Sample Size Calculator

Math

Determine the optimal statistical sample size required for survey research or evaluate the margin of error based on confidence level and population size.

Presets:

Recommended Sample Size

385

respondents needed for 95% confidence with ±5% margin of error

Confidence Level 95%

Z-Score Critical Value 1.960

Target Population Infinite / Any

Assumed Proportion 50%

📊 Sample Size Guidelines by Population Scale

Population Scale	Population Size (N)	Required Sample (n)
Small Town (1,000)	1,000	278
Medium City (10,000)	10,000	370
Large Metro (100,000)	100,000	383
State / Region (1,000,000)	1,000,000	384
Unlimited / Infinite	∞	385

Step-by-Step Statistical Calculation

1 1. Critical Value (Z-Score)

For 95% Confidence Level, α = 0.050 → Z = 1.960

Obtained from the standard normal distribution table for a two-tailed test.

2 2. Cochran's Base Formula (Infinite Population)

n₀ = [Z² · p(1 - p)] / E² = [(1.960)² × 0.5 × 0.50] / (0.05)² = 384.15

Determines the minimum sample size required without population size constraints.

3 3. Recommended Sample Size

Sample Size = ⌈384.15⌉ = 385 respondents

Always round up to the nearest integer to maintain the desired confidence level.

How This Calculator Works

Sample size determination is the act of choosing the number of observations or replicates to include in a statistical sample. An appropriate sample size balances confidence against survey costs and prevents sampling bias.

📐 Cochran's Formula & Finite Population Correction

Base: n₀ = [Z² · p(1 - p)] / E² | Finite Correction: n = n₀ / [1 + (n₀ - 1) / N]

n₀: Cochran's recommended sample size for large/infinite populations

Z: Critical value from normal distribution corresponding to selected confidence level

p: Estimated proportion of attribute present in population (0.50 gives maximum variability)

E: Acceptable margin of error (e.g. 0.05 for ±5%)

N: Total population size

Key Mathematical Rules & Properties

Why Assume 50% Population Proportion?

The product p(1 - p) peaks at p = 0.50 (giving 0.25). Using 50% generates the maximum possible variance, guaranteeing the sample will be sufficiently large regardless of the actual outcome.

When Does Finite Population Correction Apply?

FPC is typically used when the sample accounts for more than 5% of the total population (n / N > 0.05). If population size exceeds 50,000, FPC has negligible impact.

Law of Diminishing Returns

Cutting margin of error in half (e.g., from 4% down to 2%) quadruples the required sample size because error scales with 1 / √n.

✎

Worked Example: National Survey

Find sample size needed for 95% confidence and 5% margin of error for a national electorate.

1. For 95% confidence level, Z = 1.96.
2. Set p = 0.50 (maximum variability) and E = 0.05.
3. Calculate n₀ = [(1.96)² × 0.50 × 0.50] / (0.05)² = [3.8416 × 0.25] / 0.0025 = 0.9604 / 0.0025 = 384.16.
4. Round up to the nearest integer: 385 respondents.

Final Answer: 385 respondents

Frequently Asked Questions

Does population size matter for large nations?

Surprisingly, no. Whether sampling the population of the United States (330 million) or California (39 million), a sample of ~385 respondents provides ±5% margin of error at 95% confidence.

What is the relationship between confidence level and margin of error?

Confidence level reflects how sure you can be that the survey results reflect the population (e.g., 95%). Margin of error reflects the radius of error (e.g., ±3%). Higher confidence requires larger samples.

Understanding your calculation

Method, example and limitations

How the calculation works

For an estimated population proportion, a common starting point is n = z²p(1 − p)/e², optionally adjusted for a finite population.

Example to check

At 95% confidence with p = 0.5 and a five-percentage-point margin, the unadjusted size is about 384.16, requiring at least 385 completed observations.

Before using the result

The calculation assumes an appropriate random-sampling design. Nonresponse, clustering, bias and tests of means or effects require additional planning.

Calculation methodology · Report a result that needs review