Overview
On this page, we discuss how to find a confidence interval from a sample when the population standard deviation is not known, by using the Student’s tdistribution.
Basic learning objectives
These are the tasks you should be able to perform with reasonable fluency when you arrive at your next class meeting. Important new vocabulary words are indicated in italics.

Understand the definition of tscores and Student’s tdistribution, and how we can use them to derive the confidence interval for a population mean with the sample standard deviation in place of the population standard deviation.

Recall that in the formula for the sample variance \(s\), i.e. the variance of the actually observed data in a sample of size \(n\), we divide by \(n1\) instead of \(n\), giving: \(s^2=\frac{\sum (x_i\overline{x})^2}{n1}\). (This is sometimes called Bessel’s correction, and ensures that \(s\) is an unbiased estimator for the unknown population variance \(\sigma\). You may want to rewatch the last video on the page about estimators for an intuitive explanation.)
Advanced learning objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:
 Be able to construct confidence intervals for a population mean at any confidence level when the population standard deviation is unknown.
To prepare for class

Watch the following video (by jbstatistics) which introduces the Student’s tdistribution:

Watch the following video (by jbstatistics) which shows how to calculate confidence intervals for the population mean when the population standard deviation is not known:
After class

Play around with the following interactive web apps to better understand how the degrees of freedom affect the shape of a tdistribution:

Comparing the tdistribution with the normal distribution (by Kristoffer Magnusson)

Normal distribution app (by Matt Bognar) (note that here, the degrees of freedom are denoted by \(\nu=n1\))


Watch the following video (by jbstatistics) which discusses guidelines on when using a tdistribution is appropriate or not: