Student’s T Distribution


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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.

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

PDF and CDF

The probability density function is:

And the cumulative distribution function is:

Where

  • 2F1 is the hypergeometric function,
  • Γ is the gamma function,
  • Ν is degrees of freedom.

A related distribution is 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].

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.  

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—Guiness—prohibited its employees from publishing under their real names due to some trade secret leaks in earlier scientific publications [3].

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