Use the binomial distribution when your scenario meets all of these criteria: (1) You have a fixed number of trials, (2) Each trial is independent of the others, (3) Each trial has exactly two possible outcomes (success or failure), and (4) The probability of success remains constant across all trials. If trials aren’t independent or if the probability of success varies between trials, other distributions like the hypergeometric or Poisson-binomial may be more appropriate. For rare events or when dealing with rates over time or space, the Poisson distribution is often used instead. If your random variable is continuous rather than discrete (count-based), distributions like the normal or exponential would be applicable.

As the number of trials (n) increases, the binomial distribution approaches a normal distribution with mean μ = np and variance σ² = np(1-p). This approximation is generally considered good when both np ≥ 5 and n(1-p) ≥ 5. This relationship is a specific case of the Central Limit Theorem and is especially useful for calculating binomial probabilities with large values of n, where direct calculation using the binomial formula becomes computationally intensive. When using the normal approximation, a continuity correction is often applied to improve accuracy, typically by adding or subtracting 0.5 from boundary values. The approximation is most accurate when p is close to 0.5 and becomes less reliable as p approaches 0 or 1, unless n is very large.

Determining the appropriate sample size for a binomial experiment typically involves balancing statistical power with practical constraints. The process generally follows these steps: First, specify the effect size you want to detect (the minimum difference in proportions that is meaningful). Second, set your desired confidence level (typically 95%) and power (typically 80% or 90%). Third, estimate the expected proportion based on prior information or a pilot study. Then, use a sample size formula that incorporates these parameters, or use a sample size calculator specifically designed for proportion-based studies. As a general rule, larger samples are needed when: (1) The expected proportion is close to 0.5, (2) You need to detect smaller differences, (3) You require higher confidence levels or statistical power, or (4) You expect higher variability in your data. In practice, logistical and cost constraints often influence the final sample size decision.

These three distributions are related but answer different questions about sequences of independent trials. The binomial distribution counts the number of successes in a fixed number of trials. For example, “How many heads will I get in 10 coin tosses?” The negative binomial distribution counts the number of trials needed to achieve a fixed number of successes. For example, “How many coin tosses will I need to get 10 heads?” The geometric distribution is a special case of the negative binomial, counting the number of trials until the first success occurs. For example, “How many coin tosses until I get my first head?” All three distributions assume independent trials with constant probability of success, but they differ in what’s fixed (number of trials or number of successes) and what’s random (what you’re counting). The choice between them depends on the specific question you’re trying to answer about your process or experiment.

Verifying whether data follows a binomial distribution involves both conceptual and statistical approaches. Conceptually, assess whether your data collection process satisfies the four binomial conditions: fixed number of trials, independent trials, constant success probability, and binary outcomes. Statistically, several methods can help: (1) Compare the observed frequencies with the expected frequencies using a chi-square goodness-of-fit test, (2) Plot the observed distribution against the theoretical binomial distribution with the same parameters, (3) Check if the sample mean and variance are consistent with binomial properties (for a binomial, the variance should be approximately n×p×(1-p)), (4) Examine the index of dispersion (variance-to-mean ratio), which should be close to (1-p) for binomial data. Significant deviations may indicate overdispersion (more variability than expected) or underdispersion (less variability than expected), suggesting that a different distribution might be more appropriate. Remember that real-world data rarely follows any theoretical distribution perfectly, so minor deviations may be acceptable depending on your application.