Confidence Interval Calculator
Calculate confidence intervals for a mean, a proportion or a difference between two groups, using the t distribution for small samples and the Wilson interval for proportions.
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What a confidence interval actually claims
A 95 percent confidence interval does not mean there is a 95 percent probability that the true value lies inside this particular interval. The true value is fixed; the interval is what varies from sample to sample. The correct statement is that a procedure producing intervals this way captures the true value in 95 percent of repeated samples. The distinction sounds pedantic and matters, because the common misreading invites treating the interval as a probability distribution over the parameter, which it is not.
Small samples need the t distribution
When the population standard deviation is unknown and estimated from the sample, the extra uncertainty in that estimate makes the interval wider than the normal distribution would suggest. The t distribution accounts for it, and the effect is large at small sample sizes: with ten observations the multiplier is 2.26 rather than 1.96, which is fifteen percent wider. Above about thirty observations the difference becomes negligible, which is where the well known rule of thumb comes from.
Proportions need a different interval
The textbook interval for a proportion behaves badly when the proportion is near zero or one, or the sample is small. It can produce a lower bound below zero, and its actual coverage can fall well below the nominal level. The Wilson score interval, used here, is asymmetric, always stays within zero and one, and has far more reliable coverage. It is what most statisticians recommend and is unfortunately not what most textbooks teach.
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Frequently Asked Questions
What does 95 percent confidence mean?
That the procedure captures the true value in 95 percent of repeated samples. It is a property of the method, not of this particular interval. Saying there is a 95 percent probability the truth lies in this interval treats a fixed value as random, which is the standard misreading.
When should I use the t distribution?
Whenever the population standard deviation is unknown and estimated from the sample, which is nearly always. The effect is large at small sample sizes: ten observations need a multiplier of 2.26 rather than 1.96. Above about thirty it barely matters.
Why the Wilson interval for proportions?
Because the textbook interval breaks near zero and one, can give a lower bound below zero, and its real coverage can fall well below the stated level for small samples. The Wilson interval stays inside the valid range and has far more reliable coverage.
What makes an interval narrower?
A larger sample, less variability in the data, or a lower confidence level. The width falls with the square root of the sample size, so halving the width requires four times as much data, which is the fact that governs the cost of most experiments.
Does a wide interval mean the result is useless?
No, it means the data supports a wide range of values. That is genuine information: it says the experiment cannot distinguish between those possibilities, which is more honest than reporting a point estimate that appears more precise than it is.
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How to Use
Choose what you are estimating, enter your sample figures and read the interval.
Disclaimer: This tool is provided "as is" without warranty of any kind. Results are for educational and utility purposes.
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