Sample Confidence Intervals

Sample Confidence Intervals is an interactive visualization tool for exploring Normal and Student's t distributions.

Adjust parameters, CI levels, and tail modes to see how confidence intervals change — and compare both distributions side-by-side. You'll see how the fatter tails of the Student's t distribution can make it easier to detect statistically significant differences between means or confidence intervals, for examples.

By examining the one-tailed tests on either the negative or positive side of the mean, it becomes easier to examine, for example, the likelihood of significant downside performance of a financial asset or the likelihood of significant upside performance of an investment opportunity. This is especially useful for investors and analysts who are trying to make informed decisions about where to allocate their capital.

Risk assessment is another real-world application of confidence intervals. For example, a company may want to know if a new product is more likely to be a success or a failure. Or, a medical professional may want to know if a new treatment is more likely to be effective or not. Different models of outcomes can lead to significantly differnt conclusions. Confidence intervals can help to quantify the uncertainty of these conclusions.

While the Gaussian Normal distribution is the most common distribution used in statistics and certainly the one most commonly taught in school, it is not the only one. The Student's t distribution is used when the sample size is small and the tails of the distribution are fatter than tails of the Normal distribution, which is the case for many real-world data.

The Cauchy distribution is a special case of the Student's t distribution with a Degree of Freedom of 1.0. It is also known as the Laplace distribution. Additionally, the Cauchy has no defined mean or variance. It is recommended that the mode of the distribution be used as an estimate of the mean. Interestingly, the larger the sample size, the more the Cauchy distribution's apparent variance increases towards infinity.