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Confidence Interval

General

Confidence Interval Estimate

A range of values, calculated from sample data, that is likely to contain the true population parameter at a stated confidence level such as 90%, 95%, or 99%.

Definition

A confidence interval is a range of values, calculated from sample data, that is likely to contain the true population parameter — such as a mean or proportion — at a stated confidence level like 90%, 95%, or 99%. Instead of reporting a single point estimate (for example, "52% of voters support the measure"), a confidence interval communicates the uncertainty inherent in sampling by reporting a range (for example, "49% to 55%, with 95% confidence").

Confidence intervals are central to survey research, clinical trials, quality control, and A/B testing, wherever a sample is used to estimate something about a larger population. The width of the interval depends on the variability in the data, the sample size, and the chosen confidence level, and it is calculated using the Confidence Interval Calculator.

Confidence intervals and hypothesis testing are two sides of the same coin: a confidence interval that excludes a hypothesized value (like zero difference between two groups) corresponds to a statistically significant result at the complementary p-value threshold.

Formula

Confidence Interval = Point Estimate ± Margin of Error

For a sample mean, this expands to:

CI = x̄ ± z × (σ ÷ √n)

Where x̄ is the sample mean, z is the z-score for the chosen confidence level (1.96 for 95%), σ is the standard deviation, and n is the sample size.

Worked Example

A researcher samples 100 students and finds a mean test score of 78, with a standard deviation of 10. For a 95% confidence interval:

Margin of Error = 1.96 × (10 ÷ √100) = 1.96 × 1.0 = 1.96

Confidence Interval = 78 ± 1.96 = 76.04 to 79.96

The researcher can state with 95% confidence that the true mean score of the full student population falls between 76.04 and 79.96.

Key Things to Know

  • Wider is not better: a very wide confidence interval signals high uncertainty (often from a small sample), while a narrow interval signals a more precise estimate.
  • The confidence level is not a probability about the specific interval: it describes the long-run reliability of the method used to construct intervals, not a single interval's chance of being correct.
  • Larger samples produce tighter intervals: use the Sample Size Calculator to plan how many respondents are needed to hit a target margin of error.
  • Related to Margin of Error: the confidence interval is simply the point estimate plus and minus the margin of error.
  • Depends on Standard Deviation: more variable underlying data widens the interval for any given sample size and confidence level.

Related Calculators

Confidence Interval Calculator →Calculate a confidence interval from a sample mean, standard deviation, and sample size in seconds. Get the lower bound, upper bound, and margin of error instantly.
Sample Size Calculator →Find the minimum sample size needed for a survey or experiment in seconds. Enter confidence level, margin of error, and estimated standard deviation to get your answer.
Margin of Error Calculator →Calculate the margin of error for a survey or sample estimate in seconds. Enter confidence level, standard deviation, and sample size to get the exact margin instantly.

Related Terms

Margin of ErrorSampling Margin of ErrorStandard DeviationStatistical Measure of Data VariabilityP-ValueProbability Value

Frequently Asked Questions

A 95% confidence interval means that if you repeated the same sampling process many times and built an interval each time, about 95% of those intervals would contain the true population parameter. It does not mean there is a 95% probability the true value falls in this one specific interval — the true value is fixed, only the interval varies from sample to sample.
A higher confidence level (99% vs 95% vs 90%) requires a wider interval to be more certain of capturing the true parameter, since the corresponding z-score increases from 1.645 (90%) to 1.96 (95%) to 2.576 (99%). Researchers trade off precision (a narrower interval) against confidence (a higher percentage) depending on how much certainty the situation demands.
Larger sample sizes shrink the confidence interval because the margin of error is divided by the square root of the sample size — quadrupling the sample size halves the margin of error. This is why polls and studies with small samples report much wider, less precise intervals.
The margin of error is the ± amount added to and subtracted from the point estimate; the confidence interval is the full range that results from that calculation. For example, if a poll estimate is 52% with a margin of error of 3%, the confidence interval is 49% to 55%.
The standard z-based confidence interval formula assumes a reasonably large sample (commonly n ≥ 30) or a known population standard deviation, thanks to the central limit theorem. For smaller samples with an unknown population standard deviation, a t-distribution based interval is used instead of the normal (z) distribution.