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The displayed pointwise control limits follow from the normal approximation to an observed binomial proportion. Clip plotted limits to the probability range. Their nominal coverage is approximate and can be poor for small expected counts; exact binomial tails give a discrete alternative. Pointwise coverage does not provide simultaneous coverage of all hospitals; use Bonferroni correction or another familywise error rate procedure when that is the required guarantee.

For $D\sim \mathrm{Bin}(n,p)$ with $n>0$, the sample binomial proportion is $D/n$. Its expectation is p and its variance is $p(1-p)/n$. The central limit theorem gives a normal approximation when both expected outcome counts are sufficiently large. These moments determine a standard error and model-based control limits.

Under the target model $p_0=0.20$, the normal approximation givesThe pointwise 95% binomial funnel control limits are thereforeDraw these curves around the horizontal 20% target, as in the figure. At $n=100$, the approximate limits are $0.1216$ and $0.2784$, so B and H are outside the nominal bands; that is a signal for investigation, not proof of inferior or superior care. The bands use the target variance rather than each hospital's observed rate. They are control limits for possible observations under the target model, not confidence intervals centered on the observed points.

Normal coverage is approximate, especially for the small hospitals; clipping does not repair that approximation. Exact binomial tail limits can be used if exact calibration is needed.

Four advantages of the hospital mortality funnel plot are:

It does not automatically adjust for patient severity, referral patterns or data quality, and an outlying point does not establish a causal explanation.

Statistical process control compares observed process outcomes with a stable reference model and control limits reflecting expected sampling variation. An observation outside those limits is a signal to investigate, not proof of a specific cause. Varying sample volumes require varying limits, and multiple testing changes the chance of a chance signal somewhere in the display.