Confidence Interval Calculator

Confidence Interval Calculator

The Confidence Interval Calculator estimates a range of plausible values for a population mean using a sample mean, population standard deviation, sample size, and selected confidence level. Instead of providing only a single estimate, a confidence interval shows a range around the sample mean that reflects sampling uncertainty.

This calculator uses the z-based confidence interval for a population mean when the population standard deviation is known or when a z-based approximation is appropriate. You can calculate intervals at the 90%, 95%, or 99% confidence level.

What Is a Confidence Interval?

A confidence interval is an interval estimate constructed from sample data. The sample mean provides a point estimate of the population mean, while the confidence interval provides a lower and upper bound that reflect the uncertainty associated with estimating a population parameter from a sample.

A higher confidence level requires a wider interval because greater confidence in the estimation procedure requires allowing for a larger range of possible sampling variation.

Confidence Interval Formula

For a population mean with a known population standard deviation, the z-based confidence interval is calculated using the following formula:

Here, x̄ is the sample mean, σ is the population standard deviation, n is the sample size, and z* is the critical value associated with the selected confidence level.

Standard Error

The standard error measures the expected variability of the sample mean from sample to sample. When the population standard deviation is known, the standard error of the mean is calculated as:

As the sample size increases, the standard error decreases. This means that larger samples generally provide a more precise estimate of the population mean.

Margin of Error

The margin of error determines how far the confidence interval extends above and below the sample mean. It is calculated by multiplying the critical value by the standard error.

A larger confidence level increases the critical value and therefore produces a wider margin of error. Increasing the sample size generally has the opposite effect because it reduces the standard error.

Confidence Levels

Common confidence levels include 90%, 95%, and 99%. Each level uses a different critical value from the standard normal distribution.

Higher confidence requires a wider interval when the other inputs remain unchanged. For example, a 99% confidence interval will generally be wider than a 95% confidence interval calculated from the same sample information.

Example

Suppose a sample has a mean of 50, a population standard deviation of 10, and a sample size of 100. At the 95% confidence level, the critical value is 1.96.

The resulting interval is therefore approximately 48.04 to 51.96.

How to Interpret a Confidence Interval

A 95% confidence interval should not be interpreted as saying that there is a 95% probability that the fixed population mean lies inside the particular interval calculated from the sample. In the frequentist interpretation, the 95% confidence level describes the long-run performance of the method: if the same sampling procedure were repeated many times and an interval were constructed each time, approximately 95% of those intervals would contain the true population parameter under the required assumptions.

Effect of Sample Size

Sample size has an important effect on the width of a confidence interval. Holding the standard deviation and confidence level constant, increasing the sample size reduces the standard error and usually produces a narrower interval.

Effect of Confidence Level

Increasing the confidence level generally makes the interval wider. This happens because a higher confidence level requires a larger critical value, which increases the margin of error.

Z-Based vs. T-Based Confidence Intervals

The method used by this calculator is based on the standard normal distribution and assumes that the population standard deviation is known or that a z-based approximation is suitable.

When the population standard deviation is unknown and the sample standard deviation is used instead, a t-distribution is generally used for a confidence interval for the population mean. This distinction is particularly important for smaller samples because the t-distribution accounts for the additional uncertainty introduced by estimating the standard deviation from the sample.

How to Use the Calculator

Enter the sample mean, population standard deviation, and sample size. Then select the desired confidence level. The calculator will determine the standard error, critical value, margin of error, lower bound, and upper bound of the confidence interval.

The result also provides the calculation steps and an interpretation of the interval so that you can understand how the final range was obtained.