Student-t distribution | Vose Software

Student-t distribution

Format: Student(n)


The Student distribution (often called the T distribution) plays an important part in statistics, as explained below. Examples of the Student-t distribution:


The most common use of the Student distribution is for the estimation of the mean of a (assumed Normally distributed) population where random samples from that population have been observed, and its standard deviation is unknown. The relationship:

Student(n) = Normal(0,SQRT(n/ChiSq(n)))

is at the centre of the method, discussed in more detail here. This is equivalent to a t-test in classical statistics.

Other sample statistics can be approximated to a Student distribution, and thus a Student distribution can be used to describe one's uncertainty about the parameter's true value: in regression analysis, for example.


First discovered by the English statistician William Sealy Gossett (1876-1937), whose employer (the brewery company, Guinness) forbade employees from publishing their work, so he wrote a paper introducing the distribution under the pseudonym 'Student'.

As n increases, the Student-t distribution tends to a Normal(0, 1) distribution. By the time n is 30+, the two distributions are virtually identical. This may explain why people think a statistical rule of thumb is that one needs at least 30 data points to have a robust analysis - which would mean that, were one performing the analysis to estimate the population mean, not knowing the population standard deviation has become unimportant.

ModelRisk functions added to Microsoft Excel for the Student-T distribution

VoseStudent generates random values from this distribution for Monte Carlo simulation, or calculates a percentile if used with a U parameter.

VoseStudentObject constructs a distribution object for this distribution.

VoseStudentProb returns the probability density or cumulative distribution function for this distribution.

VoseStudentProb10 returns the log10 of the probability density or cumulative distribution function. 

VoseStudentFit generates values from this distribution fitted to data, or calculates a percentile from the fitted distribution.

VoseStudentFitObject constructs a distribution object of this distribution fitted to data.

VoseStudentFitP returns the parameters of this distribution fitted to data.


Student distribution equations