Confidence Interval Calculator
A confidence interval gives you a likely range for a true value, based on a sample. It tells you how much to trust a number pulled from limited data.
What a confidence interval really tells you
Say you want to know the average height of every adult in a city. Measuring everyone is not realistic. So you measure a sample of 200 people instead.
That sample gives you an average, but it will not match the true city wide average exactly. A confidence interval gives you a range instead of one guess. It says the true average is likely somewhere between two numbers, not exactly one.
Formula for the mean confidence interval
x̄ is your sample mean, the average you measured.z is the z score for your chosen confidence level.s is your sample standard deviation, how spread out the data is.n is your sample size, how many people or items you measured.
In plain words, you take your average, then add and subtract a small buffer. That buffer grows if your data is spread out, and shrinks if your sample is large.
Formula for the proportion confidence interval
p̂ is your sample proportion, written as a decimal.z is the z score for your chosen confidence level.n is your total sample size.
Common z scores for confidence levels
| Confidence level | Z score |
|---|---|
| 80% | 1.282 |
| 90% | 1.645 |
| 95% | 1.960 |
| 99% | 2.576 |
A mistake almost everyone makes
A 95% confidence interval does not mean there is a 95% chance the true value sits inside your specific range. It means that if you repeated your survey many times, about 95% of the resulting ranges would contain the true value. This distinction is subtle, but it matters for correct interpretation.
Why sample size changes everything
A small sample gives a wide, less useful range. A larger sample tightens that range and gives a more precise answer. This is why pollsters and researchers push for bigger sample sizes whenever possible.
Frequently asked questions
A single average from a sample is only an estimate. It rarely matches the true population value exactly. A confidence interval shows how much uncertainty surrounds that estimate.
A higher confidence level, like 99% instead of 95%, gives you a wider range. You gain more certainty, but the range becomes less precise and less useful.
This calculator assumes a roughly normal distribution, which is a safe assumption once your sample size is around 30 or more, thanks to the central limit theorem.
Yes. This is exactly how businesses estimate customer satisfaction, average order value, or approval rates from surveys without asking every single customer.
Want to check if a difference between two groups is real?
Try a Sample Size Calculator next, or head back to the toolkit.
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