Determine the minimum required sample size or evaluate the margin of error and confidence interval for survey research, academic studies, and finite population models.
• Find Sample Size: Enter desired confidence level (typically 95%), acceptable margin of error (e.g. 5%), and expected proportion to calculate minimum required sample size $n$.
• Find Margin of Error: Enter an existing sample size $n$ to determine the survey precision $\pm E\%$.
• Leave Population Size blank if surveying an unlimited or unknown population.
This calculator computes the minimum number of necessary samples to meet the desired statistical constraints.
Principles of statistical inferencing, Cochran's formula, finite population correction, and confidence interval theory.
In clinical research, opinion polling, and quality assurance, surveying an entire population (a census) is often physically impossible, costly, or time-prohibitive. By taking a representative, randomly selected sample of size $n$, statisticians can extrapolate findings to the whole population with quantifiable mathematical certainty.
The probability that the sample results accurately capture the true population parameter within the margin of error. A 95% confidence level means that if the survey were repeated 100 times, 95 of those surveys would yield intervals containing the real population value.
The maximum expected difference between the true population parameter and the sample estimate (e.g., “$\pm 3\%$”). A narrower margin of error requires a substantially larger sample size.
The product $p(1 - p)$ reaches its absolute mathematical maximum when $p = 0.50$ ($0.5 \times 0.5 = 0.25$). Setting $p = 50\%$ guarantees the most conservative (largest) necessary sample size, ensuring sufficient statistical power regardless of the true split.
When a sample size represents a noticeable percentage of the total population ($n / N > 5\%$), sampling without replacement reduces variability. FPC adjusts the required sample size downward according to $n = n_0 / [1 + (n_0 - 1) / N]$.
Suppose a national polling organization wants to measure public opinion with a 95% confidence level ($Z = 1.960$), a 5% margin of error ($E = 0.05$), and assumes maximum uncertainty ($p = 0.50$).
A high school principal surveys a student body of $N = 1,000$ at a 95% confidence level and 5% margin of error.
In a customer satisfaction survey, $n = 100$ customers were polled, with $60\%$ reporting positive satisfaction ($p = 0.60$) at a 95% confidence level ($Z = 1.960$) with an unlimited customer base.
Once a population is large (e.g. over 50,000 or 100,000), the Finite Population Correction factor approaches 1. For a 95% confidence level and 5% margin of error, the required sample size for a city of 100,000 is 383, and for an entire country of 300 million it is 384.
A 95% confidence level (Z = 1.960) has become the gold standard in scientific and social research because it strikes an ideal pragmatic balance between statistical precision and realistic sample collection costs.
Always default to 50% (0.50). This produces the maximum possible standard error p(1 - p) = 0.25, ensuring your sample size is sufficiently large to maintain your desired margin of error no matter the survey outcome.
No. When sampling without replacement from a finite population, the sample size n cannot exceed N. The calculator will automatically alert you if n > N.