Confidence Interval Calculator
Calculate the margin of error and the range within which a population parameter is likely to fall based on your sample data.
Calculation Results
Understanding Confidence Intervals
A Confidence Interval (CI) is a range of values, derived from sample statistics, that is likely to contain the value of an unknown population parameter. It provides a measure of uncertainty associated with an estimate.
The Confidence Interval Formula
CI = x̄ ± Z * (σ / √n)
- x̄ (Sample Mean): The average value of your data sample.
- Z (Z-score): The number of standard deviations a point is from the mean, determined by the confidence level (e.g., 1.96 for 95%).
- σ (Standard Deviation): The measure of variation or dispersion in the data.
- n (Sample Size): The total number of observations in your sample.
- σ / √n (Standard Error): The standard deviation of the sampling distribution.
How to Interpret the Results
If you calculate a 95% confidence interval for a weight measurement to be between 150 lbs and 160 lbs, it means that if you were to repeat the sampling process many times, 95% of the calculated intervals would contain the true population mean weight.
Common Z-scores
| Confidence Level | Z-Score |
|---|---|
| 90% | 1.645 |
| 95% | 1.960 |
| 99% | 2.576 |
Practical Example
Imagine a light bulb manufacturer tests 100 bulbs (n = 100) and finds an average lifespan of 1,200 hours (x̄ = 1200) with a standard deviation of 50 hours (σ = 50). To find the 95% confidence interval:
- Find Z-score for 95%: 1.96
- Calculate Standard Error: 50 / √100 = 5
- Calculate Margin of Error: 1.96 * 5 = 9.8
- Lower Bound: 1200 – 9.8 = 1190.2
- Upper Bound: 1200 + 9.8 = 1209.8
The manufacturer is 95% confident the true average lifespan of all bulbs is between 1190.2 and 1209.8 hours.