Signature
E_R = n * p; V_R = n * p * (1 - p); SD_R = sqrt(V_R)
| Inputs | Definition | Unit |
|---|---|---|
n | Number of patients at risk: a study sample, or the N patients entering a state in a simulated cohort | patients |
p | Probability that any one patient has the event within the period | probability |
E_R | Expected number of patients with the event | patients |
|---|---|---|
V_R | Variance of the number of patients with the event | patients squared |
SD_R | Standard deviation of the number of patients with the event | patients |
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 count mean, variance and standard deviation in three cells
With named cells Patients and Prob, the formulas return the mean, the variance and the standard deviation; the third refers to the variance held in a cell named VarCount.
=Patients*Prob; =Patients*Prob*(1-Prob); =SQRT(VarCount)
Assumptions
Binomial conditions for the moments of an event count
A fixed number of patients, a common probability and independent outcomes. Clustering of patients within centres inflates the variance beyond n p (1 minus p).
Fixed probability behind the chance variation of a simulated binomial count
V_R describes chance variation among identical patients when p is held fixed. Uncertainty in p itself, parameter uncertainty, is described separately, usually by a beta distribution whose width comes from the study's sampling variance p (1 minus p)/n.
Worked examples
Expected events and chance variation for 1,000 simulated patients
For 1,000 patients at a probability of 0.15, a cohort model records 150 events in the first cycle. A microsimulation with the same fixed probability gives a count with a standard deviation of about 11.29, and Excel BINOM.INV places its 2.5% and 97.5% points at 128 and 172 events, as in the article's step 5.
n = 1000; p = 0.15; E_R = 150; V_R = 127.5; SD_R = 11.2916
Expected events in the 40-patient adverse event study
At a probability of 0.15, 40 patients give an expected 6 events with a variance of 5.1 and a standard deviation of about 2.26.
n = 40; p = 0.15; E_R = 6; V_R = 5.1; SD_R = 2.2583
Common errors
Simulated binomial count variance used as parameter uncertainty
The variance n p (1 minus p) of a simulated count describes chance variation among patients, not how well p is known. For 1,000 simulated patients at 0.15 it implies about 128 to 172 events, whereas the Beta(7, 35) distribution from the 40-patient study implies about 72 to 292 events per 1,000. Using the narrower range as parameter uncertainty understates the uncertainty carried through to costs and QALYs.
Sources
NIST handbook mean and standard deviation of the binomial
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. Common statistics: mean np and standard deviation equal to the square root of np(1 minus p).
Stochastic and parameter uncertainty in the ISPOR-SMDM task force report
Briggs AH, Weinstein MC, Fenwick EAL, Karnon J, Sculpher MJ, Paltiel AD. Model parameter estimation and uncertainty analysis: a report of the ISPOR-SMDM Modeling Good Research Practices Task Force Working Group-6. Medical Decision Making. 2012;32(5):722-732. Table 1 and pp. 723-725, which define stochastic (first-order) uncertainty as random variability in outcomes between identical patients, separate it from parameter uncertainty, and state that in patient-level simulations assessing parameter uncertainty requires stochastic uncertainty to be eliminated.
Canonical Identity
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