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Confidence Interval Calculator

Calculate confidence intervals to estimate population parameters from sample data. Perfect for research, data analysis, and statistical inference.

How to Use This Calculator

  1. Enter your sample mean (average of your data)
  2. Enter the standard deviation of your sample
  3. Enter your sample size (number of observations)
  4. Choose your desired confidence level (90%, 95%, or 99%)
  5. Select whether to use Z-score (large samples) or T-score (small samples)
  6. Click "Calculate Confidence Interval" to see the results

For large samples (n ≥ 30), use Z-scores. For smaller samples, use T-scores to account for the additional uncertainty in estimating population parameters from small samples.

Understanding Confidence Intervals

A confidence interval provides a range of values that likely contains the true population parameter. It helps quantify the uncertainty in sample estimates and is crucial for statistical inference.

Interpreting Confidence Levels

The confidence level represents the reliability of the interval:

  • 95% Confidence: If we repeated the sampling process many times, about 95% of the intervals would contain the true population parameter.
  • 90% Confidence: Narrower interval, but less confident that it contains the true value.
  • 99% Confidence: Wider interval, but more confident that it contains the true value.

Z-scores vs. T-scores

The choice between Z-scores and T-scores depends on your sample size:

  • Z-scores: Use when n ≥ 30 or when population standard deviation is known. Based on the normal distribution.
  • T-scores: Use when n < 30 and population standard deviation is unknown. Accounts for additional uncertainty in small samples.

Formula and Calculation

The confidence interval is calculated as:

CI = x̄ ± (critical value × (s/√n))

Where:

  • • x̄ is the sample mean
  • • s is the sample standard deviation
  • • n is the sample size
  • • Critical value is from Z or T distribution

Example Calculation

For a sample with:

  • • Mean = 75
  • • Standard deviation = 8
  • • Sample size = 36
  • • 95% confidence level (Z = 1.96)
  • • Margin of error = 1.96 × (8/√36) = 2.61
  • • Confidence interval: 75 ± 2.61 = (72.39, 77.61)

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