Math & Engineering
Sample Size Calculator
Calculate the required sample size for your research or survey based on population size, confidence level, and margin of error.
Enter values to calculate the required sample size
Related to Sample Size Calculator
The Sample Size Calculator determines the minimum number of subjects required for a study to achieve statistically significant results. It uses statistical formulas that consider several key parameters: population size (if finite), confidence level, margin of error, and response distribution.
The Formula
For infinite populations, the formula is: n = (Z² × p × (1-p)) / e²
For finite populations, we use the correction: n = (N × Z² × p × (1-p)) / (e² × (N-1) + Z² × p × (1-p))
Where:
n = Sample size
Z = Z-score from confidence level
p = Response distribution
e = Margin of error
N = Population size
The calculator automatically selects the appropriate formula based on whether a population size is provided. For infinite populations (when no population size is specified), it uses the standard formula. When a finite population size is given, it applies the finite population correction to provide a more accurate sample size.
The calculated sample size represents the minimum number of respondents or subjects needed to achieve your desired level of precision and confidence in your research results. Understanding the output requires consideration of several factors:
Confidence Level
The confidence level (typically 95%) indicates how certain you can be that your results reflect the true population value. A 95% confidence level means that if you repeated the study multiple times with the same sample size, 95% of the time your results would fall within your specified margin of error.
Margin of Error
The margin of error represents the maximum expected difference between the true population value and your sample estimate. A smaller margin of error requires a larger sample size but provides more precise results.
The response distribution affects your required sample size - a response distribution of 50% typically requires the largest sample size and is often used as a conservative estimate when the true distribution is unknown.
1. What is a good sample size for my research?
A good sample size depends on several factors, including your research goals, population size, and required precision. Generally, larger sample sizes provide more accurate results but require more resources. For most academic studies, a 95% confidence level and 5% margin of error are considered standard.
2. Why does the calculator ask for response distribution?
Response distribution represents the expected spread of responses in your population. If unknown, use 50% as it provides the most conservative (largest) sample size. If you expect responses to be more homogeneous, you can use a different value to potentially reduce the required sample size.
3. How does population size affect the sample size?
For very large or infinite populations, the required sample size tends to stabilize. However, for smaller populations, the finite population correction reduces the required sample size. This is why the calculator asks for population size - it can help you avoid over-sampling in smaller populations.
4. What confidence level should I use?
The most commonly used confidence level is 95%, which provides a good balance between precision and practicality. Higher confidence levels (e.g., 99%) require larger sample sizes and might be necessary for critical research where errors could have serious consequences. Lower confidence levels might be acceptable for preliminary or exploratory research.
5. What is the scientific source for this calculator?
This calculator implements standard statistical sampling theory formulas that are widely used in research methodology. The formulas are derived from the Central Limit Theorem and probability theory, as documented in standard statistical textbooks and research methodology literature. The Z-scores used for confidence levels are based on the standard normal distribution table, which is a fundamental tool in statistics. The finite population correction formula is derived from sampling theory principles established by statistical researchers and is documented in works such as Cochran's "Sampling Techniques" (1977) and other statistical sampling methodology textbooks.