How to Determine Sample Size

Calculating the sample size is one of the first and most important steps in designing a high-quality survey or study. It's a balancing act: you need enough participants to get reliable results, but not so many that the study becomes too expensive or time-consuming. This calculator helps you find that optimal number.

Key Concepts Explained

• Confidence Level: This tells you how confident you can be that your results reflect the true views of the population. A 95% confidence level means that if you were to repeat the survey 100 times, you would get the same results within the margin of error 95 of those times. It corresponds to a Z-score from a standard normal distribution.
• Margin of Error: Also called the confidence interval, this is the "+/-" figure usually reported in survey results. It represents the range within which you expect the true population value to lie. For example, if your margin of error is 5% and 60% of your sample picks an answer, you can be confident that between 55% (60-5) and 65% (60+5) of the entire population would pick the same answer.
• Population Proportion: This is the expected distribution of a characteristic in your population. For example, if you're surveying voters, you might expect a 50/50 split on a certain issue. If you are unsure, use 50%, as it is the most conservative choice and will yield the largest required sample size.
• Population Size: This is the total number of individuals in the group you are studying. If your population is very large (e.g., the entire population of a country) or unknown, you can leave this field blank. The calculator will use a formula for an infinite population. If the population is smaller and known, filling this in will apply a "finite population correction" for a more accurate sample size.

The Formulas for Sample Size

This calculator uses two primary formulas:

1. Cochran's Formula (for large populations):

This is the standard formula used when the population is large or infinite.

n₀ = (Z² × p × (1-p)) / e²

• n₀ is the initial sample size.
• Z is the Z-score from your chosen confidence level.
• p is the population proportion.
• e is the margin of error.

2. Finite Population Correction (FPC) Formula:

If you provide a population size, the calculator adjusts the initial sample size with this formula:

n = n₀ / (1 + (n₀ - 1) / N)

• n is the final, adjusted sample size.
• n₀ is the initial sample size from Cochran's formula.
• N is the population size.

Once you collect your data, you can analyze its distribution with our Standard Deviation Calculator.

Related Statistical Tools

• Math Calculators - Our full suite of mathematical and statistical tools.
• Rounding Calculator - For rounding your final results.
• Scientific Notation Calculator - Useful for working with very large population numbers.