Student t Distribution: Definition & Example

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  • 0:05 T Distribution
  • 0:44 Degrees of Freedom
  • 1:37 Important Properties
  • 2:15 Example
  • 3:44 Lesson Summary
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Lesson Transcript
Instructor: Artem Cheprasov

Artem has a doctor of veterinary medicine degree.

In this lesson, you're going to learn about the t-distribution, t-curves, their important properties, and differences from the standard normal distribution as well as how to find the value of t.

T Distribution

The t distribution also known as the Student's t distribution is a kind of symmetric, bell-shaped distribution that has a lower height but a wider spread than the standard normal distribution. The units of a t distribution are denoted with a lower case 't'.

If you look at the image on your screen, you can see how each corresponding curve is bell-shaped and are symmetric about 0, but the t distribution has a bigger spread than the standard normal distribution curve, in essence the t distribution has a large standard deviation.

A standard deviation refers to the variability of individual observations around their mean.

Let's learn a bit more about the t distribution.

Degrees of Freedom

The only parameter of the t distribution is the number of degrees of freedom. The degrees of freedom (df) are simply n-1. Meaning df = n - 1, where n is our sample size.

The shape of each individual t distribution curve depends on the degrees of freedom, but all t-curves still resemble the standard normal curve nonetheless. Why does a t-curve have more spread than the standard normal curve?

It's because the standard deviation for a t-curve with v degrees of freedom, where v > 2, is the square root of v divided by v - 2. Because this value is always greater than 1, which is the standard deviation of the standard normal distribution curve, the spread is thus larger for a t-curve.

Important Properties

There are several important properties you should be aware of with respect to t-curves.

Property #1: The total area under a t distribution curve is 1.0: that is 100%.

Property #2: A t-curve is symmetric around 0.

Property #3: While a t-curve extends infinitely in either direction, it approaches, but never touches the horizontal axis.

Property #4: As the number of df increases, the t distribution curve will look more and more like the standard normal distribution curve.


Let's solidify our knowledge of the t distribution with an example. Using 15 degrees of freedom and a 0.05 area in the right tail of a t-curve, find the value of t.

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