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Student's t-distribution is a sampling distribution derived from the normal distribution. Suppose we take a random sample of size n from a normal distribution with mean μ and unknown variance. The sample mean and variance are X̅ and S². The distribution of (X̅ − μ)/(S/√n) is a t-distribution with ν = n − 1 degrees of freedom.

The function Γ(s) in the pdf denotes the gamma function. The function B(a, b) in the cdf denotes the beta function, while the function B(x; a, b) in the cdf denotes the incomplete beta function.

Parameter Range Description
ν ν > 0 Degrees of freedom

Probability Density Function

f ( x ; ν ) = Γ ( ν + 1 2 ) Γ ( ν 2 ) 1 ( ν π ) 1 ( 1 + x 2 ν ) ( ν + 1 ) / 2
F ( x ; ν ) = 1 − 1 2 B ( ν x 2 + ν ; ν 2 , 1 2 ) B ( ν 2 , 1 2 )

Support

− ∞ < x < ∞

Mean

Variance

Example ν
The length of human pregnancies is normally distributed with mean 266 days. A random sample of four new mothers found a mean pregnancy length of Y̅ with sample variance S². Then X = (Y̅ − 266)/(S/2) is a t(3) random variable. 3.000
Y₁, Y₂, ..., Y9 is a sample taken from a normal(0, 4) distribution, with sample mean Y̅ and sample variance S². Then X = Y̅/(S/3) is a t(8) random variable. 8.000
100 students take the math SAT, which has a normal distribution with a mean score of 500. Their scores have mean Y̅ and variance S². Then X = (Y̅ − 500)/(S/10) is a t(99) random variable. 99.00

X ∼ T(ν)

ν =
Chart of the t distribution Chart area for displaying the t pdf, cdf, and simulation

E(X) = , Var(X) =

Student's t-distribution can be used to estimate a confidence interval for the true population mean μ of a normally distributed population based on a sample mean X̅ and sample standard deviation S.

Since X̅ and S are independent, the t-distribution is constructed as the ratio of two independent random variables. The mean and variance of such ratio distributions is often undefined. In this case, the mean is only defined for ν > 1, and the variance is only defined for ν > 2.

Student's t-distribution can be used to estimate a confidence interval for the true population mean μ of a normally distributed population based on a sample mean X̅ and sample standard deviation S.

Since X̅ and S are independent, the t-distribution is constructed as the ratio of two independent random variables. The mean and variance of such ratio distributions is often undefined. In this case, the mean is only defined for ν > 1, and the variance is only defined for ν > 2.

The illustration above shows a sample of n points chosen independently and at random from a normal(μ, σ2) distribution. The random variable X = (X̅ − μ)/(S/√n) has a t(n − 1) distribution, where X̅ is the sample mean and S is the sample standard deviation.

The simulation above shows a sample of n points (marked red) chosen independently and at random from a normal(μ, σ2) distribution. The light blue line shows the value of X = (X̅ − μ)/(S/√n), where X̅ is the sample mean and S is the sample standard deviation. The random variable X has a t(n − 1) distribution. The histogram accumulates the results of each simulation.

Y = ν/(ν + X²) ∼ Beta(ν/2, 1/2) X² ∼ F(1, ν) T(1) ∼ Standard Cauchy limν→ ∞ X ∼ Standard Normal

Chart of the related distribution Chart area for displaying the related pdf, cdf, and simulation

E(Y) = , Var(Y) =

Note that:

  • X² is an F(1, ν) random variable, so W = 1/X² is an F(ν, 1) random variable.
  • Y = ν/(ν + X²) = (ν/X²)/(ν/X² + 1) = νW/(νW + 1). From the relationship between the F and beta distributions, Y is therefore a beta(ν/2, 1/2) random variable.

Note that, from the definition of a t random variable, Y = X² = (W2/1)(V/ν) where:

  • W = (X̅ − μ)/(σ/√n) ∼ standard normal, so W2 ∼ chi-squared(1).
  • V = νS²/σ² ∼ chi-squared(ν).
  • X̅ and S² are independent, so W and V are independent.

Hence, Y is the ratio of independent scaled chi-squared(1) and chi-squared(ν) random variables, and is therefore an F(1, ν) random variable.

In the graph of the pdf of the t-distribution, it can be seen that as ν approaches 1, the properties of the distribution approach that of a standard Cauchy distribution:

  • The tails of the t-distribution become heavier, as with the Cauchy distribution.
  • The variance becomes infinite when ν ≤ 2.
  • The expected value is not defined when ν = 1.

The t(ν) distribution is shown in red superimposed over the standard Cauchy. It can be seen that these are identical when ν = 1.

In the graph of the pdf of the t-distribution, it can be seen that as ν increases:

  • The tails of the pdf become lighter.
  • The variance approaches 1.
  • The shape of the pdf approaches that of a standard normal random variable.

The t(ν) distribution is shown in red. This can be seen to converge to a standard normal distribution as ν → ∞.