Sample Size Calculator
Free Sample size Calculator for biostatistics. Enter variables to compute results with formulas and detailed steps. Enter your values for instant results.
Calculator
Adjust values & calculateAdjusted for Response Rate
Sample Size by Margin of Error
Sample Size by Confidence Level
Formula
Where n is the unadjusted sample size, z is the z-score for the desired confidence level, p is the estimated population proportion (use 0.5 for maximum), e is the margin of error (as decimal), and N is the population size. The second formula applies finite population correction, reducing n when sampling a significant fraction of the population.
Last reviewed: December 2025
Worked Examples
Example 1: Survey of Hospital Patients
Example 2: Ecological Bird Population Study
Background & Theory
The Sample Size Calculator applies the following established principles and formulas. Biology is the scientific study of life, encompassing the structure, function, growth, evolution, and distribution of living organisms. At the cellular level, all life is composed of cells, the basic structural and functional units of organisms. Prokaryotic cells lack a membrane-bound nucleus, while eukaryotic cells possess a nucleus and membrane-bound organelles including mitochondria, which generate ATP through oxidative phosphorylation, and ribosomes, which synthesize proteins. Genetics quantifies the inheritance of traits. Gregor Mendel's laws describe how alleles segregate during gamete formation and assort independently for genes on different chromosomes. Punnett squares provide a visual method for calculating the probability of offspring genotypes and phenotypes from known parental genotypes. For a monohybrid cross of two heterozygotes (Aa ร Aa), the expected phenotypic ratio is 3 dominant to 1 recessive. The Hardy-Weinberg equilibrium principle states that allele and genotype frequencies in a population remain constant from generation to generation in the absence of evolutionary forces. If p and q are the frequencies of two alleles at a locus, then p + q = 1 and genotype frequencies are pยฒ, 2pq, and qยฒ for the three possible genotypes. Deviations from equilibrium signal the action of natural selection, genetic drift, mutation, migration, or non-random mating. Population growth follows two primary models. Exponential growth, N = Nโeสณแต, describes unlimited growth where Nโ is the initial population, r is the intrinsic rate of increase, and t is time. Logistic growth incorporates carrying capacity K, describing how growth slows as population approaches the environment's maximum sustainable size: dN/dt = rN(1 โ N/K). Enzyme kinetics describes the rate of enzyme-catalyzed reactions. The Michaelis-Menten equation, v = Vmax[S]/(Km + [S]), relates reaction velocity v to substrate concentration [S], maximum velocity Vmax, and the Michaelis constant Km, which equals the substrate concentration at half-maximal velocity. DNA replication relies on complementary base pairing: adenine pairs with thymine (two hydrogen bonds) and guanine with cytosine (three hydrogen bonds), ensuring faithful copying of genetic information.
History
The history behind the Sample Size Calculator traces back through the following developments. The systematic study of living things began with Aristotle (384โ322 BCE), who classified over 500 animal species and wrote foundational texts on anatomy, reproduction, and animal behavior. His scala naturae ranked organisms in a hierarchy from simple to complex and influenced biological thought for two millennia. Theophrastus, his student, applied similar methods to plants. Carl Linnaeus established modern taxonomy in Systema Naturae (1735), introducing the binomial nomenclature system that assigns each organism a genus and species name. His hierarchical classification system โ species, genus, family, order, class, phylum, kingdom โ provided the organizational framework that biologists still use, now extended to seven ranks and supplemented by cladistics. Charles Darwin and Alfred Russel Wallace independently developed the theory of evolution by natural selection, which Darwin published in On the Origin of Species in 1859. Darwin argued that heritable variation exists within populations, that organisms with advantageous traits survive and reproduce at higher rates, and that this differential reproduction gradually changes the character of populations over generations. This unified all of biology under a single explanatory framework. Gregor Mendel's meticulous pea plant experiments, conducted from 1856 to 1863 and published in 1866, established the particulate nature of inheritance and the laws of segregation and independent assortment. Overlooked until 1900, when three botanists independently rediscovered his work, Mendel's laws laid the foundation for the science of genetics. James Watson and Francis Crick, building on Rosalind Franklin's X-ray crystallography data, determined the double-helix structure of DNA in 1953, revealing the physical basis of heredity and the mechanism by which genetic information is stored and copied. The Human Genome Project, a 13-year international collaboration, published the complete sequence of the human genome in 2003, comprising approximately 3.2 billion base pairs. The development of CRISPR-Cas9 gene editing by Jennifer Doudna, Emmanuelle Charpentier, and colleagues from 2012 onward opened an era of precise genome modification with transformative implications for medicine, agriculture, and basic research.
Key Features
- Computes a full descriptive statistics summary from a data set, including mean, median, mode, range, variance, standard deviation, skewness, and interquartile range.
- Constructs confidence intervals for population proportions and means at any confidence level, displaying the margin of error, standard error, and critical value used.
- Calculates p-values and test statistics for z-tests, one- and two-sample t-tests, and chi-square goodness-of-fit and independence tests, with automatic two-tailed or one-tailed selection.
