Confidence Interval Calculator
Confidence Interval Calculator
Calculate confidence intervals for population means with known or unknown standard deviation
Confidence Interval Calculator: Complete Statistical Guide
A confidence interval provides a range of values that likely contains the true population parameter.Confidence intervals are fundamental in statistics for quantifying uncertainty in estimates and are essential for research, quality control, and decision-making across all scientific and business fields.
Our professional confidence interval calculator supports both z-distribution (known population standard deviation) and t-distribution (unknown population standard deviation) calculations with multiple confidence levels and detailed statistical interpretation.
Quick Answer
To calculate a confidence interval: Use the formula CI = x̄ ± (critical value × standard error). For a 95% confidence interval with sample mean 50, standard deviation 10, and sample size 25: CI = 50 ± (2.064 × 2) = (45.9, 54.1). This means we're 95% confident the true population mean lies between 45.9 and 54.1.
Mathematical Foundation
Confidence interval formula where x̄ is sample mean, tα/2 is critical value, and SE is standard error
Key Components:
Standard Error (SE)
SE = σ/√n where σ is standard deviation and n is sample size. Measures the precision of the sample mean as an estimate of population mean.
Critical Value
From z-distribution (σ known) or t-distribution (σ unknown). Determines the width of the confidence interval based on desired confidence level.
Margin of Error (E)
E = critical value × SE. Half the width of the confidence interval. Represents the maximum expected difference between sample and population mean.
Z vs T Distribution
When to Use Z-Distribution
Use when population standard deviation is known or large sample sizes.
• Population σ is known
• Sample size n ≥ 30 (Central Limit Theorem)
• Normal population distributionWhen to Use T-Distribution
Use when population standard deviation is unknown (most common case).
• Population σ is unknown
• Using sample standard deviation (s)
• Small to moderate sample sizes
• Degrees of freedom = n - 1Understanding Confidence Levels
Common Confidence Levels
90% Confidence
Lower precision, narrower interval. Used for preliminary analysis or when precision less critical.
95% Confidence
Standard choice for most research. Good balance between precision and confidence.
99% Confidence
High confidence, wider interval. Used for critical decisions or safety-related applications.
Interpretation Guidelines
What 95% Confidence Means
If we repeated the sampling process many times, 95% of calculated intervals would contain the true population parameter.
Common Misconceptions
The interval does NOT mean there's a 95% probability the true value is in this specific interval.
Practical Use
Use confidence intervals to assess precision of estimates and make informed decisions about populations.
Applications of Confidence Intervals
Research & Science
Clinical Trials
Estimate treatment effects, assess drug efficacy, determine sample sizes for studies
Environmental Studies
Pollution levels, species populations, climate measurements with uncertainty quantification
Psychology Research
Behavioral measurements, cognitive testing, survey analysis with confidence bounds
Business & Quality
Quality Control
Manufacturing specifications, defect rates, process capability analysis
Market Research
Customer satisfaction, market share estimates, survey results with margin of error
Financial Analysis
Investment returns, risk assessment, performance evaluation with uncertainty
Related Statistical Tools
Confidence Interval Calculator
Calculate confidence intervals for population means with known or unknown standard deviation