Confidence Interval Calculator

What Is a Confidence Interval?

A confidence interval is a range of values calculated from sample data to estimate an unknown population parameter. Instead of providing only a single estimate, a confidence interval provides an estimated lower and upper bound that reflects uncertainty arising from sampling.

This confidence interval calculator can calculate intervals for a population mean or population proportion. For means, it supports both situations where the population standard deviation is known and where the sample standard deviation is used instead. Common confidence levels include 90%, 95% and 99%.

Why Use a Confidence Interval Calculator?

Confidence interval calculations require an appropriate critical value and standard error. This calculator performs those calculations automatically from the sample information you provide.

How to Use the Confidence Interval Calculator

Select the type of confidence interval required and enter the corresponding sample information.

1. Select the parameter — Choose a confidence interval for a mean or population proportion.
2. Choose the confidence level — Select 90%, 95%, 99% or enter a custom confidence level.
3. Enter the sample information — Provide the sample mean, sample size and appropriate standard deviation for a mean, or the number of successes and sample size for a proportion.
4. Calculate the interval — Click Calculate Confidence Interval to display the estimate, lower bound, upper bound, margin of error and critical value.

How to Calculate a Confidence Interval

A confidence interval is commonly calculated by taking a sample estimate and adding and subtracting a margin of error. The margin of error is determined by multiplying an appropriate critical value by the standard error of the estimate.

For a population mean with an unknown population standard deviation, the t-distribution is generally used. When the population standard deviation is known, a z critical value can be used under the appropriate assumptions. For a population proportion, the normal approximation can be used when its conditions are sufficiently satisfied.

95% Confidence Interval for a Mean

When the population standard deviation is unknown, a common confidence interval for a mean is calculated as the sample mean plus or minus the t critical value multiplied by the sample standard deviation divided by the square root of the sample size. The degrees of freedom for this calculation are n − 1.

95% Confidence Interval for a Proportion

For a population proportion, the sample proportion is calculated by dividing the number of successes by the sample size. The standard error is then based on the estimated proportion and sample size. The normal approximation should only be used when the underlying conditions for that approximation are appropriate.

Understanding Confidence Interval Results

A confidence level describes the long-run performance of the interval-producing procedure. For example, a 95% confidence interval means that if the same sampling procedure were repeated many times and an interval were calculated each time, approximately 95% of those intervals would contain the true population parameter, assuming the statistical model and procedure are appropriate.

A higher confidence level generally produces a wider interval because greater coverage requires a larger critical value. Increasing the sample size generally reduces the standard error and therefore produces a narrower interval, all else being equal.

The confidence interval should not be interpreted as saying that there is a 95% probability that a fixed population parameter lies between the two calculated endpoints. In the conventional frequentist interpretation, the parameter is treated as fixed and the interval-producing procedure has the stated long-run coverage.

Frequently Asked Questions