Sample Size Calculator

Find Out The Sample Size

This calculator computes the minimum number of necessary samples to meet the desired statistical constraints.

Confidence Level:	70% 75% 80% 85% 90% 95% 98% 99% 99.9% 99.99% 99.999%
Margin of Error:
Population Proportion:		Use 50% if not sure
Population Size:		Leave blank if unlimited population size.

Find Out the Margin of Error

This calculator gives out the margin of error or confidence interval of observation or survey.

Confidence Level:	70% 75% 80% 85% 90% 95% 98% 99% 99.9% 99.99% 99.999%
Sample Size:
Population Proportion:
Population Size:		Leave blank if unlimited population size.

Statistical inference generally relies on examining a limited subset of members drawn from a larger group, treating that subset as a representative sample of the whole. Suppose a community contains a fraction p that differs in some characteristic from the remaining fraction 1‑p – for instance, the share of people with brown hair versus those with other hair colors. By selecting n individuals and computing the sample proportion p̂ for brown‑haired subjects, we obtain an estimate of p. Because we rarely sample the entire population, p̂ will typically deviate from the true p due to sampling variability. Nevertheless, confidence intervals can be derived to express how close p̂ is likely to be to the actual proportion.

Statistics of a Random Sample

Confidence Level

A confidence level expresses how confident we are that a chosen interval correctly reflects the underlying population within a specified margin. The most frequent levels are 90 %, 95 % and 99 %, each associated with a particular z‑score that can be obtained from standard tables or formulas. These z‑scores presuppose a normally distributed sampling distribution, as mentioned earlier under “Statistics of a Random Sample.” When an experiment or survey is repeated many times, the confidence level indicates the proportion of those repetitions for which the calculated interval will include the true population value.

Confidence Interval

In statistical terminology, a confidence interval is a predicted span of plausible values for a population parameter—for example, 40 ± 2 or 40 ± 5 %. Using the common 95 % confidence level, if we were to draw many independent samples from the same population and compute an interval each time, about 95 % of those intervals would enclose the actual parameter. The 95 % figure refers to the reliability of the interval‑construction method, not to any single interval. After an interval is computed, it either does or does not contain the true value. Factors that influence interval width include the sample size, the chosen confidence level, and the variability observed in the data.

In the context of statistics, a population denotes the entire set of elements or events relevant to a particular research question or experiment. This may be a concrete collection of objects, systems, or even a hypothetical assemblage. Most often, the term refers to a group of people—for instance, the workforce of a company, residents of a specific age bracket within a region, or the patrons present in a university library at a given moment.

When the population is finite, the standard formula must be adjusted by incorporating the finite‑population correction factor (N‑n)/(N‑1). This factor is required because the assumption of independent draws breaks down for small populations. Imagine a room with ten individuals aged 1 to 100; if one selected person is 100 years old, the chance that the next picked person is also very old diminishes. The correction factor compensates for such dependence. See the example below for computing a confidence interval with an effectively infinite population.

EX: Given that 120 people work at Company Q, 85 of which drink coffee daily, find the 99% confidence interval of the true proportion of people who drink coffee at Company Q on a daily basis.

Sample Size Calculation

Sample size refers to the number of observations or repetitions needed in a study to reliably estimate a phenomenon’s variability. Determining an appropriate sample size is crucial for any empirical investigation that draws conclusions about a larger group from a subset. In practice, researchers decide on an acceptable margin of error ε—the maximum deviation they are willing to tolerate between the sample estimate and the true value. By setting the term to the right of the ± sign in the confidence‑interval formula equal to ε and solving for n, one obtains the required sample size. The resulting formula is displayed below.

Example: Estimate how many shoppers at a U.S. supermarket are vegans with 95 % confidence and a ±5 % margin of error. Assume an initial proportion of 0.5 and an effectively unlimited population. For a 95 % confidence level, the z‑value is 1.96. Consult the table in the confidence‑level section for z‑scores corresponding to other confidence levels.

Applying the numbers from the example, a minimum of 385 respondents would be required. In practice, surveys suggest that roughly 6 % of the U.S. population identifies as vegan, so using 0.06 instead of 0.5 for p̂ would be more realistic. If, on a particular day, 40 out of 500 supermarket customers were vegan, the observed proportion p̂ would be 0.08.