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Comprehensive Confidence Interval Calculator

Calculate 95% confidence intervals for means, proportions, and differences. Free CI calculator with margin of error, sample size recommendations, and bootstrap methods. Excel compatible with APA formatting.

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Confidence Interval for Population Mean

Calculate confidence intervals for population means using t-distribution. Handles both known and unknown population standard deviation with appropriate critical values and standard error calculations.

  • 95% confidence interval for mean
  • Custom confidence levels (90%, 99%)
  • T-distribution for small samples
  • Z-distribution for large samples
  • Standard error calculation

Confidence Interval for Proportion

Calculate confidence intervals for population proportions using normal approximation and Wilson score methods. Includes sample size recommendations and margin of error for survey research.

  • Normal approximation method
  • Wilson score confidence intervals
  • Adjusted Wald intervals
  • Exact binomial intervals
  • Survey margin of error calculator

Difference Between Two Means

Calculate confidence intervals for the difference between two population means. Supports both independent samples and paired data with appropriate pooled or separate variance estimates.

  • Independent samples CI
  • Paired samples CI
  • Pooled variance estimation
  • Welch's method for unequal variances
  • Effect size confidence intervals

Bootstrap Confidence Intervals

Non-parametric bootstrap confidence intervals for any statistic. Ideal when distribution assumptions are violated or for complex statistics like medians, correlations, or regression coefficients.

  • Percentile bootstrap method
  • Bias-corrected bootstrap
  • Bootstrap-t intervals
  • Custom statistic support
  • Resampling visualization

How to Calculate Confidence Intervals

Step 1: Choose Confidence Level

Select your desired confidence level (typically 95%). This determines the critical value and the width of your interval. Higher confidence levels produce wider intervals.

Step 2: Calculate Standard Error

Compute the standard error of your statistic. For means: SE = s/√n. For proportions: SE = √[p(1-p)/n]. This measures the precision of your estimate.

Step 3: Find Critical Value

Determine the appropriate critical value from t-distribution (small samples) or z-distribution (large samples). For 95% CI: t₀.₀₂₅ or z₀.₀₂₅ = 1.96.

Step 4: Calculate Margin of Error

Multiply the critical value by the standard error: ME = critical value × SE. This represents the maximum expected sampling error.

Step 5: Construct the Interval

Add and subtract the margin of error from your point estimate: CI = estimate ± ME. This gives you the lower and upper bounds of your confidence interval.

Confidence Interval Methods Comparison

Parameter Method When to Use Formula
Population Mean (σ known) Z-interval Large samples, known population SD x̄ ± z₍α/2₎ × (σ/√n)
Population Mean (σ unknown) T-interval Any sample size, unknown population SD x̄ ± t₍α/2₎ × (s/√n)
Population Proportion Normal Approximation np ≥ 5 and n(1-p) ≥ 5 p̂ ± z₍α/2₎ × √[p̂(1-p̂)/n]
Population Proportion Wilson Score Small samples or extreme proportions More complex formula, better coverage
Difference in Means Two-sample t Independent groups comparison (x̄₁-x̄₂) ± t₍α/2₎ × SE_diff

Sample Size Calculator for Desired Margin of Error

Determine the required sample size to achieve your desired margin of error and confidence level:

For Population Mean

Formula: n = (z₍α/2₎ × σ / E)²
Where E is the desired margin of error and σ is the population standard deviation (or use sample SD as estimate).

For Population Proportion

Formula: n = (z₍α/2₎)² × p(1-p) / E²
Use p = 0.5 for maximum sample size when proportion is unknown. This gives the most conservative estimate.

Survey Research Guidelines

Typical Margins:
• Political polls: ±3% (n ≈ 1,067)
• Market research: ±5% (n ≈ 384)
• Academic research: ±2% (n ≈ 2,401)

Frequently Asked Questions

How to calculate 95% confidence interval?
For a 95% confidence interval: CI = x̄ ± (t₀.₀₂₅ × SE) where x̄ is the sample mean, t₀.₀₂₅ is the critical t-value (1.96 for large samples), and SE is the standard error (s/√n). This means we're 95% confident the true population parameter lies within this range.
What does a confidence interval tell you?
A confidence interval provides a range of plausible values for a population parameter. A 95% CI means that if we repeated the study 100 times, about 95 of the intervals would contain the true population value. It quantifies uncertainty in our estimate and provides more information than a point estimate alone.
How to interpret margin of error?
Margin of error is the maximum expected difference between the sample estimate and true population value. For example, if a poll shows 52% ± 3%, the true value is likely between 49% and 55%. Smaller margins indicate more precise estimates and require larger sample sizes.
When to use Wilson score vs normal approximation?
Use Wilson score confidence intervals for proportions when sample size is small or proportion is near 0 or 1. Use normal approximation when np ≥ 5 and n(1-p) ≥ 5. Wilson score provides better coverage for extreme proportions and is generally more robust.

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