**Student’s T-Distribution** (sometimes called the* Student Distribution* or *T-Distribution*) is a family of probability distributions that are similar in shape to the normal distribution. Student’s t-distribution is used instead of the normal distribution when you have small samples or don’t know the population standard deviation. In real life, you usually *don’t* know the population standard deviation, so Student’s t is seen more frequently in real-world practical applications than the normal distribution.

## Student’s T-distribution properties

The probability density function (PDF) is:

And the cumulative distribution function (CDF) is:

Where

_{2}F_{1 }is the hypergeometric function,- Γ is the gamma function,
- Ν is degrees of freedom.

## What is Student’s T-distribution used for?

Student’s T Distribution is used in hypothesis testing (the “t-test”) to figure out if you should accept or reject the null hypothesis.

In most cases, you won’t have to resort to hand calculations: technology can help you.

**TI-83 Example problem**: Find the area under a T curve with degrees of freedom 10 for P( 1 ≤ X ≤ 2 ).

- Press “2nd” then “VARS” and “5” to select
**tcdf(**. - Enter the lower/upper bound given in the problem and the degrees of freedom. The lower bound is the lowest number and the upper bound is the highest number. So, enter “1”, “2” and “10”. Your screen should now read
**tcdf(1,2,10)**. - Press ENTER. to get
**.133752549**, or**13.38%**.

**TI-89 Example problem: **Find the area under a T distribution curve with degrees of freedom 10 for P( 1 ≤ X ≤ 2 ).

- Install the STAT/LIST editor.
- Press “APPS” then press “ENTER” twice to enter the STATS/LIST Editor.
- Press “F5” for
**F5Distr**. - Choose 6 for
**6:t Cdf**. - Enter “1” for the
**Lower Valu**e, “2” for**Upper Value**and “10” in the box for**Deg of Freedom, df**. These figures were given in the question above. - Press “ENTER” to get
**.133753**or 13.38%.

## How Student’s T-Distribution got its Name

The distribution was developed by William Gosset under the pseudonym “Student”. Gosset used a pseudonym because the company he worked for—Guinness—prohibited its employees from publishing under their real names due to some trade secret leaks in earlier scientific publications [3].

## Related distributions

- Hotelling’s distribution, which for
*q*= 1 is the positive half of Student’s T; Hotelling’s is sometimes called the*generalized Student*[1]. - The standard probability density function of Cauchy distribution is a t-distribution with 1 degree of freedom [2].

## References

[1] Haight, F. (1958). Index to the Distributions of Mathematical Statistics. National Bureau of Standards Report.

[2] Johnson, Kotz, and Balakrishnan, (1994), Continuous Univariate Distributions, Volumes I and II, 2nd. Ed., John Wiley and Sons.

[3] Josic, K. No. 3072. William Gosset. Online: https://www.uh.edu/engines/epi3072.htm