Sample Size Calculator

TL;DR: To calculate the confidence interval for a sample proportion, use the formula p ± z × √(p(1-p)/n), where p is your sample proportion, z is the z-score from your confidence level (typically 1.96 for 95% confidence), and n is your sample size; for a sample size of 100 with a proportion of 0.5 and 95% confidence, the standard error is 0.05, the margin of error is 0.098, and your 95% confidence interval is [0.402, 0.598].

What Is the Sample Size Calculator?

The Sample Size Calculator is a statistical tool that computes the precision of a sample proportion by deriving the standard error, margin of error, and the resulting confidence interval. This is essential for anyone conducting surveys, A/B tests, opinion polls, or quality control checks. The tool evaluates how much trust you can place in the percentage (proportion) you observed in your sample, relative to the entire population.

For example, if you survey 100 customers and 50% say they are satisfied, this calculator tells you whether that "50%" is a solid estimate or a statistical fluke. The primary audience includes market researchers, data analysts, product managers, and social scientists who need to report findings with statistical rigor. By entering just four values—tamano_muestra (sample size), proportion (observed success rate), margen_error (desired precision), and confianza (confidence level)—you receive the exact numerical bounds within which the true population proportion likely falls.

This calculator is distinct from a "sample size determination" tool, which tells you how many respondents you need before collecting data. Instead, this online tool works after data collection to verify the reliability of your observed percentage. It is particularly critical in medical trials where a 5% difference in efficacy could be meaningful, or in political polling where a 2% shift in support changes the projected winner.

How to Use the Calculator

Using this calculator requires only four numeric inputs. Follow these steps precisely to get accurate confidence intervals:

1. Enter the sample size (tamano_muestra): Input the total number of observations in your sample. For example, if you collected 100 survey responses, enter 100. This value must be a positive integer greater than 1.
2. Enter the observed proportion (proportion): Input the percentage of "successes" or "yes" responses as a decimal. For 50% success, enter 0.5. If 30 out of 100 people answered "yes," enter 0.3. The value must be between 0 and 1 (inclusive).
3. Enter the margin of error (margen_error): Input the desired precision level as a decimal. This represents how much ± error you are willing to accept. For a ±5% margin, enter 0.05. Standard values range from 0.01 (very precise) to 0.10 (lenient).
4. Enter the confidence level (confianza): Input the statistical confidence as a decimal. For 95% confidence, enter 0.95. For 99%, enter 0.99. Common values are 0.90, 0.95, and 0.99.
5. Click Calculate: The tool instantly computes four outputs: standard_error (the standard deviation of the sampling distribution), margin_of_error (the maximum expected difference from the true value), ci_lower (the lower bound of your confidence interval), and ci_upper (the upper bound).
6. Interpret results: The confidence interval [ci_lower, ci_upper] defines the range where the true population proportion probably resides. If the interval is too wide, increase your sample size or lower your confidence level.

Formula and Calculation Method

The underlying mathematics for this calculator relies on the Central Limit Theorem, which states that sample proportions follow a normal distribution for large samples. The tool calculates the standard error first, then uses that to derive the margin of error and the confidence interval bounds.

The formula for the standard error (SE) of a proportion is:

SE = √(p × (1 - p) / n)

Where p is the proportion (as a decimal) and n is the sample size. The margin of error (ME) is then calculated by multiplying the standard error by the z-score corresponding to your confidence level:

ME = z × SE

For 95% confidence, z = 1.96. For 90% confidence, z = 1.645. For 99% confidence, z = 2.576. Finally, the confidence interval is:

CI = p ± ME, which produces the lower bound (p - ME) and upper bound (p + ME).

Worked Example: Suppose you have a sample size (tamano_muestra) of 100, a proportion (proportion) of 0.5, a margin of error input of 0.05, and a confidence level (confianza) of 0.95.

Step 1: Calculate the standard error.
SE = √(0.5 × (1 - 0.5) / 100) = √(0.25 / 100) = √0.0025 = 0.05.

Step 2: Calculate the margin of error using the z-score for 95% confidence (1.96).
ME = 1.96 × 0.05 = 0.098.

Step 3: Calculate the confidence interval bounds.
Lower bound (ci_lower) = 0.5 - 0.098 = 0.402.
Upper bound (ci_upper) = 0.5 + 0.098 = 0.598.

Step 4: Interpret the result. You can state with 95% confidence that the true population proportion falls between 40.2% and 59.8%. Notice that the margin of error passed as an input (0.05) is ignored for the calculation; the actual computed margin of error is 0.098, which is wider than your target. This discrepancy signals that your sample size is too small to achieve ±5% precision at 95% confidence. To achieve ±5%, you would need a sample size of approximately 385.

Practical Examples

Here are three realistic scenarios showing how the inputs change the outputs and what they mean for decision-making.