- Performs ordinary least squares linear regression on paired data, returning the slope, intercept, R-squared value, and a residual summary to assess model fit.
- Evaluates the CDF and PDF for major probability distributions including the normal, binomial, and Poisson distributions, given user-supplied parameters and input values.
- Determines the required sample size to achieve a specified margin of error and confidence level for both proportion and mean estimation problems.
- Computes the Pearson and Spearman correlation coefficients between two variables, indicating the strength and direction of their linear or monotonic relationship.
- Applies Bayes' theorem to calculate posterior probabilities given a prior probability, likelihood, and marginal likelihood, with a clear breakdown of each term in the formula.
Frequently Asked Questions
Formula
n = (z^2 * p * (1-p)) / e^2, adjusted: n_adj = n / (1 + (n-1)/N)
Where n is the unadjusted sample size, z is the z-score for the desired confidence level, p is the estimated population proportion (use 0.5 for maximum), e is the margin of error (as decimal), and N is the population size. The second formula applies finite population correction, reducing n when sampling a significant fraction of the population.
Worked Examples
Example 1: Survey of Hospital Patients
Problem: A hospital with 2,000 patients wants to survey satisfaction with 95% confidence and 5% margin of error. Assume 50% response proportion and 70% response rate.
Solution: z = 1.96 for 95% confidence\nn0 = (1.96^2 * 0.5 * 0.5) / 0.05^2 = 384.16\nFinite correction: n = 384.16 / (1 + (384.16-1)/2000) = 384.16 / 1.1916 = 322.5\nRound up: n = 323\nAdjust for 70% response: 323 / 0.70 = 462
Result: Sample size: 323 patients (462 invitations needed at 70% response rate)
Example 2: Ecological Bird Population Study
Problem: Estimate the proportion of a bird species with a specific trait in a population of 500 birds. Need 90% confidence with 8% margin of error.
Solution: z = 1.645 for 90% confidence\nn0 = (1.645^2 * 0.5 * 0.5) / 0.08^2 = 105.7\nFinite correction: n = 105.7 / (1 + (105.7-1)/500) = 105.7 / 1.2094 = 87.4\nRound up: n = 88
Result: Sample size: 88 birds (17.6% of population) with finite population correction
Frequently Asked Questions
How do I determine the right sample size for my study?
Sample size depends on four key factors: (1) Confidence level - typically 95% for biological research, meaning you want to be 95% confident your results reflect the true population. (2) Margin of error - the acceptable range of uncertainty, usually 3-5% for surveys. (3) Population proportion - if unknown, use 50% as it gives the maximum (most conservative) sample size. (4) Population size - for small populations, a finite population correction reduces the required sample. For clinical trials, you also need to consider effect size, power, and expected dropout rate. Start with the statistical requirements and then add a buffer of 10-20% for non-response or data quality issues.
What is margin of error and how does it affect sample size?
Margin of error (also called confidence interval width) defines how close your sample estimate will be to the true population value. A margin of error of plus or minus 3% means if your sample shows 60%, the true value is likely between 57% and 63%. Reducing margin of error dramatically increases required sample size: going from 5% to 3% nearly triples the sample size, and going from 5% to 1% increases it 25-fold. In biological research, acceptable margins depend on the precision needed. Drug efficacy studies may need plus or minus 2%, while ecological surveys may accept plus or minus 10%.
Why does using p=50% give the most conservative sample size?
The formula includes p*(1-p), which is maximized when p=0.50 (giving 0.25). If you know the true proportion is near 10% or 90%, p*(1-p)=0.09, requiring much fewer samples. Using p=50% guarantees your sample is large enough regardless of the actual proportion. However, if you have strong prior evidence about the proportion (from pilot studies or previous research), using a more realistic estimate can significantly reduce your required sample size and save resources. For biostatistics studies where the outcome prevalence is known to be rare (e.g., 5%), using p=0.05 can reduce sample requirements by 75%.
How should I account for non-response or dropout in my sample size?
Always inflate your calculated sample size to account for anticipated non-response or dropout. The adjusted size is n_adjusted = n / response_rate. Common response rates: mailed surveys 30-50%, online surveys 10-30%, clinical trials 70-90%, in-person interviews 60-80%. For a 12-month clinical trial expecting 20% dropout, multiply your sample by 1.25 (divide by 0.80). For multi-year longitudinal studies, compound the dropout rate: if 10% drop out each year over 3 years, retention is 0.9^3 = 72.9%. Always report both your target and achieved sample sizes in publications.
How accurate are the results from Sample Size Calculator?
All calculations use established mathematical formulas and are performed with high-precision arithmetic. Results are accurate to the precision shown. For critical decisions in finance, medicine, or engineering, always verify results with a qualified professional.
How do I interpret the result?
Results are displayed with a label and unit to help you understand the output. Many calculators include a short explanation or classification below the result (for example, a BMI category or risk level). Refer to the worked examples section on this page for real-world context.
References
Reviewed by Daniel Agrici, Founder & Lead Developer ยท Editorial policy