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Free Confidence Interval Calculator
Get the 90%, 95% or 99% confidence interval for a sample mean or for a proportion, such as a conversion rate. Enter your numbers and the interval, margin of error and critical value update as you type.
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Calculate a confidence interval
95% confidence interval
46.8018 to 53.1982
50 ± 3.1982 (t = 2.023, 39 degrees of freedom)
What is a confidence interval?
A confidence interval is a range of values that is likely to contain the true value you are estimating, such as the average order value of all customers or the true conversion rate of a page. You only see a sample, so your estimate has some uncertainty. The interval shows how big that uncertainty is.
The confidence level says how often the method catches the true value. A 95% interval uses a method that contains the true value in about 95 out of 100 repeated samples. A 99% interval is wider, because it has to be right more often.
Confidence interval for a mean
The formula is: mean ± t × (s / √n).
Here s is the sample standard deviation, n is the sample size, and t is the critical value of Student's t distribution with n - 1 degrees of freedom. Example: a sample of 40 orders has a mean of $50 and a standard deviation of $10. At 95% confidence t is about 2.023, so the margin of error is 2.023 × 10 / √40 ≈ 3.20. The interval is about $46.80 to $53.20.
With small samples the t value is much larger than 1.96, which is why using the normal z value for a small sample gives an interval that is too narrow.
Confidence interval for a proportion
For a proportion, such as 35 conversions out of 1,000 visitors, this calculator uses the Wilson score interval. The simpler “p ± z × √(p(1 - p) / n)” formula can give limits below 0% or above 100% and is too narrow when the rate is close to 0, which is common for conversion rates. Wilson does not have those problems.
Example: 35 conversions from 1,000 visitors is a 3.5% conversion rate. The 95% Wilson interval is about 2.53% to 4.83%.
Confidence intervals in A/B testing
In an A/B test each variant has its own conversion rate and its own interval. When the intervals overlap a lot, the test has not told the variants apart yet. To decide a test, compare the two rates directly with a significance calculator, and plan the traffic you need up front with the sample size calculator.
FAQ
How do you calculate a 95% confidence interval?
For a mean: mean ± t × (standard deviation / √n), where t is the 95% critical value of Student's t with n - 1 degrees of freedom. With a large sample t is close to 1.96. For a proportion, this calculator uses the Wilson score interval with z = 1.96.
What does a 95% confidence interval mean?
If you repeated the same sampling many times and built an interval each time, about 95% of those intervals would contain the true value. It does not mean there is a 95% chance that this one interval contains it.
Why does a bigger sample give a narrower interval?
The margin of error shrinks with the square root of the sample size. Four times as many observations cut the margin of error roughly in half.
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