Sample Size Calculator

About Sample Size Calculator

Sample size is the number of observations, participants, responses, or other units that should be included in a statistical sample. Choosing an appropriate sample size is important because the number of observations affects the precision of an estimate. A sample that is too small may produce a wide margin of error, while increasing the sample size generally improves precision at the cost of additional time, effort, and resources.

This Sample Size Calculator estimates the number of observations required to estimate a population proportion using a selected confidence level, margin of error, expected proportion, and population size. The calculation first determines an initial sample size for a large or effectively unlimited population and then applies a finite population correction when a specific population size is provided.

The calculator is particularly useful for surveys, questionnaires, market research, polling, quality studies, and other situations where a proportion or percentage of a population needs to be estimated from a sample.

How Sample Size Is Calculated

The calculation begins with an initial sample-size formula based on the desired confidence level, expected population proportion, and margin of error. The result is then adjusted when the population has a known finite size.

Finite Population Correction

When the population has a known and limited size, the initial sample size can be adjusted using a finite population correction. This is important because observations are selected from a finite group rather than from an effectively unlimited population.

The correction generally reduces the required sample size when the population is finite. The smaller the population relative to the initial sample-size requirement, the more important this correction becomes.

Confidence Level

The confidence level describes the reliability of the estimation procedure. Common choices include 90%, 95%, and 99%. A higher confidence level requires a larger critical Z-score and therefore generally requires a larger sample size when all other assumptions remain unchanged.

For a two-sided confidence interval, the critical value is determined from the standard normal distribution using the tail probability associated with the selected confidence level. The calculator can also handle confidence levels other than the three common values listed above.

Margin of Error

The margin of error describes how far the estimated sample proportion is allowed to differ from the true population proportion under the selected confidence level and statistical assumptions.

For example, a margin of error of 5% means that the desired estimate is intended to have a maximum error of approximately ±5 percentage points under the specified assumptions.

The margin of error has a particularly strong effect on sample size because it is squared in the denominator:

Expected Population Proportion

The expected proportion represents the estimated percentage of the population that has the characteristic being studied. For example, if a survey is expected to find that approximately 30% of respondents have a particular characteristic, an expected proportion of 30% can be entered.

The proportion is used together with its complement:

If the true proportion is unknown, 50% is commonly used. This is a conservative assumption because p(1 − p) reaches its maximum value when p = 0.5, producing the largest required sample size for the other parameters held constant. similarly recommends using 50% when the population proportion is not known.

Example Calculation

Why 50% Is Used When the Proportion Is Unknown

The sample-size formula contains the term p(1 − p). This term measures the variability associated with estimating a population proportion. Its maximum value occurs at p = 0.50:

Because this produces the largest initial sample-size requirement, using 50% provides a conservative estimate when there is no reliable prior information about the population proportion.

Effect of Population Size

Population size affects the calculation when the population is finite. However, once a population becomes sufficiently large relative to the required sample, increasing the population size further may have only a limited effect on the required sample size.

This is why surveys involving thousands or millions of people do not necessarily require samples that are thousands or millions in size. The required sample is primarily determined by the desired precision, confidence level, and expected variability, with finite population size providing an additional correction.

Relationship Between Confidence and Sample Size

Increasing confidence generally increases the required sample size. For example, keeping the population proportion and margin of error unchanged, a 99% confidence level normally requires more observations than a 95% confidence level.

Sampling Fraction

Sampling fraction is the percentage of the population included in the sample. It can be calculated as:

This provides a useful way to understand how large the selected sample is relative to the entire population.

Rounding the Sample Size

Sample size calculations frequently produce a decimal value. Because a sample cannot contain a fraction of an observation, the calculated result is rounded up to the next whole number. Rounding downward could produce a sample smaller than the calculated requirement.

Achieved Margin of Error

Because the final sample size is rounded upward, the actual margin of error associated with the rounded sample can be slightly smaller than the requested margin of error. The calculator displays an approximate achieved margin of error so that the effect of rounding can be seen.

Nonresponse and Usable Sample Size

The calculated sample size represents the number of usable observations required under the selected assumptions. If some participants are expected not to respond or some responses are expected to be unusable, additional participants may need to be invited or selected.

For example, if a study requires 370 completed responses but expects a 10% nonresponse rate, recruiting exactly 370 people would likely leave fewer than 370 usable responses. The recruitment target would therefore need to be increased.

Sample Size and Study Design

This calculator uses a proportion-based formula and is most appropriate when the objective is to estimate a population proportion. Other study designs can require different sample-size methods.

Studies comparing two groups, estimating a population mean, testing a hypothesis, measuring an effect size, or planning an experiment may require calculations based on statistical power, expected variability, significance level, effect size, and other study-specific parameters.

Important Statistical Considerations

A larger sample does not automatically guarantee a representative sample. Sampling method, selection bias, nonresponse, measurement quality, and study design can all affect the reliability of statistical conclusions.

The confidence level also should not be interpreted as the probability that one specific calculated interval contains the true parameter. Rather, it describes the long-run performance of the interval-estimation procedure when the underlying assumptions are appropriate.

How to Use the Sample Size Calculator

Enter the total number of individuals or observations in the Population Size field. Enter the desired confidence level, such as 90%, 95%, or 99%, followed by the desired margin of error.

Enter an expected proportion if you have a reasonable prior estimate. If the proportion is unknown, leave it at the default 50% to obtain a conservative estimate.

The calculator then displays the initial sample size, finite population correction, final rounded sample size, sampling fraction, approximate achieved margin of error, formulas, and interpretation of the result.