Confidence Interval Calculator
Calculate confidence intervals for population means with customizable confidence levels
Calculate Confidence Interval
Average value of your sample data
Number of observations in your sample
Choose based on whether you know population σ
Sample standard deviation
Confidence Interval Results
Calculation Details
Interpretation Guidelines
Example: Manufacturing Quality Control
Scenario
Problem: A brick manufacturer wants to estimate the average mass of bricks
Sample size: 100 bricks
Sample mean: 3.0 kg
Sample standard deviation: 0.5 kg
Desired confidence level: 95%
Solution Steps
1. Standard Error = 0.5 ÷ √100 = 0.05
2. Critical Value (Z₀.₀₂₅) = 1.960
3. Margin of Error = 1.960 × 0.05 = 0.098
4. 95% CI = [3.0 - 0.098, 3.0 + 0.098] = [2.902, 3.098]
Conclusion: We are 95% confident that the true average mass is between 2.902 kg and 3.098 kg
Common Confidence Levels
Less precise, more confident
Most commonly used
More precise, less confident
Z-score vs t-score
Use Z-score when:
- • Sample size ≥ 30
- • Population σ is known
- • Normal distribution
Use t-score when:
- • Sample size < 30
- • Population σ unknown
- • Using sample standard deviation
Quick Tips
Larger samples give narrower confidence intervals
Higher confidence levels give wider intervals
The sample mean doesn't affect interval width
Lower standard deviation gives narrower intervals
Understanding Confidence Intervals
What is a Confidence Interval?
A confidence interval is a range of values that likely contains the true population parameter (such as the mean) based on sample data. It provides both an estimate and a measure of the uncertainty in that estimate.
Key Components
- •Point Estimate: The sample mean (x̄)
- •Margin of Error: How much we expect our estimate to vary
- •Confidence Level: How often the method works correctly
Mathematical Formula
CI = x̄ ± (critical value × SE)
SE = σ / √n or s / √n
Interpretation
Correct Interpretation: If we repeated our sampling process many times and calculated a confidence interval each time, about 95% of those intervals would contain the true population mean.
Factors Affecting Interval Width
Sample Size (n)
Larger samples → Smaller standard error → Narrower intervals
Confidence Level
Higher confidence → Larger critical value → Wider intervals
Standard Deviation
More variability → Larger standard error → Wider intervals