Confidence Interval for Mean Calculator (Z-Interval & T-Interval

Statistical Foundations of Confidence Intervals for the Mean

A confidence interval provides a range of plausible values for an unknown population parameter (such as population mean μ) based on sample statistics:

• T-Interval vs. Z-Interval:

• T-Interval (Student's t): Used whenever the population standard deviation σ is unknown and estimated using sample standard deviation s (the standard real-world case).

• Z-Interval (Standard Normal Z): Used only when the true population standard deviation σ is known from extensive census or baseline data.

• Margin of Error Drivers: Margin of error shrinks (increasing estimation precision) when:

• Sample size n increases (inversely proportional to √(n)).

• Population variability (s or σ) is smaller.

• Confidence level is reduced (e.g. 90% produces a narrower interval than 99%).

• Frequentist Meaning: A 95% confidence level does not mean there is a 95% chance that the true mean is in this specific fixed numerical interval; rather, it means that 95% of intervals generated across repeated random samples will contain the true population mean μ.

How to Use This Calculator

Enter your Sample Mean (x̄) (e.g. 52.4), Standard Deviation Known / Type — choose from Sample Standard Deviation (s) - Uses Student's T-Interval (Standard), Known Population Standard Deviation (σ) - Uses Z-Interval, Standard Deviation Value (s or σ) (e.g. 8.5), Sample Size (n) (e.g. 35 (Number of observations)), and Confidence Level (1 - α) — choose from 95% Confidence Level (α = 0.05 - Standard), 90% Confidence Level (α = 0.10), 99% Confidence Level (α = 0.01). Results update instantly as you type — adjust any value to compare scenarios.

The Formula

Confidence Interval for Population Mean μ equations:

Standard Error of the Mean: SE = (s) / (√(n)) (or (σ) / (√(n)))

Margin of Error: ME = t^ × SE (with df = n - 1) or ME = z^ × SE

Confidence Interval: CI = [ x - ME, x + ME ] = x ± ME

Total Interval Width: W = 2 × ME = 2 × t^* × (s) / (√(n))

Worked Example

For example, evaluating a sample of \ observations with sample mean \ and sample standard deviation \ at 95\% confidence: Standard error is . With critical . Margin of Error is . The 95% confidence interval is .

Understanding Your Result

Statistical results require context: a mean alone is misleading without knowing the standard deviation (spread). Normal distribution reference: 68% of values fall within ±1 SD of the mean, 95% within ±2 SD, 99.7% within ±3 SD. For small sample sizes (n < 30), use t-distributions rather than z-scores. Statistical significance (p < 0.05) does not imply practical significance — effect size (Cohen's d, R², odds ratio) measures real-world importance. Correlation coefficient r: |r| < 0.3 weak, 0.3–0.7 moderate, > 0.7 strong.