What is a confidence level?

The confidence level tells you how sure you can be that your sample results reflect the true attitudes 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.

What is a margin of error?

The margin of error is the degree to which your sample results might differ from the actual population's views. A smaller margin of error (e.g., ±3%) means your results are more precise, but it requires a larger sample size.

What if my population size is very large or unknown?

If your population is very large (e.g., over 100,000) or you don't know its size, you can leave the population size field blank or enter a very large number. The required sample size tends to plateau for very large populations, so the result will still be accurate.

Sample Size Calculator (Confidence Level & Margin of Error)

How Does This Calculator Work?

This calculator determines the minimum number of samples you need to survey to get results that reflect the target population with a desired level of accuracy and confidence. It uses a standard formula for sample size calculation:

Sample Size = (Z² * P * (1-P)) / E²

Z-score (Z): This value is determined by your chosen confidence level. For example, a 95% confidence level corresponds to a Z-score of 1.96.
Population Proportion (P): This is the expected proportion of the population that has the attribute you are studying. Since this is often unknown, the calculator uses a conservative value of 0.5 (50%), which provides the largest possible sample size.
Margin of Error (E): This is your desired level of precision, expressed as a decimal (e.g., 5% is 0.05).
Finite Population Correction: If you provide a specific population size, the calculator applies a correction formula to adjust the sample size, as smaller populations require a proportionally smaller sample.

The Surprising History of the "Margin of Error"

The concept of "margin of error" in surveys and polls is a cornerstone of modern statistics, but it became a household term thanks to political polling in the 20th century. Before its widespread use, poll results were often presented as absolute certainties, leading to major public surprises, like the infamous "Dewey Defeats Truman" headline in 1948.

Pioneering pollsters like George Gallup worked to make polling more scientific. They began to publicly state a "margin of error" to communicate that their results were not perfect predictions, but rather estimates with a known range of statistical uncertainty. This was a revolutionary step in transparency and public education. It taught people that a sample could never perfectly represent a whole population, but that we could be a certain level of confident (e.g., 95%) that the "true" result lies within a specific range. This concept is now fundamental to understanding the accuracy of any survey or scientific study.

Explore More Related Tools

While you're here, check out some of our other popular math and statistics calculators:

Statistics Calculator: Analyze your survey data with this tool.
Standard Deviation Calculator: Measure the spread of your data.
Probability Calculator: Understand the likelihood of events.
Percentage Calculator: Essential for working with margins of error.
Fraction Calculator: For calculations involving fractional data.
Ratio Calculator: Compare different groups in your data.
Permutation & Combination Calculator: For advanced probability problems.
Scientific Calculator: For advanced mathematical calculations.
Exponent & Root Calculator: Used in some statistical formulas.
Logarithm Calculator: Useful for data transformations.

Frequently Asked Questions (FAQ)

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

Confidence Level tells you how often you can expect your results to be accurate (e.g., 95% of the time). Margin of Error tells you how *much* your results might differ from the true population value (e.g., by ±5%). They work together to define the reliability of your survey.

What are the standard confidence levels to use?

The most common confidence level used in research is 95%. This provides a strong balance between confidence and a manageable sample size. 90% is used when less confidence is acceptable, while 99% is used for research where very high certainty is required.

What if I can't afford the recommended sample size?

If the required sample size is too large for your budget or resources, you have two options: you can either decrease your confidence level (e.g., from 95% to 90%) or increase your margin of error (e.g., from ±5% to ±7%). Both of these will reduce the required number of samples, but at the cost of less confidence or precision in your results.

Sample Size Calculator

Required Sample Size: