What is Cochran's formula for sample size?

Cochran's formula is `n₀ = (Z² × p × q) / e²`, where `Z` is z-score, `p` is estimated proportion, `q = 1 - p`, and `e` is desired margin of error.

Why is a 50% proportion assumption standard in survey sizing?

A proportion of `p = 0.5` maximizes the product `p(1 - p) = 0.25`, producing the most conservative (largest) sample size recommendation to guarantee statistical power.

How many survey responses are needed for a population of 100,000 at 95% confidence and 5% margin of error?

Applying the finite population formula yields exactly 383 completed responses.

What is the difference between response rate and sample size?

Sample size is the number of completed responses needed. If your survey response rate is 20% and you need 400 responses, you must invite `400 / 0.20 = 2,000` participants.

Does a larger population require a proportionally larger sample size?

No! Beyond a population of ~50,000, required sample size plateaus around 384 for a 5% margin of error, whether surveying a city of 100,000 or a country of 300 million.

How does margin of error affect required survey sample size?

Halving the margin of error (e.g. from 10% down to 5%) requires quadrupling the total sample size due to the inverse square relationship in statistical sampling formulas.

Math & Statistics Cluster

Sample Size Calculator (Survey & Research Sample Sizing)

Calculate statistically significant sample sizes for surveys, polls, and scientific research.

TL;DR: Sample Size Calculator determines the minimum sample size needed for statistical validity using Cochran's formula: $n = [Z^2 \times p(1-p)] / e^2$.

Confidence level

Margin of error %

Estimated proportion %

Population size (optional)

How Do You Calculate the Sample Size Needed for a Survey?

To calculate sample size for an infinite population: n = [Z² × p(1 - p)] / e², where Z is the critical value (1.96 for 95%), p is estimated proportion (0.5 for maximum variance), and e is the margin of error (e.g., 0.05 for ±5%). For a finite population (N), apply the finite population correction.

How to Use the Sample Size Calculator

Our Sample Size Calculator performs high-precision mathematical operations directly in your browser with zero latency and complete client-side privacy.

Select your Confidence Level (90%, 95%, or 99%).
Enter your acceptable Margin of Error percentage (e.g., 3% or 5%).
Optionally enter Total Population Size ($N$) for finite population correction.
Enter Estimated Proportion (50% default provides conservative maximum sizing).
Click 'Calculate Sample Size' to see required completed survey responses.

Mathematical Formula & Equations

Determines the minimum survey sample size required to achieve a specified confidence level and margin of error.

\[ n = \frac{Z^2 \cdot p(1-p)}{E^2} \]

Calculation Example

For 95% confidence ($Z = 1.96$), 5% margin of error ($E = 0.05$), and maximum variability ($p = 0.5$): $$n = \frac{1.96^2 \times 0.25}{0.0025} = 384.16 \implies 385 \text{ respondents}$$

100% Client-Side Privacy & Data Security

All calculations, amortization schedules, variables, and sensitive numerical datasets execute 100% locally in your web browser memory. Your financial, medical, and personal values are never transmitted, logged, or uploaded to any external server.

Frequently Asked Questions

: Cochran's formula is `n₀ = (Z² × p × q) / e²`, where `Z` is z-score, `p` is estimated proportion, `q = 1 - p`, and `e` is desired margin of error.
: A proportion of `p = 0.5` maximizes the product `p(1 - p) = 0.25`, producing the most conservative (largest) sample size recommendation to guarantee statistical power.
: Applying the finite population formula yields exactly 383 completed responses.
: Sample size is the number of completed responses needed. If your survey response rate is 20% and you need 400 responses, you must invite `400 / 0.20 = 2,000` participants.
: No! Beyond a population of ~50,000, required sample size plateaus around 384 for a 5% margin of error, whether surveying a city of 100,000 or a country of 300 million.
: Halving the margin of error (e.g. from 10% down to 5%) requires quadrupling the total sample size due to the inverse square relationship in statistical sampling formulas.

Cite This Calculator

Need to reference this online calculator in a financial proposal, academic paper, blog post, or classroom curriculum? Copy a citation below.

NexLove.org. (2026). Sample Size Calculator [Online Calculation Tool]. https://nexlove.org/tools/sample-size-calculator

“Sample Size Calculator.” NexLove.org, 2026, nexlove.org/tools/sample-size-calculator.

NexLove.org (2026) Sample Size Calculator. Available at: https://nexlove.org/tools/sample-size-calculator (Accessed: 2026).

@misc{nexlove_sample_size_calculator,
  title = {Sample Size Calculator},
  author = {{NexLove.org}},
  year = {2026},
  url = {https://nexlove.org/tools/sample-size-calculator}
}

Embed This Tool

Add the live Sample Size Calculator to your own website with this snippet — it loads the real, working calculator in an iframe.

<iframe src="https://nexlove.org/embed/sample-size-calculator.html" width="100%" height="600" style="border:1px solid #e2e8f0;border-radius:12px" title="Sample Size Calculator — NexLove.org" loading="lazy"></iframe>

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