Confidence Interval Calculator
Confidence Level
Confidence Interval48.0400 - 51.9600
Margin of Error1.9600
Lower Bound48.0400
Upper Bound51.9600
A confidence interval estimates the range in which a population mean is likely to fall, based on a sample. This calculator takes your sample mean, standard deviation, and sample size, multiplies the standard error by a z-score for the chosen confidence level, and reports the lower and upper bounds plus the margin of error. It uses z-scores of 1.645, 1.96, and 2.576 for the 90%, 95%, and 99% levels.
Formula
CI = x̄ ± z · (σ / √n)
- x̄
- Sample mean
- z
- Z-score for the confidence level (1.645, 1.96, or 2.576)
- σ
- Standard deviation
- n
- Sample size
How it works
- Enter the sample mean, the standard deviation, and the sample size n.
- Pick a confidence level — 90%, 95%, or 99% — which selects the matching z-score (1.645, 1.96, or 2.576).
- The calculator computes the standard error as σ/√n, multiplies it by the z-score to get the margin of error, then subtracts and adds that margin to the mean for the interval bounds.
Worked example
A sample of 36 has a mean of 100 and a standard deviation of 15. Find the 95% confidence interval.
- Standard error: 15 ÷ √36 = 15 ÷ 6 = 2.5.
- Margin of error: 1.96 × 2.5 = 4.9.
- Bounds: 100 − 4.9 = 95.1 and 100 + 4.9 = 104.9.
The 95% confidence interval is 95.1000 to 104.9000, with a margin of error of 4.9.
Frequently asked questions
- What does a 95% confidence level actually mean?
- It means that if you repeated the sampling many times and built an interval each time, about 95% of those intervals would contain the true population mean. It is not the probability that this single interval contains it.
- Why does a higher confidence level give a wider interval?
- Greater confidence requires a larger z-score (2.576 for 99% versus 1.96 for 95%), which widens the margin of error. You trade precision for a higher chance of capturing the true mean.
- How does sample size affect the interval?
- The standard error is σ divided by the square root of n, so larger samples shrink the margin of error and tighten the interval. Quadrupling n halves the margin of error.
- Which confidence levels does this calculator support?
- It supports the three most common levels — 90%, 95%, and 99% — using z-scores of 1.645, 1.96, and 2.576 respectively, which assume a normal distribution.