Confidence Interval Calculator

What Is a Confidence Interval?

A confidence interval (CI) is a range of values, derived from sample data, that is likely to contain the true population parameter. It quantifies the uncertainty inherent in estimation: rather than providing a single point estimate, a CI gives an interval within which the parameter is expected to fall with a given level of confidence.

For example, a 95% confidence interval means that if you were to repeat the sampling process many times and compute a CI each time, approximately 95% of those intervals would contain the true population mean.

The Formula

• x̄ = sample mean
• t_{α/2, n-1} = critical value from the t-distribution with n-1 degrees of freedom
• s = sample standard deviation
• n = sample size
• s / √n = standard error of the mean (SE)

When to Use Z vs. t Distribution

This calculator uses the t-distribution, which is always valid and automatically approaches the normal distribution as sample size increases.

How to Interpret a Confidence Interval

A common misconception is that a 95% CI means there is a 95% probability that the true parameter lies within the interval. In frequentist statistics, the correct interpretation is:

• If the experiment were repeated many times, 95% of the computed intervals would contain the true population mean
• The true parameter is a fixed (but unknown) value, not random
• A wider interval indicates greater uncertainty in the estimate
• A narrower interval indicates a more precise estimate

Factors That Affect the Width of a CI

• Sample size (n): Larger samples produce narrower intervals (more precision)
• Variability (s): Lower variability in the data leads to narrower intervals
• Confidence level: Higher confidence (e.g. 99% vs. 95%) produces wider intervals

Worked Example

Suppose we measure the heights (in cm) of 8 randomly selected students and get: 165, 170, 168, 172, 175, 169, 171, 174.

• Step 1: Compute the sample mean: x̄ = (165 + 170 + 168 + 172 + 175 + 169 + 171 + 174) / 8 = 170.5
• Step 2: Compute the sample standard deviation: s ≈ 3.16
• Step 3: Calculate the standard error: SE = 3.16 / √8 ≈ 1.118
• Step 4: Find the t-critical value for 95% confidence with df = 7: t ≈ 2.365
• Step 5: Compute the margin of error: E = 2.365 × 1.118 ≈ 2.644
• Step 6: CI = 170.5 ± 2.644 = (167.856, 173.144)

We are 95% confident that the true mean height of the student population falls between approximately 167.86 cm and 173.14 cm.

References

The formulas and methods used in this calculator are based on established statistical theory from the following sources:

• NIST/SEMATECH e-Handbook of Statistical Methods - Confidence Intervals for the Mean
• Penn State STAT 200 - Confidence Intervals
• Khan Academy - Confidence Intervals
• Devore, J. L. (2015). Probability and Statistics for Engineering and the Sciences (9th ed.). Cengage Learning. Chapter 7: Statistical Intervals Based on a Single Sample.
• Wackerly, D., Mendenhall, W., & Scheaffer, R. (2014). Mathematical Statistics with Applications (7th ed.). Cengage Learning. Chapter 8: Estimation.

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