Signature
p_hat = r / n; SE_p = sqrt(p_hat * (1 - p_hat) / n)
| Inputs | Definition | Unit |
|---|---|---|
r | Number of patients in the study who had the event | patients |
n | Number of patients in the study, each followed for the whole period | patients |
p_hat | Observed proportion of patients with the event, the maximum likelihood estimate of p | probability |
|---|---|---|
SE_p | Estimated standard error of p_hat | probability |
Function
Binomial distribution of an event count among patients at risk
Gives the probability that exactly r of n patients have an event within a fixed period when each patient has the same probability p of the event and outcomes are independent. Read as a function of p for an observed count, the same expression is the binomial likelihood, proportional to p^r (1 minus p)^(n minus r), which peaks at the observed proportion r/n. The formulae below give the probability function, the chance of a zero count, the mean and variance of the count, the estimated probability with its standard error, the Wald and Wilson intervals and the logit model for binomial data in NICE DSU evidence synthesis. The conjugate beta update of a binomial probability is set out on the Beta Distribution page (HE-FM-BETA-002), and the general Wald interval for any estimate on the Asymptotic Normality page (HE-FM-AN-001).
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Implementations
Excel
Binomial proportion and its standard error in two cells
With named cells Events and Patients, the first formula returns the estimate into a cell named PHat and the second its standard error.
=Events/Patients; =SQRT(PHat*(1-PHat)/Patients)
Assumptions
Binomial sampling behind an observed proportion
The count arises from a fixed number of patients followed for the whole period, with a common probability and independent outcomes. With incomplete follow-up or censoring, r/n is a biased estimate and survival analysis is the usual route.
Plug-in standard error unreliable with few binomial events
The standard error uses p_hat itself, so it is zero when r is 0 or n and imprecise with small counts. With few events the Wilson interval, HE-FM-BINOM-006, or a beta distribution describes uncertainty better.
Worked examples
Estimated probability and standard error for 6 events among 40 patients
The estimate is 6/40 = 0.15 and the standard error about 0.0565, as in the article's steps 1 and 2.
r = 6; n = 40; p_hat = 0.15; SE_p = 0.0565
Zero standard error for no events among 40 patients
With no events the estimate is 0 and the plug-in standard error is also 0, although the data are consistent with an annual probability of up to about 9%.
r = 0; n = 40; p_hat = 0; SE_p = 0
Common errors
Model cohort size used in a binomial standard error
The standard error depends on the study's n, not on the number of patients simulated in the model. Using 1,000 instead of 40 for 6 events among 40 patients gives about 0.0113 instead of 0.0565, a fivefold understatement.
Sources
NIST maximum likelihood estimator of the binomial probability
NIST/SEMATECH. e-Handbook of Statistical Methods. Section 1.3.6.6.18, Binomial Distribution. National Institute of Standards and Technology; accessed 2 October 2026. Parameter estimation: the maximum likelihood estimator of p for fixed n is x/n.
Canonical Identity
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