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03 // Bidirectional STEM Calculators // STATISTICSn = (z² × p × (1 - p)) / (E / 100)²

Survey Sample Size Estimator (Proportion Poll Solver)

Cochran's formula calculates the minimum survey sample size n required to estimate a population proportion p within percentage margin of error E at critical confidence score z.

n = (z² × p × (1 - p)) / (E / 100)²
Pick which value you want to calculate (or type into any field):Solving for: n — Required Sample Size
Calculated

Minimum number of survey respondents

10100,000.00
unit

Acceptable polling error in percentage points (e.g., 5 for ±5%)

0.525
%

1.96 for 95% confidence, 1.645 for 90%, 2.576 for 99%

13.5
unit

Expected proportion (0.05 to 0.95; 0.50 is most conservative)

0.050.95
unit
Calculated Result · Required Sample Size (n)
384.16

Step-by-Step Calculation

01Canonical Law: n = (z² × p × (1 - p)) / (E / 100)²
02Given Inputs (Base Units): E (±%) = 5, z* = 1.96, p = 0.5
03Solve for n (Required Sample Size): n = 384.16

Cochran's formula calculates the minimum survey sample size n required to estimate a population proportion p within percentage margin of error E at critical confidence score z.

Real-World Reference Values

±5% Margin @ 95% Confidencen = 385 Respondents

Standard product & UX research sample size

±3% Margin @ 95% Confidencen = 1,068 Respondents

National political polling benchmark

±1% Margin @ 95% Confidencen = 9,604 Respondents

High-precision epidemiological survey

20-Row Comparison Table

Required Sample Size (n) · Margin of Error (%) (E (±%))

Survey Sample Size Estimator (Proportion Poll Solver) — Margin of Error (%) → Required Sample Size
#Margin of Error (%) (E (±%))Required Sample Size (n)Δ %
011.85 2,806.14 +630.46%
022.2184 1,951.48 +407.99%
032.5868 1,435.20 +273.59%
042.9553 1,099.66 +186.25%
053.3237 869.386 +126.31%
063.6921 704.5373 +83.40%
074.0605 582.4887 +51.63%
084.4289 489.6109 +27.45%
094.7974 417.2977 +8.63%
105 Current384.16 0.00%
115.5342 313.5745 -18.37%
125.9026 275.6518 -28.25%
136.2711 244.2144 -36.43%
146.6395 217.8637 -43.29%
157.0079 195.5586 -49.09%
167.3763 176.5116 -54.05%
177.7447 160.1175 -58.32%
188.1132 145.9057 -62.02%
198.4816 133.5054 -65.25%
208.85 122.6212 -68.08%
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Frequently Asked Questions

Why is p = 0.50 used when the true proportion is unknown?+

The Bernoulli variance term p(1 - p) reaches its mathematical maximum of 0.25 at p = 0.50, guaranteeing your sample size is sufficient for any outcome.

How do I check the margin of error of an existing 1,000-person poll?+

Click SOLVE FOR on E (±%), enter n = 1000, z* = 1.96, and p = 0.50 to obtain ±3.10%.

Does population size affect required sample size?+

For large populations (N > 20,000), sample size n is virtually independent of population size; finite-population correction only matters when n > 5% of N.