Scenario 1: A company surveys 400 customers and finds 60% are satisfied. At 95% confidence, the interval is [55.2%, 64.8%]. The company can reliably claim that the majority of all customers are satisfied because the lower bound exceeds 50%.

Scenario 2: A political poll surveys only 100 voters, finding 52% support Candidate A. At 90% confidence, the interval is [43.8%, 60.2%]. Because this interval includes 50%, the poll is statistically inconclusive—Candidate A could be ahead or behind. This demonstrates why reliable polls require larger samples.

Scenario 3: A factory tests 500 units and finds a 2% defect rate. At 99% confidence, the interval is [0.4%, 3.6%]. The upper bound of 3.6% is critical: if the acceptable defect threshold is 5%, the process is in control. If the threshold were 3%, the process would need adjustment.

Tips for Accurate Results

To obtain trustworthy calculations, you must avoid common input errors and understand the assumptions behind the formula.

• Never enter zero or negative values: The formula requires a positive sample size (n > 1) and a proportion strictly between 0 and 1 (0 < p < 1). Entering n = 0 causes division by zero; entering p = 0 or p = 1 produces a zero standard error, implying false certainty. If your observed proportion is 0% or 100%, use a correction (e.g., 0.5/n or 1 - 0.5/n).
• Use decimal format, not percentages: Enter proportion 0.5, not 50. Enter confidence 0.95, not 95. The calculator assumes decimal inputs; entering whole numbers will produce wildly incorrect intervals (e.g., entering 95 for confidence will attempt to compute a z-score for a 9500% confidence level).
• Do not confuse desired margin with actual margin: The input margen_error is not used in the computation. It is provided as a comparison target. The calculator always computes the actual margin based on your sample size and confidence. If your actual margin exceeds your desired margin, you need a larger sample.
• Check the confidence level z-score: The tool uses standard normal z-scores. For 95% it uses 1.96, for 99% it uses 2.576, for 90% it uses 1.645. If you pass an unusual confidence (e.g., 0.92), the calculator must interpolate a z-score; standard online tools typically round to the nearest common confidence level.
• Be aware of the sample size assumption: The normal approximation formula works well when n × p > 5 and n × (1-p) > 5. If your sample is small (n < 30) or your proportion is extreme (near 0 or 1), consider the Wilson score interval instead. Your calculated ci_lower might be negative or ci_upper might exceed 1; in such cases, clip the bounds to 0 and 1 for interpretation.

Frequently Asked Questions

1. Why is my calculated margin of error different from the margin of error I entered?
The margin of error input (margen_error) serves as a target for your study design, not a variable in the calculation. The calculator computes the achieved margin of error based directly on your sample size and proportion. For a sample size of 100 with p = 0.5 at 95% confidence, the achieved margin is ±9.8%, regardless of whether you entered 0.05 or 0.10. If you need ±5%, you must increase your sample size to roughly 385 (calculated as (1.96/0.05)² × 0.25). The input margin is simply a benchmark for you to compare against the calculated margin_of_error output.

2. What does it mean if my confidence interval includes 0 or 1?
If the ci_lower is negative or ci_upper exceeds 1, it indicates that the normal approximation is unreliable for your data. This typically occurs when the proportion is very close to 0 or 1, or when the sample size is small. For example, with n = 20 and p = 0.98, the standard error formula gives √(0.98 × 0.02 / 20) = 0.0313, and at 95% confidence, the upper bound becomes 0.98 + 1.96 × 0.0313 = 1.041. Proportions cannot exceed 100%, so interpret the upper bound as effectively 1.0. Statistically, you should switch to the Wilson score interval or the Clopper-Pearson exact method for reliable bounds.

3. Can I use this calculator to determine how large my sample should be?
No, this specific calculator is purely descriptive—it evaluates the precision of an existing sample. It does not solve for the required sample size (n). To determine sample size, you would rearrange the formula: n = (z² × p(1-p)) / ME². For instance, with 95% confidence (z=1.96), margin of error 0.05, and an assumed proportion of 0.5, you need n = (1.96² × 0.5 × 0.5) / 0.05² = 384.16, rounded up to 385. However, a standard "Sample Size Determination Calculator" would use inputs for confidence, margin, and estimated proportion to produce this directly. Use this tool only to quantify the uncertainty you already have.

4. Why does the standard error get larger when the proportion is close to 0.5?
This is a mathematical property of binomial variance. The product p(1-p) is maximized when p = 0.5 (yielding 0.25) and approaches zero as p nears 0 or 1. For example, with n = 100, the standard error at p = 0.5 is √(0.25/100) = 0.05, but at p = 0.1 it is √(0.09/100) = 0.03. This means that surveys reporting results near 50% have the largest possible uncertainty—a finding of 50% is less precise than a finding of 90% or 10%, given the same sample size. This is why pollsters typically assume p = 0.5 when calculating required sample sizes; it represents the worst-case (most conservative) scenario.