Calculate the confidence interval for a sample mean using the normal approximation. See the range where the true population mean likely falls.
Confidence Interval (95%)
95.8422 to 104.1578
Margin of Error±4.1578
Standard Error2.1213
Lower Bound95.8422
Upper Bound104.1578
Z-Score1.96
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What is the Confidence Interval Calculator?
A confidence interval is a range of values around a sample mean that is likely to contain the true population mean with a specified level of confidence. For example, a 95% confidence interval means you can be 95% confident that the true mean lies within the calculated range. It's a fundamental tool in statistics for expressing uncertainty in estimates.
How it works
The calculator uses the formula: Confidence Interval = Sample Mean ± (Z-Score × Standard Error). The standard error is the sample standard deviation divided by the square root of the sample size. The Z-score changes based on your chosen confidence level (90%→1.645, 95%→1.96, 99%→2.576). The result is a lower and upper bound around your sample mean.
CI = Mean ± (Z × SD / √n)
CI is the confidence interval. Mean is your sample mean. Z is the critical value for your confidence level. SD is sample standard deviation. n is sample size. This formula assumes a normal distribution and is appropriate for large samples (typically n ≥ 30).
Examples
Input
Result
Notes
Mean 100, SD 15, n=50, 95% confidence
95.84 to 104.16
You can be 95% confident the true population mean is between 95.84 and 104.16. Margin of error is ±4.16.
Mean 72.5, SD 8.2, n=100, 90% confidence
70.55 to 74.45
90% CI for test scores: margin of error ±1.95. Wider confidence increases margin.
Mean 250, SD 45, n=64, 99% confidence
233.53 to 266.47
99% CI requires a larger margin (±16.47) than 95% to achieve higher confidence.
How to use the Confidence Interval Calculator
Enter your sample mean (the average of your data)
Enter your sample standard deviation (a measure of spread)
Enter your sample size (how many observations you collected)
Select your desired confidence level (90%, 95%, or 99%)
Read the confidence interval from the results—it shows the likely range for the true population mean
Benefits
Quantifies uncertainty—express how confident you are in your estimate, not just a point estimate
Supports decision-making—use the interval to set realistic expectations or bounds on a parameter
Fast computation—avoid manual Z-score lookups and arithmetic errors
Multiple confidence levels—compare 90%, 95%, and 99% to see how width changes with confidence
Essential for research—publish confidence intervals alongside means to meet scientific reporting standards
Tips & common mistakes
Common mistakes
Misinterpreting the interval—a 95% CI does not mean 95% of your data falls in the range; it means the true population mean has a 95% chance of lying within it
Using wrong standard deviation—make sure you use sample SD, not population SD
Forgetting to check sample size—this method assumes a reasonably large sample (n ≥ 30); for very small samples, use t-distribution instead
Assuming a normal distribution when data is skewed—this formula works best with normally distributed data
Tips
Remember: wider intervals = higher confidence. A 99% CI will always be wider than a 90% CI for the same data
For larger samples, the standard error decreases, making your interval narrower and your estimate more precise
Use 95% confidence as a standard in most research contexts unless your field demands otherwise
If you have fewer than 30 observations, consider using a t-distribution instead of the Z-distribution for a more conservative estimate
Frequently asked questions
What does a 95% confidence interval really mean?
It means that if you repeated your sampling many times and calculated a CI each time, about 95% of those intervals would contain the true population mean. It does not mean 95% of the data falls in the range.
Why does a higher confidence level make the interval wider?
A higher confidence level (e.g., 99% vs. 95%) requires a larger Z-score multiplier, which increases the margin of error. You trade width for certainty.
What if my sample size is very small (less than 30)?
For small samples, the t-distribution is more appropriate than the normal distribution (Z-test). Your true margin of error will be slightly larger than this calculator shows.
Can I use this for non-normally distributed data?
The method assumes normality. For large samples (n ≥ 30), the Central Limit Theorem makes the distribution of the sample mean approximately normal, so this works well. For smaller, skewed samples, use caution.
What is standard error and how is it different from standard deviation?
Standard deviation measures spread in your data. Standard error measures the spread of the sample mean across repeated samples. SE = SD / √n. As n increases, SE decreases.
How do margin of error and confidence interval relate?
The margin of error (±value) is half the width of the interval. CI = Mean ± Margin of Error. A smaller margin means a more precise estimate.