Free tool Confidence interval calculator and formula
A rate or an average from a sample is an estimate. The confidence interval shows how far off it could reasonably be. Enter successes and sample size for a rate, or the mean, standard deviation and sample size for an average.
- Proportions: Wilson interval
- Means: Student t interval
- Runs locally
Choose what you are measuring
A proportion is a share: converted visitors, yes answers. A mean is an average of a quantity: order value, minutes on page.
Proportion for a rate or share; mean for an average.
Conversions, yes answers or any count of hits in the sample. Cannot exceed the sample size.
Visitors, respondents or orders in the sample. A mean needs at least 2.
The average you measured, in its own unit.
The sample standard deviation, in the same unit as the mean (not the standard error).
95% is the usual choice. 99% gives a wider interval, 90% a narrower one.
Observed rate
21.17%
Successes ÷ sample size.
Sample mean
—
The centre of the t interval.
Share these inputs: the values go after the # in the link, so they never reach a server.
https://tapcub.com/tools/confidence-interval-calculator
Version v1.0Updated Update history
What the interval says
The point estimate is a single guess. The interval is the honest version: every value inside it is reasonably consistent with your data.
Observed rate or mean
What the sample shows. 508 conversions from 2,400 visitors is 21.17%. On its own it hides how many people it is based on.
Lower and upper bound
The plausible range for the true value: 19.58% to 22.85% in the example. If a target or a competitor’s figure sits inside the range, your data cannot tell them apart.
Margin of error
Half the width of the interval, ±1.63 percentage points here. This is the "±3 points" figure surveys quote. It shrinks with the square root of the sample: four times the sample, half the margin.
Confidence level
How often the method captures the true value over many samples. 99% intervals are wider than 95%, which are wider than 90%: 19.10%–23.39% against 19.83%–22.57% for the same data.
The interval only covers sampling error. It says nothing about bots in the data, a tracking gap or a badly defined conversion; those shift the whole interval.
Three choices, then read the range
The same calculator handles a rate and an average. Pick the mode first; the fields change with it.
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1
Pick the mode
Proportion for anything counted as yes or no per person: converted, clicked, answered yes. Mean for a measured amount: revenue per order, session length.
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2
Enter the sample
For a proportion, successes and sample size. For a mean, the mean, the standard deviation and the sample size, all from the same data.
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3
Choose the level
95% unless you have a reason. Report the interval with the level, for example "21.2% (95% CI 19.6%–22.8%)".
Wilson for proportions, t for means
The familiar "rate ± 1.96 × standard error" (the Wald interval) breaks down for small samples and for rates near 0% or 100%: it can go below zero, above 100%, or shrink to nothing. The Wilson interval fixes this at no extra cost, so the calculator uses it for proportions. For means it uses the Student t distribution.
- z is 1.645 at 90%, 1.960 at 95% and 2.576 at 99%
- Wilson centres the interval slightly toward 50%, so it is not symmetric around the observed rate
- For large samples away from the edges, Wilson and Wald agree closely
- The t interval uses n − 1 degrees of freedom and the sample standard deviation
Wilson interval (proportion)
( p̂ + z²/2n ± z × √( p̂(1 − p̂)/n + z²/4n² ) ) ÷ (1 + z²/n), p̂ = successes ÷ n
508 of 2,400 at 95%: p̂ = 21.17% → 19.58% to 22.85%
Wald interval, for comparison
p̂ ± z × √( p̂(1 − p̂) ÷ n )
Same data: 19.53% to 22.80%. At 1 of 50: −1.88% to 5.88%
t interval (mean)
x̄ ± t(n − 1) × s ÷ √n
Order value $74, s = $38, n = 120: $74 ± 1.980 × $38 ÷ √120 = $74 ± $6.87 → $67.13 to $80.87
Wilson against the simple Wald interval
Five samples, both methods at 95%. With plenty of data they agree; with small samples or rates near the edges, Wald produces ranges that cannot be right.
| Sample | Observed rate | Wilson (this tool) | Wald | Problem with Wald |
|---|---|---|---|---|
| 508 of 2,400 | 21.17% | 19.58% – 22.85% | 19.53% – 22.80% | None worth noting |
| 12 of 40 | 30.00% | 18.07% – 45.43% | 15.80% – 44.20% | Ignores the skew of a small sample |
| 1 of 50 | 2.00% | 0.35% – 10.50% | −1.88% – 5.88% | Lower bound below zero |
| 0 of 50 | 0.00% | 0.00% – 7.13% | 0.00% – 0.00% | Zero width: claims certainty |
| 48 of 50 | 96.00% | 86.54% – 98.90% | 90.57% – 101.43% | Upper bound above 100% |
The conversion rate calculator shows the simpler Wald interval, which is why its range for 508 of 2,400 (19.5%–22.8%) differs slightly from this one. Sample figures.
