An empty success tray beside a full observation tray and a blank notebook. Original AI-generated artwork produced for GreenCalc; not a measured diagram or empirical data display.
A sample endpoint makes the plug-in expression vanish
If zero successes occur in twenty trials, p̂ = 0/20 = 0. Substituting that estimate into √[p̂(1−p̂)/n] gives zero. The same happens when every trial succeeds and p̂ = 1. This is a limitation of the plug-in standard-error expression at a sample endpoint, not proof that the population probability is exactly zero or one.
A nonzero population probability can produce zero observed successes
If the actual independent success probability is 5%, the chance of zero successes in twenty trials is 0.95²⁰, approximately 35.85%. Zero observed successes are therefore entirely possible under a positive population probability. The observation and the parameter must not be treated as interchangeable.
Do not turn a zero standard error into an automatic zero-width interval
Using a naive normal interval p̂ ± z×SE at this endpoint collapses the interval and can give a misleading impression of certainty. Methods designed for binomial-proportion intervals handle boundaries differently. The P-Hat Calculator deliberately reports no confidence interval, leaving interval selection to a method appropriate for the study and inferential goal.
Keep the sample size and model visible
Report “0 of 20 observed” alongside the fraction, rather than saying the event is impossible. Larger sample sizes can provide more information, but do not repair dependence, missing outcomes or biased selection by themselves. Read sample proportion versus population probability for the estimate–parameter distinction. The binomial probability calculator can check the zero-success probability under an explicitly supplied p; it cannot infer p with certainty from the endpoint observation.
Check the model behind your result
Use the P-Hat Calculator for the supported calculation and review its stated assumptions before interpreting the output. Find related tools in the Statistics category, or browse the GreenCalc homepage for other calculation topics.
By GreenCalc Editorial Team · Updated 2026-10-26