Confidence Interval Calculator

What the confidence interval calculator does

When you measure a percentage from a sample, like the share of people who prefer one option, the figure you get is only an estimate of the true value in the whole population. A confidence interval puts a sensible range around that estimate. This calculator works it out from your sample size, your sample percentage, and the confidence level you choose.

It is the range you see quoted after a poll result, the "plus or minus" that says how much the estimate might be off. Below is how it works and what it does, and does not, mean.

How to use it

1. Choose a confidence level of 90, 95, or 99 percent.
2. Enter the sample size and the sample percentage you measured. You can add the population size, for a small, known population.
3. Press Calculate for the margin of error, which you add and subtract from your percentage to form the interval, or Reset to clear it.

How the interval is worked out

The interval is your sample percentage, give or take a margin of error. That margin is built from two things: a z-score set by your confidence level, and the standard error, which measures how much a sample percentage tends to wobble from sample to sample.

Margin of error = z × √(p × (1 minus p) ÷ n)

Here p is your percentage written as a proportion and n is the sample size. The z-score is 1.645 for 90 percent confidence, 1.96 for 95 percent, and 2.576 for 99 percent. The confidence interval then runs from your percentage minus the margin to your percentage plus the margin.

What a confidence interval really means

This is the part almost everyone gets slightly wrong, so it is worth stating carefully. A 95 percent confidence interval does not mean there is a 95 percent chance the true value sits inside your particular interval. The true value is a fixed number, either in your interval or not.

What the 95 percent describes is the method. If you repeated the whole sampling process many times and built an interval each time, about 95 percent of those intervals would capture the true value. Your one interval is a single draw from that process, and you are trusting the method that produced it. In everyday terms, it is a range of plausible values for the truth, produced by a procedure that is right 95 percent of the time.

What makes an interval wider or narrower

Two things move the width. The first is the sample size. A bigger sample gives a smaller standard error, and so a narrower, more precise interval. This is why polls want thousands of responses, not dozens. The catch is that the standard error falls with the square root of the sample size, so cutting the margin in half takes four times the data.

The second is the confidence level. Asking for more confidence, say 99 percent instead of 95, widens the interval, because being more sure of catching the truth means casting a wider net. So there is always a trade-off between how confident you are and how precise your range is, and choosing a level is really choosing where to sit on that balance.

A worked example

Suppose a survey of 1,000 people finds 60 percent support a proposal, and you want a 95 percent confidence interval.

The margin of error is 1.96 × √(0.6 × 0.4 ÷ 1000) = 1.96 × √0.00024 = about 0.030, or 3.0 percentage points. So the confidence interval is 60 percent give or take 3 points, running from about 57 percent to 63 percent. The honest headline is not "60 percent support it" but "somewhere around 57 to 63 percent, and our method is right 95 times in 100."

Entering your values

Enter the sample size and the sample percentage, and pick a confidence level. The calculator uses the standard normal z-score method for a proportion, which is the usual approach when the sample is a reasonable size. It returns the margin of error, which you add to and subtract from your percentage to get the lower and upper ends of the interval.

Questions people ask

What is a confidence interval?

A range around an estimate that is likely to contain the true population value. It is your sample result plus and minus a margin of error set by the confidence level.

Does 95 percent mean a 95 percent chance the truth is in my interval?

Not quite. It means that if you repeated the sampling many times, about 95 percent of the intervals produced would contain the true value. Your single interval comes from that reliable method.

How do I get a narrower interval?

Use a larger sample, which shrinks the margin of error, or accept a lower confidence level. Since the margin falls with the square root of the sample size, halving it needs four times the data.

Which confidence level should I use?

95 percent is the common default. Higher confidence like 99 percent gives a wider interval, and lower confidence like 90 percent gives a narrower one, so it is a trade-off between sureness and precision.

References

A quick note on where the methods here come from. Confidence intervals for a proportion and their interpretation are set out in the NIST/SEMATECH e-Handbook of Statistical Methods, the US government's public statistics reference. OpenStax Introductory Statistics is a free, widely used textbook covering confidence intervals.

1. NIST/SEMATECH e-Handbook of Statistical Methods (confidence intervals). https://www.itl.nist.gov/div898/handbook/
2. OpenStax, Introductory Statistics (confidence intervals). https://openstax.org/details/books/introductory-statistics-2e