Signature
p_hat = r / n; L_Wald = p_hat - z * sqrt(p_hat * (1 - p_hat) / n); U_Wald = p_hat + z * sqrt(p_hat * (1 - p_hat) / n)
| Inputs | Definition | Unit |
|---|---|---|
r | Number of patients with the event | patients |
n | Number of patients in the study | patients |
z | Standard normal quantile for the chosen confidence level, 1.96 for a 95% interval | none |
p_hat | Observed proportion r/n | probability |
|---|---|---|
L_Wald | Lower limit of the Wald interval for p | probability |
U_Wald | Upper limit of the Wald interval for p | 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 Wald interval limits in two cells
With named cells PHat, Z and Patients, the two formulas return the lower and upper limits.
=PHat-Z*SQRT(PHat*(1-PHat)/Patients); =PHat+Z*SQRT(PHat*(1-PHat)/Patients)
Assumptions
Large sample and probability away from the bounds for the binomial Wald interval
The approximation assumes n is large and p is not close to zero or one. Brown, Cai and DasGupta found that common textbook rules about when the interval is safe cannot be trusted, so meeting such a rule gives no guarantee of coverage.
Worked examples
Wald interval for 6 adverse events among 40 patients
The standard error of about 0.0565 times 1.96 gives a half-width of about 0.1107, so the interval runs from about 0.039 to 0.261, as in the article's step 2.
r = 6; n = 40; z = 1.96; p_hat = 0.15; L_Wald = 0.0393; U_Wald = 0.2607
Wald interval of zero width for no events among 40 patients
With no events the standard error is zero and the interval collapses to the single point 0, the article's step 3.
r = 0; n = 40; z = 1.96; p_hat = 0; L_Wald = 0; U_Wald = 0
Common errors
Binomial Wald interval reported after zero or few events
For no events among 40 patients the Wald interval is the single point 0, and for one event it runs from about minus 0.023 to 0.073, below zero. In the article's exact calculation for 40 patients and a true probability of 0.02, the nominal 95% Wald interval covers the true value in about 55% of samples, against about 95% for the Wilson interval.
Sources
NIST formula for the usual interval for a proportion and its negative lower limit
NIST/SEMATECH. e-Handbook of Statistical Methods. Section 7.2.4.1, Confidence intervals for a proportion. National Institute of Standards and Technology; accessed 2 October 2026. Gives the usual interval p_hat plus or minus z times the square root of p_hat (1 minus p_hat)/n and notes that, unlike the Wilson interval, its lower limit can be negative.
Erratic coverage of the Wald interval for a binomial proportion
Brown LD, Cai TT, DasGupta A. Interval estimation for a binomial proportion. Statistical Science. 2001;16(2):101-133. Abstract, which reports that the chaotic coverage of the Wald interval is far more persistent than appreciated and that textbook prescriptions on its safety cannot be trusted.
Canonical Identity
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