Confidence Interval Calculator
Free Confidence interval Calculator for biostatistics. Enter variables to compute results with formulas and detailed steps.
Formula
CI = x-bar +/- z * (s / sqrt(n)) | CI(p) = p-hat +/- z * sqrt(p*q/n)
For means: the margin of error is the critical z-value (or t-value for small samples) times the standard error (SD / sqrt of sample size). For proportions: the standard error uses p-hat*(1-p-hat)/n. The Wilson score interval provides improved coverage for proportions.
Frequently Asked Questions
What is a confidence interval and what does it tell us?
A confidence interval (CI) provides a range of plausible values for a population parameter based on sample data. A 95% CI means that if you repeated the sampling process many times and computed a CI each time, approximately 95% of those intervals would contain the true population parameter. It does NOT mean there is a 95% probability the true value lies in any single interval. The CI width reflects precision: narrower intervals indicate more precise estimates. Confidence intervals are more informative than p-values alone because they show both the direction and magnitude of an effect, along with the uncertainty in the estimate.
When should I use a z-interval versus a t-interval?
Use a z-interval when the population standard deviation is known (rare in practice) or when the sample size is large (n >= 30), as the Central Limit Theorem ensures the sampling distribution is approximately normal. Use a t-interval when the population standard deviation is unknown and you must estimate it from the sample, especially with small samples (n < 30). The t-distribution has heavier tails than the normal distribution, producing wider confidence intervals that account for additional uncertainty from estimating the standard deviation. As sample size increases, the t-distribution approaches the normal distribution, and z and t intervals become virtually identical above n = 100.
How does sample size affect the confidence interval width?
The margin of error is inversely proportional to the square root of the sample size: MOE = z * sigma / sqrt(n). This means quadrupling the sample size cuts the margin of error in half. For example, with sigma=10 at 95% confidence: n=25 gives MOE=3.92, n=100 gives MOE=1.96, n=400 gives MOE=0.98. This diminishing-returns relationship means that beyond a certain point, large increases in sample size yield small improvements in precision. Researchers use sample size calculators to find the minimum n needed for a desired margin of error, balancing precision against cost and feasibility.
What is the Wilson score interval for proportions?
The Wilson score interval is an improved method for computing confidence intervals for proportions, developed by Edwin Wilson in 1927. Unlike the standard Wald interval (p-hat +/- z*SE), the Wilson interval performs better when the sample size is small or the proportion is near 0 or 1. The Wald interval can produce impossible values (below 0 or above 1) and has poor coverage probability for extreme proportions. The Wilson interval is centered not at p-hat but at a value pulled slightly toward 0.5, and its width adjusts based on the estimated proportion. It is now recommended as the default method by many statisticians and is used in medical research and survey analysis.
How do I interpret overlapping confidence intervals?
A common misconception is that non-overlapping confidence intervals indicate a statistically significant difference. While non-overlap does imply significance, overlapping CIs do NOT necessarily mean the difference is non-significant. Two 95% CIs can overlap by as much as 25% of their width and still indicate a significant difference at the 0.05 level. This is because the CI for the difference between two means is not simply the overlap of individual CIs. To properly compare two groups, construct a CI for the difference (mean1 - mean2) and check if it contains zero. If zero is not in the interval, the difference is statistically significant at the corresponding alpha level.
What formula does Confidence Interval Calculator use?
The formula used is described in the Formula section on this page. It is based on widely accepted standards in the relevant field. If you need a specific reference or citation, the References section provides links to authoritative sources.