The total number of individuals in the group you want to study. Use a large number if unknown or infinite.
90%
95%
99%
The probability that the sample statistic (e.g., mean) is within the confidence interval.
The acceptable range of error in your results. A smaller margin of error requires a larger sample size.
An estimate of the variability in the population. 0.5 is often used when there's no prior information, representing maximum variability.
Required Sample Size (n):
—
Understanding Sample Size Calculation
Determining the appropriate sample size is a crucial step in research design, statistical analysis, and data collection. A sample that is too small may not yield statistically significant results, while a sample that is too large can be unnecessarily costly and time-consuming. The formula used here is a common approach for estimating the sample size required for a population proportion, assuming a finite population.
The Formula
The formula for calculating sample size (n) for a proportion, considering a finite population, is often derived from the formula for an infinite population and then adjusted:
For an infinite population, the sample size n₀ is calculated as:
n₀ = (Z² * p * (1-p)) / E²
Where:
Z is the Z-score corresponding to the desired confidence level.
p is the estimated proportion of the attribute in the population (standard deviation of the proportion).
E is the desired margin of error (expressed as a proportion, e.g., 5% = 0.05).
For a finite population, the adjusted sample size n is calculated using Cochran's formula (or a similar adaptation):
n = n₀ / (1 + (n₀ - 1) / N)
Substituting n₀:
n = (Z² * p * (1-p)) / (E² + (Z² * p * (1-p)) / N)
Or, more commonly applied by first calculating n₀ and then adjusting:
n = n₀ / (1 + (n₀ / N)) (a simplified adjustment for large populations where n₀ - 1 is approximated by n₀)
This calculator uses the adjusted formula for finite populations. If the population size (N) is very large, the adjustment has a minimal effect, and n will approach n₀.
Key Components Explained:
Population Size (N): The total number of individuals in the group you are interested in studying. If the population is extremely large or unknown, a sufficiently large number can be used, and the sample size will primarily depend on the confidence level and margin of error.
Confidence Level: This indicates how confident you want to be that the true population parameter falls within your calculated confidence interval. Common levels are 90%, 95%, and 99%. The higher the confidence level, the larger the sample size needed. For example, a 95% confidence level means that if you were to repeat the study many times, 95% of the results would contain the true population value.
Margin of Error (E): This is the maximum amount of error you are willing to tolerate in your results. It's often expressed as a percentage or proportion. A smaller margin of error (e.g., ±3% instead of ±5%) means your sample results are expected to be closer to the actual population value, thus requiring a larger sample size.
Estimated Standard Deviation (p): For proportions, this represents the expected variability of the attribute being measured. If you have no prior knowledge, using p = 0.5 (50%) is the most conservative approach, as it yields the largest possible sample size for a given confidence level and margin of error.
When to Use This Calculator
This sample size calculator is useful for researchers, statisticians, marketers, and anyone planning a study or survey involving proportions. It helps ensure that the collected data will be representative of the population and that the conclusions drawn will have a desired level of confidence and precision.
Example Use Case: A marketing firm wants to conduct a survey to understand customer satisfaction with a new product. They aim for a 95% confidence level and a 4% margin of error. Based on previous surveys, they estimate that about 60% of customers will be satisfied (p=0.6). If the total customer base is 10,000, they can use this calculator to determine the minimum number of customers they need to survey.
function getZScore(confidenceLevel) {
if (confidenceLevel === '0.90') return 1.645;
if (confidenceLevel === '0.95') return 1.960;
if (confidenceLevel === '0.99') return 2.576;
return 1.960; // Default to 95%
}
function calculateSampleSize() {
var populationSize = parseFloat(document.getElementById("populationSize").value);
var confidenceLevel = parseFloat(document.getElementById("confidenceLevel").value);
var marginOfError = parseFloat(document.getElementById("marginOfError").value) / 100; // Convert % to decimal
var standardDeviation = parseFloat(document.getElementById("standardDeviation").value);
var resultElement = document.getElementById("result-value");
resultElement.innerText = "–"; // Reset
// Input validation
if (isNaN(populationSize) || populationSize <= 0) {
alert("Please enter a valid Population Size (greater than 0).");
return;
}
if (isNaN(marginOfError) || marginOfError <= 0) {
alert("Please enter a valid Margin of Error (greater than 0).");
return;
}
if (isNaN(standardDeviation) || standardDeviation 1) {
alert("Please enter a valid Estimated Standard Deviation between 0.01 and 0.99.");
return;
}
var zScore = getZScore(confidenceLevel);
// Calculate n0 (sample size for infinite population)
var n0 = (Math.pow(zScore, 2) * standardDeviation * (1 – standardDeviation)) / Math.pow(marginOfError, 2);
// Adjust for finite population
var n = n0 / (1 + (n0 – 1) / populationSize);
// Ensure sample size is a whole number and not larger than population
var finalSampleSize = Math.ceil(n);
if (finalSampleSize > populationSize) {
finalSampleSize = populationSize;
}
if (finalSampleSize < 1) {
finalSampleSize = 1;
}
resultElement.innerText = finalSampleSize.toLocaleString();
}