Student’s T-Distribution

The t-distribution’s shape changes with degrees of freedom, usually denoted as n or ν.

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

  • 2F1 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.


The blue tail on this graph is the rejection region. The null hypothesis will be rejected if your calculated t-score falls into this area.  

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 ).

  1. Press “2nd” then “VARS” and “5” to select tcdf(.
  2. 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).
  3. 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 ).

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

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