Mathematics, Statistics & Geometry

Sample Size Calculator

Determine the required survey sample size to achieve your desired confidence level and margin of error, with optional finite population correction.

Required Sample Size
385
Calculation StepsConfidence Level: 95% => Z-Score = 1.96 Margin of Error (E) = 5% = 0.05 Population Proportion (P) = 50% = 0.5 1. Calculate Infinite Population Sample Size (n₀): n₀ = (Z² * P * (1 - P)) / E² n₀ = (1.96² * 0.5 * (1 - 0.5)) / 0.05² n₀ = (3.8416 * 0.2500) / 0.002500 n₀ = 384.1600 No finite population correction applied (Population Size left blank/zero). 3. Final Sample Size: Always round up to the nearest whole person to guarantee the margin of error. Sample Size = 385

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Statistical Significance Design

The Sample Size Calculator is the starting point for any rigorous scientific study or market research. By balancing your desired confidence level against your acceptable margin of error, it calculates the exact minimum number of participants required.

n=Z2p(1−p)E2\begin{aligned} n = \frac{Z^2 p(1-p)}{E^2} \end{aligned}

Where:
n=
The total number of people or items you need to test
Z=
The statistical value associated with your desired Confidence Level
E=
Your acceptable threshold of inaccuracy (e.g., ±5%)

The Myth of Population Size

The most common misconception in statistics is that if a population is 10 times larger, you need a sample size 10 times larger. The math proves this false.

Imagine taking a sip of soup to see if it needs salt. It doesn't matter if you are tasting a small bowl or an industrial restaurant vat; one well-stirred spoonful gives you the exact same information. The same principle applies to surveying populations.

Real-World Applications

  • Medical Research: Determining the exact number of patients needed in a Phase III clinical trial to statistically prove a new drug is safe without endangering unnecessary participants.
  • Manufacturing: Calculating how many microchips out of a batch of 100,000 must be rigorously tested to guarantee a 99% defect-free rate.
  • Political Polling: News organizations determining that surveying precisely 1,068 registered voters will accurately predict a national election within a ±3% margin.

Frequently Asked Questions

It depends entirely on how accurate you want to be. If you want a 95% confidence level with a tight 3% margin of error, you need about 1,068 people. If you accept a looser 5% error, you only need 385.

Surprisingly, no! Whether you are surveying a city of 100,000 or a country of 300 million, the required sample size is almost identical. The math depends on the sample, not the population.

If your total population is very small (e.g., surveying a company of 500 employees), the standard formula will overestimate how many people you need. FPC mathematically reduces your required sample size.

If you expect 80% of people to answer 'Yes' to a survey, P=0.80. If you have no idea what people will say, you MUST use P=0.50 (50%). This is the 'worst-case scenario' and guarantees your sample size is large enough.

You cannot survey 0.4 of a person. If the math says you need 384.1 people, rounding down to 384 would technically put your margin of error slightly above your limit. You must always round up to 385.

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