Confidence bounds and sampling uncertainty
Bounds describe a repeated-sampling estimation procedure. Proportion bounds and distances use percentage points. Wilson bounds are not generally symmetric around the observed proportion, so the reported distance is the larger side.
To plan how many observations you need before collecting data, use the Sample Size Calculator .
The formula
For a mean, standard deviation must be a known population value. For a proportion, p̂ = successes/n; the Wilson adjustment keeps endpoints within zero and one, including when no successes or all successes are observed. Presets use two-sided normal critical values.
Worked example
A sample mean of 80, known population standard deviation 12, and sample size 36 give a 95% interval of approximately 76.080072 to 83.919928. The standard error is 2 and the margin is 3.919928.
How to use this calculator
- Choose the interval type and confidence level.
- Enter a mean and known population spread, or an integer success count, together with sample size.
- Calculate and interpret the bounds in the context of the sampling process.
Known spread is a specific assumption
A mean interval based on z assumes the population standard deviation is known. If you estimate spread from the same sample, a t-based interval is usually needed instead. This calculator labels that distinction in the controls and does not silently treat sample spread as known. Normal populations support the small-sample mean formula; a large-sample approximation needs an appropriate sampling distribution.
Why use a Wilson interval for proportions?
The observed proportion is simply the success count divided by sample size. The interval additionally reflects uncertainty about the population proportion. Wilson construction avoids the zero-width result that a basic normal approximation can produce when all observations have the same outcome. It is still an approximate frequentist interval, rather than an exact binomial or Bayesian credible interval.
What the confidence percentage means
If the same sampling procedure were repeated many times, a nominal 95% interval method would cover the fixed population parameter about 95% of the time under its assumptions. It does not mean that 95% of individual observations fall inside this interval. Increasing confidence generally widens bounds; increasing sample size generally narrows them when other inputs stay fixed.
Assumptions & limitations
What this calculation assumes
- Independent representative observations; no finite-population correction.
What to keep in mind
- No t interval, paired data, cluster adjustment, exact binomial method, or causality assessment.
Common questions
Can a proportion interval start below zero?
This Wilson implementation constrains rounded numerical endpoints to 0% through 100%. Mean intervals are not constrained to that range.
What if successes exceed sample size?
That is impossible for one binary outcome per observation. The tool asks you to correct the success count.