Confidence Interval Calculator
for Mean and Proportion

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Computing margin of error…
Constructing interval…

Solution Output

Sample data recognized ✓
Method: Confidence Interval for Mean (t-distribution)
Step 1: n = 40, x̄ = 52, s = 8, confidence = 95%

df = n − 1 = 39 Critical value: t* = 2.023 (95% CI, df = 39, two-tailed) Standard Error: SE = s/√n = 8/√40 = 8/6.324 = 1.265 Margin of Error: ME = t* × SE = 2.023 × 1.265 = 2.56 Lower bound: x̄ − ME = 52 − 2.56 = 49.44 Upper bound: x̄ + ME = 52 + 2.56 = 54.56 95% CI: (49.44, 54.56) Interpretation: We are 95% confident the population mean lies between 49.44 and 54.56.

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What Is a Confidence Interval?

A confidence interval is a range of values that is likely to contain the true population parameter with a specified level of confidence. A 95% confidence interval means that if you repeated the sampling process many times, 95% of the intervals you constructed would contain the true parameter. It does not mean there is a 95% probability the parameter falls in this specific interval.

Confidence intervals are reported in research papers, clinical trials, opinion polls, and quality control studies. They give more information than a point estimate alone because they indicate how precise the estimate is — a narrow CI indicates high precision; a wide CI indicates high uncertainty.

CI for Population Mean

When estimating the population mean from a sample, the formula is CI = x̄ ± t* × (s/√n), where t* is the critical value from the t-distribution for the chosen confidence level and df = n−1. For large samples (n > 30), z* can substitute for t*.

CI for Population Proportion

For proportion data, the formula is CI = p̂ ± z* × √(p̂(1−p̂)/n). This applies when np̂ ≥ 10 and n(1−p̂) ≥ 10 — the normal approximation conditions.

What Affects the Width of a CI

Confidence level: Higher confidence (99% vs 95%) produces a wider interval
Sample size: Larger n produces a narrower interval (more precise)
Standard deviation: Greater variability in the data produces a wider interval

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