Confidence Interval Calculator
Calculate the confidence interval or margin of error for a sample mean, assuming the sample follows a normal distribution. Use the Standard Deviation Calculator if you have raw data only.
About Confidence Intervals
A confidence interval is a range of values that is likely to contain an unknown population parameter. It is calculated from a sample of data and provides a measure of the uncertainty or precision of an estimate.
Understanding Confidence Intervals
When we calculate a confidence interval, we are making a statement about how confident we are that the interval contains the true population parameter. For example, a 95% confidence interval means that if we were to take 100 different samples and compute a 95% confidence interval for each sample, then approximately 95 of the 100 confidence intervals would contain the true population parameter.
Formula
Confidence Interval:
CI = X̄ ± Zα/2 × (σ/√n)
Where:
X̄ = sample mean
Zα/2 = critical value for the confidence level
σ = population standard deviation (or s for sample standard deviation)
n = sample size
Common Z-values for Confidence Levels
- 90% confidence: Z = 1.645
- 95% confidence: Z = 1.96
- 99% confidence: Z = 2.576
Example Calculation
Let's calculate a 95% confidence interval for a sample with the following characteristics:
Sample size (n) = 100
Sample mean (X̄) = 25.2
Standard deviation (σ) = 4.5
Confidence level = 95% (Z = 1.96)
Step 1: Calculate the margin of error
Margin of error = Z × (σ/√n) = 1.96 × (4.5/√100) = 1.96 × 0.45 = 0.882
Step 2: Calculate the confidence interval
CI = X̄ ± margin of error = 25.2 ± 0.882 = [24.318, 26.082]
Interpreting Confidence Intervals
A narrower confidence interval indicates a more precise estimate, while a wider interval suggests more uncertainty. Factors that affect the width of a confidence interval include:
- Sample size: Larger samples produce narrower intervals
- Confidence level: Higher confidence levels produce wider intervals
- Variability: More variable data produces wider intervals
Applications of Confidence Intervals
- Medical research: Estimating treatment effects
- Political polling: Estimating voter preferences
- Quality control: Monitoring manufacturing processes
- Market research: Estimating consumer preferences
- Scientific research: Estimating population parameters