Confidence Interval Calculator
Enter a sample mean, standard deviation, sample size, and z critical value to get the margin of error and the lower and upper bounds of a confidence interval for the mean.
Margin of error and bounds at once
Enter the four values and the confidence interval calculator returns the margin of error E = z × (σ / √n) plus the lower and upper bounds together.
Pick the right z
Use 1.645 for 90% confidence, 1.96 for 95%, and 2.576 for 99% — a higher confidence level uses a larger z and widens the interval.
What is a confidence interval?
A range that likely contains the true mean
A confidence interval calculator turns a single sample estimate into a range of plausible values for the true population mean. Instead of reporting only that the sample averaged 100, you report an interval — say 94.6 to 105.4 — together with a confidence level, most often 95%. The interval is built from four numbers: the sample mean, the standard deviation, the sample size, and a z critical value that encodes how confident you want to be. The half-width of the interval is the margin of error, the single number that tells you how precise your estimate is.
Enter a sample mean, a standard deviation, a sample size, and a z value to get the margin of error and the confidence interval instantly.
The margin of error is the z critical value multiplied by the standard error of the mean, and the interval is the sample mean plus and minus that margin.
E = z × (σ / √n)Take a sample of 30 observations with a mean of 100 and a standard deviation of 15, and choose 95% confidence, so z = 1.96. The standard error is 15 / √30 ≈ 2.7386. Multiply by z to get the margin of error: 1.96 × 2.7386 ≈ 5.3677. The confidence interval runs from 100 − 5.3677 ≈ 94.6323 up to 100 + 5.3677 ≈ 105.3677. Because the margin of error divides the standard deviation by √n, a bigger sample pulls the bounds inward and sharpens the estimate.
The formula is exact, but a few assumptions are worth keeping in mind.
Z method, random sampling, and interpretation
This calculator uses the z (normal) method, which assumes the population standard deviation is known or the sample is large (roughly n ≥ 30); small samples with an unknown standard deviation need a t interval instead. It also assumes a random, representative sample. Remember what the interval means: a 95% interval does not say there is a 95% chance the true mean is inside this one interval — it says the procedure captures the true mean 95% of the time across many samples.