infinite sample size is 385; after finite population correction for N = 10,000, minimum sample size is 370.
Summary: Cochran infinite sample size is 385; after finite population correction for N = 10,000, minimum sample size is 370.

Formula & Mathematical Methodology: Sample Size Mathematical Identity

n0 = (Z² × p(1 − p)) / e²
n = n0 / (1 + (n0 − 1) / N)
n_0 = (Z² × p(1 − p)) / e²; n = n_0 / (1 + (n_0 − 1) / N)
  • zCritical: Z Critical (numeric parameter)
  • proportionPct: Proportion Pct (%)
  • marginErrorPct: Margin Error Pct (%)
  • populationSize: Population Size (numeric parameter)

Rounding: All calculations run in double-precision IEEE-754 arithmetic; displayed outputs round to up to 4 decimal places.

Worked Example: Worked Mathematical Verification: Sample Size

Find required respondents for a target margin of error and population size.

  1. Verify the domain constraints of the input parameters.
  2. Apply the governing mathematical identity: n_0 = (Z² × p(1 − p)) / e²; n = n_0 / (1 + (n_0 − 1) / N).
  3. Cross-check the primary value against the supporting invariants and coordinate graph.
Result: Evaluates Sample Size via n_0 = (Z² × p(1 − p)) / e²; n = n_0 / (1 + (n_0 − 1) / N).

Frequently Asked Questions

What mathematical formula does the Sample Size Calculator (Cochran Finite Population Correction) use?

It directly evaluates n_0 = (Z² × p(1 − p)) / e²; n = n_0 / (1 + (n_0 − 1) / N) in double-precision floating-point arithmetic and updates all supporting invariants and graphs live.

How can I compare different parameter sets?

Click Compare in the results toolbar to pin up to three parameter combinations side by side.

Can I export the mathematical report or share a link?

Yes. Use the Share Link, Copy Summary, or Markdown/JSON export buttons in the results panel.

Mathematics & Statistics • Inferential Statistics Guide

How to Use the Sample Size Calculator (Cochran Finite Population Correction): Practical Guide & Visual Analysis

Determine minimum survey sample size using Cochran’s formula and finite population correction. Includes textbook mathematical notation, step-by-step verification, and three custom SVG visualizations. Whether you are researching sample size calculator (cochran finite population correction), sample size, sample size survey, inferential statistics calculator, this interactive tool helps you determine minimum survey sample size using Cochran’s formula and finite population correction.

Interpreting the 3 Custom Visual Graphs & Key Outputs

As you adjust Z Critical, Proportion Pct, Margin Error Pct, Population Size, the calculator immediately computes Primary exact or numerical solution, Supporting identities & invariants, 3-Panel mathematical SVG plots and updates all three domain-specific SVG charts in real time. Instead of relying on a single static figure, you can inspect the primary domain diagram, component breakdown, and multi-point trajectory curve to see how changing your inputs shifts the final result. Use the Compare (up to 3 scenarios) button to evaluate side-by-side variations before making a decision.

Real-World Benchmark & Planning Tips (Universal Mathematical & Statistical Standard (ISO 80000-2))

In our benchmark scenario (Worked Mathematical Verification: Sample Size), Evaluates Sample Size via n_0 = (Z² × p(1 − p)) / e²; n = n_0 / (1 + (n_0 − 1) / N). Keep in mind that uses standard Euclidean geometry, real/complex algebra, and NIST statistical definitions. For related calculations in Mathematics & Statistics, explore Scientific Calculator, Statistics & Standard Deviation Calculator, Geometry Calculator (Right Triangle, Circle & Cylinder).