Confidence Interval Calculator
The range around an estimate, for a mean or a proportion.
Work out Confidence Interval. The range around an estimate, for a mean or a proportion. Exact arithmetic, checked against published values.
A 95% confidence level does not mean there is a 95% chance the true value sits in this particular interval. It means that if the study were repeated many times, 95% of the intervals produced would contain it. The difference matters when a single result is being reported as if it settled something.
95% confidence interval
97.06 to 102.94
100.00 ± 2.94
The margin is the critical value times the standard error, and the standard error carries a square root of the sample size. That square root is the whole story of survey cost: to halve this margin you need 400 observations, not 200. Going from 100 to 200 only narrows it by about 29%.
How the Confidence Interval Calculator works
A confidence interval puts a range around an estimate rather than pretending a single number is exact. The margin is the critical value times the standard error, and the standard error carries a square root of the sample size — which is why quadrupling a survey only halves its margin. This handles both a mean and a proportion, since polls report the second and studies the first.
Also known as: margin of error calculator · 95 confidence interval calculator · confidence interval for a proportion · how to calculate margin of error
The calculation itself
The estimate, plus and minus the critical value times the standard error. For a mean the standard error is σ ÷ √n; for a proportion it is √(p(1−p) ÷ n).
The critical value comes from the confidence level: 1.959964 for 95%, 1.644854 for 90%, 2.575829 for 99%. Higher confidence buys a wider interval, and there is no way around that trade.
In practice
A mean of 100 with a standard deviation of 15 across 100 observations has a standard error of exactly 1.5. The margin is 1.96 × 1.5 = 2.94, giving 97.06 to 102.94.
For a proportion, the familiar case: a poll of 1,000 people splitting 50/50 has a margin of 3.099 points at 95% confidence. That is the 'plus or minus three points' in every news report of a survey.
The square root is the fact worth internalising. Going from 100 to 200 observations narrows the interval by about 29%, not 50%. Halving the margin takes 400 — four times the sample for twice the precision.
What 95% confidence does not mean
It does not mean there is a 95% chance the true value lies in this particular interval. The true value is fixed; it is either in this interval or it is not.
What it means is that the procedure works 95% of the time: repeat the study many times and 95% of the intervals produced would contain the true value. The confidence attaches to the method, not to the one result in front of you.
The distinction matters most when a single study is reported as settling a question. One interval from one sample is one draw from that long run, and roughly one in twenty of those draws misses entirely.
Small samples
This page uses the normal critical value, which is correct when the sample is large. Below about thirty observations, with an unknown population standard deviation, the t distribution applies instead and produces a genuinely wider interval.
The page says so when your sample is small enough for it to matter, because the direction of the error is the dangerous one: the normal approximation makes a small-sample interval look tighter and more conclusive than the data supports.
Frequently asked questions
How do I calculate a 95% confidence interval?
Take the estimate and add and subtract 1.96 times the standard error. For a mean the standard error is σ ÷ √n; for a proportion it is √(p(1−p)/n). A mean of 100 with a standard deviation of 15 and 100 observations gives 100 ± 2.94.
What does 95% confidence actually mean?
That if the study were repeated many times, 95% of the intervals produced would contain the true value. It is not a 95% chance that this particular interval contains it — a distinction that matters whenever a single result is reported as settling a question.
Why is 1.96 used for 95%?
Because 95% of a standard normal distribution falls within 1.96 standard deviations of the mean, leaving 2.5% in each tail. The exact figure is 1.959964.
How does sample size change the interval?
Through a square root, which is slower than most people expect. Doubling the sample narrows the interval by about 29%; you need four times the sample to halve it.
What is the margin of error on a poll of 1,000?
About ±3.1 points at 95% confidence for a 50% result — 3.099 to be exact. That is the figure behind the familiar 'plus or minus three points' in news coverage.
When should I use the t distribution instead?
When the sample is small — below roughly thirty — and the population standard deviation is unknown. The t distribution gives a wider, more honest interval. This page uses the normal critical value and says so when your sample is small enough for it to matter.
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