How intervals get misused
The formula is standard. The inputs and the reading are where it goes wrong.
"95% chance the true value is inside"
The 95% describes the method across many samples. Any single interval either contains the true value or does not.
Page views as the sample
One person reloading five times is not five observations. Count people or sessions, whichever the rate is defined on.
Standard error in the SD box
The calculator divides by √n itself. Entering the standard error instead of the standard deviation gives an interval far too narrow.
Overlap as a significance test
Two intervals can overlap slightly and the difference can still be significant. Compare groups with a proper test.
Clean samples behind the interval
An interval only accounts for chance. TapCub takes care of the rest: one definition per converting event, funnels with a fixed time window, and bots kept out of the count.
Funnels
2,400 → 1,248 → 674 → 508. Conversion per step, with the drop-off highlighted.
Learn moreGoals and events
Define the converting event once; every report uses the same definition.
Learn moreBots counted separately
Five layers and 197+ rules keep crawlers out of the denominator by default.
Learn moreConfidence interval questions
Still have a question?
Chat with the team behind TapCub — we usually reply within a few hours on business days.
What is the confidence interval formula?
Estimate ± critical value × standard error. For a mean: x̄ ± t × s ÷ √n, with t from the t distribution at n − 1 degrees of freedom. For a proportion, the textbook (Wald) formula is p̂ ± z × √(p̂(1 − p̂) ÷ n), with z = 1.645, 1.960 or 2.576 at 90%, 95% or 99%; this calculator uses the more accurate Wilson interval (see the next answers). The result is a range that, with a 95% method, contains the true value in 95% of samples.
How do I calculate a confidence interval for a proportion?
Divide successes by the sample size for the observed rate, then apply an interval formula. The Wilson interval used here is ( p̂ + z²/2n ± z√(p̂(1 − p̂)/n + z²/4n²) ) ÷ (1 + z²/n). For 508 of 2,400 at 95%, that is 19.58% to 22.85%.
How do I calculate a confidence interval for a mean?
Mean ± t × standard deviation ÷ √n, where t comes from the Student t distribution with n − 1 degrees of freedom. For an average order value of $74 with a standard deviation of $38 from 120 orders, the 95% interval is $67.13 to $80.87.
Why does the calculator use the Wilson interval?
The textbook Wald interval, rate ± z × standard error, can produce impossible ranges: below 0% or above 100%, or zero width when there are no successes. Wilson stays within 0–100% and keeps close to the stated confidence level with small samples. For large samples the two are almost identical.
What is the difference between margin of error and confidence interval?
The margin of error is half the width of the interval. A survey reporting "52% ± 3 points" is giving the interval 49% to 55%. For a Wilson interval the bounds are not exactly symmetric around the rate, but the margin is still half the width.
How can I make the interval narrower?
Collect more data, or accept a lower confidence level. Width shrinks with the square root of the sample: 30% from 150 visitors has a margin of ±7.26 points; from 600 visitors, ±3.66.
If two confidence intervals overlap, is there no significant difference?
Not necessarily. Two 95% intervals can overlap a little while the difference between them is still significant. To compare two rates, use a two-proportion test such as the one in the conversion rate calculator.
How do I calculate a confidence interval in Excel or Google Sheets?
For a mean, CONFIDENCE.T(0.05, sd, n) returns the 95% margin (t-based); the interval is mean ± that margin. CONFIDENCE.T(0.05, 38, 120) = 6.87, so $74 ± $6.87. CONFIDENCE.NORM uses z instead of t and is slightly narrower for small samples. There is no built-in function for the Wilson interval of a proportion; the Wald margin is NORM.S.INV(0.975)*SQRT(p*(1-p)/n).
Update history
New features, fixes and data refreshes for this tool, newest first.
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· v1.0
LaunchedFirst release.
Missing something or found an error? Tell us — requests that come up often go into the next update. All tool updates
From a range to a decision
An interval tells you how precise one number is. To decide between two versions, test the difference, and size the test first.