Confidence Interval Calculator
Enter sample mean, standard deviation, sample size, and confidence level.
Confidence Interval — Formula
Formula
CI = x̄ ± z* × (σ / √n)
Where z* is the critical value (1.645 for 90%, 1.960 for 95%, 2.576 for 99%).
Interpretation
A 95% CI means: if you repeated the study 100 times, 95 of the intervals would contain the true population mean.
Related Calculators
What a Confidence Interval Really Means
A 95% confidence interval does NOT mean there is a 95% probability that the true value lies in this specific interval. The true parameter is fixed — either it is in the interval or it is not. What 95% means is this: if you repeated the experiment many times and built a confidence interval each time, 95% of those intervals would contain the true parameter. This distinction matters in research and statistics communication.
Wider intervals indicate less precision (small samples or high variability). Narrower intervals indicate more precision. To halve the margin of error, you must quadruple the sample size — a key insight for study design.
Confidence Level vs Confidence Interval
| Confidence level | Z-score (z*) | Typical use |
|---|---|---|
| 90% | 1.645 | Preliminary research, quick estimates |
| 95% | 1.960 | Standard in most published research |
| 99% | 2.576 | Clinical trials, high-stakes decisions |
| 99.7% | 3.000 | Physics (3-sigma rule) |
How to Read a Confidence Interval
A 95% confidence interval means: if you repeated the study 100 times and built a CI each time, approximately 95 of those intervals would contain the true population parameter. It does NOT mean there is a 95% probability the true value is in this specific interval — once computed, the true value either is or is not inside it. The width of a CI depends on sample size, standard deviation, and confidence level: larger samples and lower confidence produce narrower intervals.
To choose between 90%, 95%, and 99% confidence: 95% is the conventional standard in most research. Use 99% when the cost of being wrong is high (medical trials, safety testing). Use 90% when wider intervals are acceptable and sample collection is expensive. Increasing sample size is the most effective way to narrow a CI without sacrificing confidence level.
Confidence Level vs. Z-Score
| Confidence Level | Z-Score | Typical Use |
|---|---|---|
| 90% | 1.645 | Preliminary research |
| 95% | 1.960 | Standard in most fields |
| 99% | 2.576 | Medical, safety studies |
| 99.9% | 3.291 | High-stakes decisions |
Interpreting Confidence Intervals in Research
A confidence interval provides a range of plausible values for a population parameter based on sample data. A 95% confidence interval, for example, means that if you repeated the sampling process many times and constructed a confidence interval each time, approximately 95% of those intervals would contain the true population value. It does not mean there is a 95% chance the true value lies in any single interval you have calculated. Wider intervals indicate less precision; narrower intervals indicate more. Increasing sample size narrows the interval for a given confidence level because larger samples estimate the population more accurately. Increasing the confidence level (from 95% to 99%, for example) widens the interval because you are requiring greater certainty. Researchers, clinicians, pollsters, and quality control engineers all use confidence intervals to communicate not just a point estimate but the uncertainty surrounding it, which is essential for sound decision-making and honest reporting of statistical results.
Common Confidence Levels Reference Table
| Confidence level | Z critical value | Common use |
|---|---|---|
| 90% | 1.645 | Exploratory research |
| 95% | 1.960 | Standard in most fields |
| 99% | 2.576 | Medical and safety studies |
| 99.9% | 3.291 | High-stakes quality control |
