Confidence Interval Calculator
Calculate confidence intervals for sample data with known or unknown population standard deviation. Supports both z-distribution (normal) and t-distribution calculations.
Confidence Interval: 95% CI [92.63, 107.37]
Detailed Results
| Sample Statistic | 100.00 |
|---|---|
| Margin of Error | 7.37 |
| Standard Error | 2.74 |
| Critical Value (z/t) | 1.960 |
| Degrees of Freedom | n/a |
| Confidence Level | 95% |
About Confidence Interval Calculator
The Confidence Interval Calculator computes the range in which the population parameter is likely to fall, based on your sample data and selected confidence level. It handles three common cases: mean with known population standard deviation (z-interval), mean with unknown population standard deviation (t-interval), and proportion (z-interval).
How Confidence Intervals Work
A confidence interval gives an estimated range of values which is likely to include an unknown population parameter. The width of the interval gives us an idea of how uncertain we are about the unknown parameter.
Key Concepts:
- Confidence Level: The probability that the confidence interval contains the true population parameter (typically 90%, 95%, or 99%).
- Margin of Error: Half the width of the confidence interval, representing the maximum expected difference between the sample statistic and population parameter.
- Critical Value: The z-score or t-score corresponding to the selected confidence level.
Formulas Used
Mean (σ Known):
CI = x̄ ± z*(σ/√n)
Where x̄ is the sample mean, σ is the population standard deviation, n is the sample size, and z is the critical value from the standard normal distribution.
Mean (σ Unknown):
CI = x̄ ± t*(s/√n)
Where s is the sample standard deviation and t is the critical value from the t-distribution with n-1 degrees of freedom.
Proportion:
CI = p̂ ± z*√(p̂(1-p̂)/n)
Where p̂ is the sample proportion (x/n), x is the number of successes, and n is the sample size.
When to Use Each Method
- Mean (σ Known): Use when you know the population standard deviation (rare in practice).
- Mean (σ Unknown): Use when you only have the sample standard deviation (most common case).
- Proportion: Use when your data is categorical (success/failure) and you want to estimate the population proportion.
Applications
Confidence intervals are widely used in:
- Scientific research and experiments
- Quality control and process improvement
- Market research and surveys
- Medical studies and clinical trials
- Economic forecasting