Confidence Interval Calculator

About the Confidence Interval Calculator

Estimates a confidence interval for the population mean from the sample mean, standard deviation, sample size and a confidence level.

How to use this calculator

1. Enter the sample mean, \(\bar{x}\).
2. Enter the population standard deviation, \(\sigma\), if it is known.
3. Enter the sample size, \(n\).
4. Choose the confidence level so the calculator can use the correct \(z\)-value and show the interval.

The formula explained

The formula \(CI = \bar{x} \pm z\frac{\sigma}{\sqrt{n}}\) computes a confidence interval for a population mean when the population standard deviation is known. The value \(z\frac{\sigma}{\sqrt{n}}\) is the margin of error, the amount added to and subtracted from the sample mean.

• \(\bar{x}\) = sample mean
• \(z\) = z-score for the chosen confidence level
• \(\sigma\) = population standard deviation
• \(n\) = sample size

Step by step method

1. Find the sample mean \(\bar{x}\), the population standard deviation \(\sigma\), and the sample size \(n\).
2. Compute the standard error, \(\frac{\sigma}{\sqrt{n}}\).
3. Multiply by the critical value \(z\) to get the margin of error.
4. Add and subtract the margin of error from \(\bar{x}\) to get the lower and upper bounds.

Worked example

Problem. A company measures the average fill amount of 64 bottles. The sample mean is 500 mL, the population standard deviation is 16 mL, and you want a 95% confidence interval.

1. For 95% confidence, use \(z = 1.96\).
2. Compute the standard error: \(\frac{16}{\sqrt{64}} = \frac{16}{8} = 2\).
3. Find the margin of error: \(1.96 \times 2 = 3.92\). The interval is \(500 \pm 3.92\), so the bounds are \(496.08\) and \(503.92\).

Tips and common mistakes

• This formula is used when the population standard deviation is known, not when it is unknown.
• A larger sample size makes the interval narrower because \(\sqrt{n}\) is in the denominator.

Frequently asked questions