Wilson interval table: the uncertainty behind a conversion rate

Wilson interval table: the uncertainty behind a conversion rate

Formula & method

p̂ = successes ÷ n; z = 1.959963984540054
Center = (p̂ + z²/(2n)) ÷ (1 + z²/n)
Half-width = z × √(p̂(1−p̂)/n + z²/(4n²)) ÷ (1 + z²/n)
Interval = center ± half-width

Each trial in this model has two outcomes, is independent and has the same underlying success probability. The observed success fraction is 10% in every row; only the sample size changes. The table uses a two-sided nominal 95% Wilson score interval and rounds the bounds to two decimal percentage points.

The confidence level describes the long-run coverage of the interval procedure under its assumptions. It is not a 95% probability statement about a fixed unknown rate after observing the data. Repeatedly checking many campaigns or selecting only winning segments requires additional care.

Worked example

Ten successes among 100 trials and 1,000 among 10,000 both show 10%. The second sample produces a much narrower interval, provided the trials satisfy the same model assumptions.

What can change the answer

Repeat users, correlated sessions, bot traffic and changing audiences can violate the binomial model. This is a one-sample interval, not an A/B test, causal estimate or sample-size stopping rule.