3 Dec 2022


T Distribution: Everything You Need to Know

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T distribution is a probability distribution which is theoretical, apart from being symmetrical it is ball-shaped and has similarity with a standard normal curve. However, it contains an additional parameter named as a degree of freedom which makes it different from the standard normal curve and the setting makes it change its shape (Winter, 2013) . T distribution is used for estimation of probabilities of small samples or data that is incomplete, and it was developed to solve the fact that standard deviation is unknown to most of the situations. When the standard deviation is known, then the standard normal curve can be used. 

The t test was developed by an Englishman named William Sealy Gosset in 1908 paper in Biometrika who published it under pseudonym student. He developed both the t distribution and the t test. T distribution belongs to a family of curves in which what specifies a specify curve is the number of degrees of freedom available. Guinness Brewery is where Gosset worked in Dublin, Ireland. His interest was trying to solve the problems involving small samples, and an example was the barley chemical properties in which the sample size could be as low as 3. One version of pseudonym origin was that the employer of Gosset did not want them to use their real names but instead use pen names (Kroese, Taimre & Botev, 2013) . Again, the Guinness since they were using the t test when determining raw material quality, they did not want their competitors to discover it. 

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The reason why it's called students t distribution is because it was published in 1908 under pseudonym's student by Gosset. Therefore, it was Gosset's distribution. T distribution plays a pivotal role in statistical analysis which includes student's t-test in determining the statistical significance of two samples which are different (Winter, 2013) . It also comes up in the Bayesian analysis of an ordinary family data. T distribution is bell-shaped and systematic just like the standard curve, but the differ in the fact that it is much prone to producing numbers that are much far from its mean because of its heavier tails. It is useful for knowing the statistical behaviors of various types of ratios of random quantities. Student's t distribution in generalized hyperbolic distribution is a particular case. 

The degree of freedom is what determines a particular form of t distribution because there are different types of t distribution. In a set of data, the value of independent observations is what is referred to the degree of freedom (Kroese, Taimre & Botev, 2013) . When estimating proportion or mean score from one sample, the amount of independent observation is the same as sample size less one. So, t distribution that has 10 degrees of freedom can be used with a sample of size 11. Therefore, the level of freedom can be calculated in a different way for various application. 


de Winter, J. C. (2013). Using the Student’s t-test with extremely small sample sizes.    Practical Assessment, Research & Evaluation ,    18 (10), 1-12. 

Kroese, D. P., Taimre, T., & Botev, Z. I. (2013).    Handbook of monte carlo methods   (Vol. 706). John Wiley & Sons. 

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