Confidence Level Calculator
| Measurement | Value |
|---|
Understanding Confidence Levels
What is a Confidence Level?
A confidence level represents the probability that a population parameter falls within a specified range of values (confidence interval). It measures the reliability of statistical estimates.
Confidence Interval = x̄ ± (z × (σ/√n))
where x̄ = sample mean, z = z-score, σ = standard deviation, n = sample size
Key Components
- Z-score: Standard normal distribution value
- Margin of Error: Maximum expected difference
- Sample Size: Number of observations
- Standard Deviation: Measure of variability
- Population Mean: True value being estimated
Statistical Properties
Sample Size Formula
n = (z²σ²)/E²
where E = margin of error
Margin of Error
E = z(σ/√n)
Standard Error
SE = σ/√n
Common Confidence Levels
| Confidence Level | Z-score | Application |
|---|---|---|
| 90% | 1.645 | Preliminary studies |
| 95% | 1.96 | Standard research |
| 99% | 2.576 | Critical decisions |
Advanced Concepts
Central Limit Theorem
Sample means follow normal distribution
Degrees of Freedom
Impact on t-distribution
Type I & II Errors
Relationship with confidence level
Real-World Applications
Medical Research
Drug efficacy studies and clinical trials
Quality Control
Manufacturing process monitoring
Market Research
Consumer behavior analysis