Sample Size Calculator

How the Sample Size Calculator Works

Determining the right sample size is crucial for any survey, poll, or study. This powerful sample size calculator helps you estimate the smallest number of responses you need to achieve reliable statistical results, given your desired confidence level, margin of error, population proportion, and population size. It’s a must-have tool for researchers in statistics, market research, and social sciences.

With this calculator, you can adjust parameters like the confidence level (often 95 %), margin of error (commonly 5 %), the population proportion (frequently estimated at 50 % for maximum variability), and total population size. It then uses standard statistical formulas to compute a precise minimum sample size—ensuring your study results are accurate and credible.

√ Formula Used

To compute the required sample size, the calculator uses the following procedure:

1. Infinite population assumption:

   n = (Z² × p × q) / e²

   • Z is the z‑score for the chosen confidence level (e.g., 1.96 for 95 %).

   • p is the population proportion (expressed as a decimal, e.g., 0.50).

   • q = 1 – p.

   • e is the margin of error (decimal form, e.g., 0.05).

2. Finite population correction:
   If the total population size N is known and finite, use:

   n_adjusted = (n × N) / (n + N – 1)

This ensures accuracy in studies where the population isn’t infinitely large.

Examples

Let’s walk through two examples to see how the calculator works in practice.

Example 1: Unlimited Population

• Confidence level: 95 % → Z = 1.96

• Margin of error: 5 % → e = 0.05

• Population proportion: 50 % → p = 0.5, q = 0.5

• Population size: Unlimited

Calculation:

n = (1.96² × 0.5 × 0.5) / 0.05²
  = (3.8416 × 0.25) / 0.0025
  = 0.9604 / 0.0025
  = 384.16

Rounded up, the required sample size is 385 responses.

Example 2: Finite Population (10,000)

Using the same initial estimate of n = 385:

n_adjusted = (385 × 10000) / (385 + 10000 – 1)
           = 3,850,000 / 10,384
           ≈ 370.6

Rounded up, you need 371 responses to maintain the same precision within a population of 10,000.

Understand Your Inputs

• Confidence Level: Indicates how sure you are that the true population value falls within your margin of error. Higher levels (e.g., 99 %) require larger samples.

• Margin of Error: The range within which the true value may lie. Smaller margins need more participants.

• Population Proportion: Best guess of response distribution. Using 50 % (maximum variability) creates a conservative, larger sample size.

• Population Size: For small or mid-sized populations, applying the finite population correction yields more realistic sample sizes.

Practical Tips

• If you don’t know p, use 50 % to ensure maximum accuracy.

• For very large populations (e.g., > 100,000), finite correction has minimal effect; your sample size stays close to the infinite-population estimate.

• The formula assumes simple random sampling. Other sampling methods (like stratified, cluster, or non-probability sampling) may require more complex formulas or adjustment factors.

• For small populations (e.g., < 1,000), rounding and finite adjustments are critical—otherwise you may overestimate the sample size.

Sampling Table Example

Here’s a handy table showing required sample sizes for different combinations of confidence level and margin of error, assuming p = 0.5 and unlimited population:

Confidence Level	Margin of Error	Sample Size
90 %	5 %	271
95 %	5 %	385
99 %	5 %	662
95 %	3 %	1,067
95 %	1 %	9,604

• Required sample size rises steeply if you narrow the margin of error or increase confidence level.

Why It Matters

Using a properly determined sample size ensures:

• Statistical validity: Results will reflect the real population within set precision.

• Cost efficiency: Avoids wasting resources on overly large samples.

• Credibility in reporting: Peer reviewers and stakeholders expect methodological rigor.

What to Do Next

• Enter your values into the calculator:

  • Confidence level (e.g. 95 %)

  • Margin of error (e.g. 5 %)

  • Population proportion (e.g. 50 %)

  • Your population size (if known)

• Hit “Calculate” to get your required minimum sample.

• Apply adjustments: If your sample design or population is more complex, consult a statistician.

Why Use Finite Population Correction?

When the population you survey isn’t extremely large, the finite correction prevents overestimating sample size. It uses:

n_adjusted = (n × N) / (n + N – 1)

This adjustment keeps sample size realistic for closed populations like classrooms, communities, or specific user groups.

What do researchers commonly ask?

When should I use a 50% population proportion?

Using p = 0.5 yields the maximum required sample size; this is safest if you're uncertain about the true distribution. If you confidently know p is much higher or lower (e.g., 10 %), plug that in to reduce required sample size.

Does the calculator work for very small surveys like N=200?

Yes. For example, with N = 200, confidence = 95% and e = 5%:

Initial n ≈ 385
n_adjusted = (385 × 200) / (385 + 200 – 1) ≈ 128.5 ≈ 129

That means you'd need around 129 responses—not the full population.

What if I’m using stratified sampling?

Simple random sampling formulas don’t apply directly. Use the calculator as a baseline, but plan additional sampling within each stratum to ensure representation. You may need expert guidance or specialized software.