Student's t Distribution

Student's t distribution, also known as the t-distribution, is commonly used to test if the difference between the means of two samples is statistically significant. It is a variation of the normal distribution that takes into account that the mean of a sample is only an estimate for the mean of the population.

The Student t distribution has one shape parameter: the degrees of freedom $\nu$. The probability density function (PDF) is:

The cumulative distribution function (CDF) is:

where ${}_2{F}_1$ is the hypergeometric function and $\nu >0$ is required.

The Student t distribution has several important applications in statistical analysis and research. In hypothesis testing, it is particularly valuable when working with small sample sizes where the population standard deviation is unknown. The distribution is frequently used in medical research, scientific experiments, and quality control to determine if the means of two datasets are significantly different from each other.

Common applications include:

• Constructing confidence intervals for population means when sample sizes are small.

• Performing t-tests to compare means between two independent or paired samples.

• Analyzing the significance of regression coefficients in statistical modeling.

• Estimating measurement uncertainty in calibration and laboratory measurements.

Notable properties include:

• The distribution is symmetric about zero.

• The tails are heavier than the normal distribution.

• The characteristic function has no closed form.

• As $\nu \to \mathrm{\infty }$, the t-distribution approaches the standard normal distribution.

• When $\nu =1$, the t-distribution is equivalent to the Cauchy distribution.

• If Z follows a standard normal distribution and V follows a chi-square distribution with $\nu$ degrees of freedom, then $\frac{Z}{\sqrt{V/\nu }}$ follows a t-distribution with $\nu$ degrees of freedom.

The Student t distribution is implemented by the StudentTDistribution class. It has one constructor with the degrees of freedom as its only argument.

The following constructs the Student t distribution with 8 degrees of freedom:

var studentT = new StudentTDistribution(8);

Dim studentT = New StudentTDistribution(8)

No code example is currently available or this language may not be supported.

let studentT = StudentTDistribution(8.0)

The StudentTDistribution class has one specific property, DegreesOfFreedom, which returns the degrees of freedom of the distribution.

StudentTDistribution has one static (Shared in Visual Basic) method, Sample, which generates a random sample using a user-supplied uniform random number generator. The second and third parameters are the location and scale parameters of the distribution.

var random = new Pcg32();
double sample = StudentTDistribution.Sample(random, 8);

Dim random = New Pcg32()
Dim sample = StudentTDistribution.Sample(random, 8)

No code example is currently available or this language may not be supported.

let random = Pcg32()
let sample = StudentTDistribution.Sample(random, 8.0)

The above example uses the MersenneTwister class to generate uniform random numbers.

For more information on the student t distribution, refer to the following sources:

• Johnson, N. L., Kotz, S., & Balakrishnan, N. (1994). "Continuous Univariate Distributions, Volume 2", Chapter 28. Wiley Series in Probability and Statistics.

• Forbes, C., Evans, M., Hastings, N., & Peacock, B. (2011). "Statistical Distributions", Chapter 43. Wiley Series in Probability and Statistics.

• Student's t-Distribution. (n.d.). In Wikipedia. Retrieved from https://en.wikipedia.org/wiki/Student%27s_t-distribution

• Wackerly, D., Mendenhall, W., & Scheaffer, R. (2008). Mathematical Statistics with Applications. Cengage Learning.

Other Resources