Student's tdistribution facts for kids
Probability density function 

Cumulative distribution function 

Parameters  ν > 0 degrees of freedom (real) 

Support  x ∈ (−∞; +∞) 
Probability density function (pdf)  
Cumulative distribution function (cdf)  where _{2}F_{1} is the hypergeometric function 
Mean  0 for ν > 1, otherwise undefined 
Median  0 
Mode  0 
Variance  for ν > 2, ∞ for 1 < ν ≤ 2, otherwise undefined 
Skewness  0 for ν > 3, otherwise undefined 
Excess kurtosis  for ν > 4, ∞ for 2 < ν ≤ 4, otherwise undefined 
Entropy 

Momentgenerating function (mgf)  undefined 
Characteristic function  for ν > 0

Student's tdistribution is a probability distribution which was developed by William Sealy Gosset in 1908. Student is the pseudonym he used when he published the paper which describes the distribution. Gosset worked at a brewery and was interested in the problems of small samples, for example the chemical properties of barley. In the problems he analyzed, the sample size might be as low as three. One version of the origin of the pseudonym is that Gosset's employer preferred staff to use pen names when publishing scientific papers instead of their real name, so he used the name "Student" to hide his identity. Another version is that the brewery did not want their competitors to know that they were using the ttest to test the quality of raw material.
Because of the small sample size, estimating the standard deviation is not possible. Also, in many cases Gosset encountered, the probability distribution of the samples was not known.
A normal distribution describes a full population, tdistributions describe samples drawn from a full population; accordingly, the tdistribution for each sample size is different, and the larger the sample, the more the distribution resembles a normal distribution.
The tdistribution plays a role in many widely used statistical analyses, including the Student's ttest for assessing the statistical significance of the difference between two sample means, the construction of confidence intervals for the difference between two population means, and in linear regression analysis. The Student's tdistribution also arises in the Bayesian analysis of data from a normal family.
If we take a sample of n observations from a normal distribution, then the tdistribution with ν = n−1 degrees of freedom can be defined as the distribution of the location of the true mean, relative to the sample mean and divided by the sample standard deviation, after multiplying by the normalizing term . In this way, the tdistribution can be used to estimate how likely it is that the true mean lies in any given range.
The tdistribution is symmetric and bellshaped, like the normal distribution, but has heavier tails, meaning that it is more prone to producing values that fall far from its mean. This makes it useful for understanding the statistical behavior of certain types of ratios of random quantities, in which variation in the denominator is amplified and may produce outlying values when the denominator of the ratio falls close to zero. The Student's tdistribution is a special case of the generalised hyperbolic distribution.