Confidence Interval for a Mean Calculator

This confidence interval calculator works out a range that is likely to contain the true population mean, based on your sample data. You enter the sample mean, the standard deviation, the sample size and the confidence level you want, and the tool returns the lower and upper bounds of the interval along with the margin of error built into it. The method uses the standard normal multiplier, so a 95 percent level uses a z value of about 1.96, a 90 percent level uses 1.645 and a 99 percent level uses 2.576. The interval is the mean plus or minus that multiplier times the standard error, where the standard error is the standard deviation divided by the square root of the sample size. Researchers, students, quality analysts and survey teams use confidence intervals to express how precise an estimate is, rather than relying on a single point figure that hides the uncertainty. A wider interval means less precision, while a narrower interval means more. Three tips help you read the result well. First, larger samples shrink the interval because the square root of the sample size sits in the denominator, so collecting more data tightens your estimate. Second, a higher confidence level widens the interval, because you are asking for more certainty that the true mean is captured. Third, this normal based method suits reasonably large samples, so for very small samples a t distribution interval is more accurate. Always report the confidence level alongside the bounds so readers know exactly what the range means and can compare it fairly with other studies.

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Sample mean

Standard deviation

Sample size (n)

Confidence level

167.14 to 172.86

Confidence interval

Lower bound167.14

Upper bound172.86

Margin of error2.86

CI = mean +/- z * sd / sqrt(n). Estimate only, not financial or tax advice.

How it works

The standard error is the standard deviation divided by the square root of the sample size. The margin of error is the chosen z multiplier times that standard error. The interval is the mean minus the margin of error to the mean plus the margin of error.

Worked example

With a mean of 170, a standard deviation of 8, a sample of 30 and 95 percent confidence, the standard error is 8 divided by the square root of 30, about 1.461. The margin of error is 1.96 times 1.461, which is 2.86. The interval runs from 167.14 to 172.86.

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Frequently asked questions

What does a 95 percent confidence interval mean?

It means that if you repeated the sampling many times and built an interval each time, about 95 percent of those intervals would contain the true population mean. It does not mean there is a 95 percent chance the one interval you have contains the mean. It is a statement about the long run method.

Why does a bigger sample give a narrower interval?

The standard error is the standard deviation divided by the square root of the sample size. As the sample grows, that denominator grows, so the standard error and the margin of error both shrink. The result is a tighter, more precise interval.

When should I use a t interval instead?

Use a t distribution interval when the sample is small, roughly under 30, and the population standard deviation is unknown. The t multiplier is larger than the z multiplier for small samples, which widens the interval to account for the extra uncertainty. For large samples the two methods give very similar results.