Confidence Interval Calculator
Calculate confidence intervals for population means.
Enter your sample data or summary statistics to find the margin of error and interval bounds.
Confidence Interval Formula:
CI = x̄ ± z* × (s / √n)
Where:
- x_bar = the sample mean (the average of your data)
- z* = the critical value corresponding to your chosen confidence level
- s = the sample standard deviation (a measure of how spread out your data is)
- n = the sample size (how many observations you collected)
The expression s / √n is called the Standard Error (SE). It estimates how much the sample mean is likely to differ from the true population mean. Larger samples produce smaller standard errors, meaning more precise estimates.
Common z values:*
| Confidence Level | z* Value | Meaning |
|---|---|---|
| 90% | 1.645 | 10% chance the true mean is outside the interval |
| 95% | 1.960 | 5% chance the true mean is outside the interval |
| 99% | 2.576 | 1% chance the true mean is outside the interval |
| 99.9% | 3.291 | 0.1% chance the true mean is outside the interval |
What the confidence interval means: A 95% confidence interval does NOT mean there is a 95% probability the true mean is inside this specific interval. It means that if you repeated the entire sampling and calculation process many times, about 95% of the resulting intervals would contain the true population mean.
Practical Example: You measure the height of 50 randomly selected adults and find a mean of 170.2 cm with a standard deviation of 8.5 cm. At 95% confidence: SE = 8.5 / √50 = 1.202 Margin of error = 1.960 x 1.202 = 2.356 CI = 170.2 ± 2.356 = 167.84 to 172.56 cm You are 95% confident the true average height of the population falls between 167.84 cm and 172.56 cm.
When to use this calculator: Confidence intervals are used in scientific research, medical trials, market research, quality control, and any situation where you measure a sample and want to estimate a population parameter.
Tips:
- Increasing your sample size is the most effective way to narrow a confidence interval
- Higher confidence levels (e.g., 99% vs 95%) produce wider intervals — there is always a tradeoff between confidence and precision
- For small samples (n < 30), a t-distribution is more appropriate than the z-distribution. This calculator uses the z-distribution, which works well for n >= 30.