Sample Size Calculator: Margin of Error Solver
Calculate statistically rigorous sample sizes and confidence intervals for market research, clinical trials, and population surveys. Supports Cochran's formula, Finite Population Correction (FPC), and two-tailed Z-score distributions.
* 50% is standard default providing the most conservative (maximum) sample size.
385 participants
1.95996
Not Required ❌
Margin of Error Sensitivity Ledger (95% Confidence)
| Sample Size (n) | Margin of Error (E) | Audit Status |
|---|---|---|
| 100 | ±9.7998% | Alternative Projection |
| 250 | ±6.1979% | Alternative Projection |
| 385 | ±4.9944% | ★ Solver Selection |
| 500 | ±4.3826% | Alternative Projection |
| 1000 | ±3.099% | Alternative Projection |
Statistical Audit Workings
The Mathematical Foundations of Sample Size Determination
Cochran's formula, continuous variance modeling, and Finite Population Correction (FPC)
In inferential statistics, sampling allows researchers to draw conclusions regarding an entire population without surveying every individual. For dichotomous proportions (e.g., yes/no responses, approval ratings, conversion rates), the required sample size for an infinite or large population is governed by Cochran's formula (1977):
Where Z represents the critical value from the standard normal distribution corresponding to the desired confidence level (1.960 for 95%, 2.576 for 99%, and 1.645 for 90%), p is the estimated proportion, and E is the acceptable margin of error expressed as a decimal. When the proportion is unknown, setting p = 0.50 produces maximum variance (0.25), guaranteeing a conservative sample size.
Required Sample Size Matrix: Confidence Level vs. Margin of Error
Baseline required completed responses (n) for large or infinite populations with maximum variance (p = 0.50)
| Margin of Error (E) | 90% Confidence (Z = 1.645) | 95% Confidence (Z = 1.960) | 98% Confidence (Z = 2.326) | 99% Confidence (Z = 2.576) |
|---|---|---|---|---|
| ± 1.0% | 6,764 | 9,604 | 13,530 | 16,587 |
| ± 2.0% | 1,691 | 2,401 | 3,382 | 4,147 |
| ± 3.0% (Polling Standard) | 752 | 1,068 | 1,503 | 1,843 |
| ± 4.0% | 423 | 601 | 846 | 1,037 |
| ± 5.0% (General Research) | 271 | 385 | 541 | 664 |
| ± 10.0% (Exploratory) | 68 | 97 | 135 | 166 |
Frequently Asked Questions
Authoritative technical answers to Finite Population Correction, confidence intervals, and variance assumptions
How does the Finite Population Correction (FPC) factor change sample size when sampling over 5% of a population?
When a survey sample exceeds 5% of the total population (n/N > 0.05), sampling without replacement reduces variance because each surveyed individual reduces the remaining unknown pool. The FPC formula n = n0 / [1 + (n0 - 1)/N] significantly shrinks the required sample size for small, defined groups (such as enterprise employee rosters or clinical patient registries) without sacrificing statistical precision.
Why is a 50% (0.5) proportion assumption standard when historical data is unavailable?
The product p(1 - p) in Cochran's formula reaches its absolute mathematical maximum at p = 0.50 (0.5 × 0.5 = 0.25). Assuming p = 50% is the conservative gold standard in polling and market research because it guarantees a sample size large enough to achieve the specified margin of error regardless of the true underlying population distribution.
When should researchers choose a 90% confidence level instead of 95% or 99%?
A 90% confidence level (Z = 1.645) requires approximately 30% fewer respondents than 95% (Z = 1.960), making it ideal for preliminary exploratory studies, agile software feedback, or budget-constrained surveys. In contrast, 95% is the standard for academic and commercial market research, while 99% (Z = 2.576) is reserved for medical trials and regulatory compliance where error tolerance is critical.
Why does doubling the sample size only reduce the margin of error by roughly 29% rather than 50%?
Margin of error is inversely proportional to the square root of the sample size (E ∝ 1/√n). Due to diminishing returns, cutting the margin of error in half (e.g., from ±6% to ±3%) requires quadrupling (4×) the sample size (1/√4 = 0.5), meaning doubling sample size reduces error by a factor of 1/√2 ≈ 0.707 (a ~29.3% reduction).
How does non-response bias impact required sample size in real-world survey fielding?
The calculated sample size represents completed, valid responses. If a survey anticipates a 20% completion rate, the initial deployment outreach must equal n_outreach = n_required / 0.20 (a 5× oversampling). Furthermore, systematic non-response among specific demographic cohorts cannot be corrected simply by increasing sample size; it requires stratified post-weighting.