Confidence Interval Calculator

What a Confidence Interval Means

A 95% confidence interval does not mean there is a 95% chance the true value lies within the range. It means that if you repeated this sampling procedure 100 times, approximately 95 of those intervals would contain the true population value. The interval is a statement about the method, not about any single result. This is a subtle but important distinction that even scientists frequently misstate.

Choosing Your Confidence Level

90% CI uses Z equals 1.645 — narrower interval, less certainty. Used when a wider margin of error is acceptable and a smaller sample size is necessary. 95% CI uses Z equals 1.96 — the standard for most research and polling. 99% CI uses Z equals 2.576 — wider interval, very high certainty. Used when false positives are costly, such as pharmaceutical trials.

Margin of Error in Practice

Election polls quoted as X% plus or minus 3% use a 95% confidence interval with a sample of approximately 1,000 respondents. Doubling precision (halving the margin of error) requires quadrupling the sample size — the main reason polls rarely use samples larger than 1,000–2,000. A margin of error of plus or minus 3% at 95% confidence with n equals 1,000 is the standard in UK political polling.

Sample Size vs Margin of Error

For proportion-based CIs at 95% confidence: n=100 gives plus or minus 10%, n=400 gives plus or minus 5%, n=1,000 gives plus or minus 3.1%, n=2,500 gives plus or minus 2%, n=10,000 gives plus or minus 1%. The diminishing returns of larger samples are clear — going from 1,000 to 10,000 only reduces margin from 3% to 1%. This is why practical research balances cost against precision.