Confidence Interval

What is a Confidence Interval?

A confidence interval (CI) is a range of plausible values for a population parameter, such as the population mean. It is calculated from sample data and reflects the uncertainty in the estimate.

If several samples are taken from a population, it is very likely that each sample will have a different mean value. However, we want to know the mean of the population, not only the mean of the sample. A confidence interval gives a range of plausible values for that population mean.

Caution! A 95% confidence level does not mean that there is a 95% probability that the parameter lies in this particular interval. In frequentist statistics, the parameter is fixed, while the interval varies from sample to sample. The precise definition is:

In statistics, parameters of the population are often estimated based on a sample, such as the mean or the variance. However, these are only estimates and will usually differ from the true population values. A confidence interval expresses the uncertainty around an estimate.

Calculating a Confidence Interval

To calculate a confidence interval, the sampling distribution of the estimator is required. When a normal-distribution critical value is appropriate, the confidence interval for the mean is given by:

Where x is the sample mean, n is the sample size and s is the sample standard deviation. Plus and minus indicate the lower and upper limits of the confidence interval respectively.

When the population standard deviation is unknown and is estimated by s, the t-distribution is generally used instead of the normal distribution. This distinction is especially important for small samples. The z value is then replaced by a t value with n − 1 degrees of freedom:

95% Confidence Interval

Before calculating a confidence interval, you choose a confidence level. Common choices are 95% and 99%. In repeated sampling, the confidence level is the proportion of intervals produced by the method that would contain the true population parameter.

For the 95% confidence interval and the 99% confidence interval, the z values are as follows:

A 95% confidence interval comes from a method that captures the true parameter in about 95% of repeated samples. For one calculated interval, the true parameter either lies within it or it does not.

Confidence Interval for the t-test

A t-test compares differences in means, e.g. you can use a t-test to test whether there is a difference in salary between men and women.

You actually want to make a statement whether there is a difference in salary in the population. Since you cannot survey the entire population, you use a sample. In that sample, you may observe a difference in salary.

In order to be able to estimate approximately in which range the mean difference in the population lies, you calculate the confidence interval.

In the t-test calculator on numiqo.com you can calculate the confidence interval of the mean difference.

When combining independent study estimates, the meta-analysis calculator displays 95% confidence intervals for the individual studies and the pooled effect in a forest plot. Enter study summary data or effect estimates with their standard errors.