Support q ∈ [0; +∞) | ||
PDF f ( q ; k , ν ) = 2 π k ( k − 1 ) ν ν / 2 Γ ( ν 2 ) 2 ν / 2 − 1 ∫ 0 ∞ x ν φ ( ν x ) × [ ∫ − ∞ ∞ φ ( u ) φ ( u − q x ) ( Φ ( u ) − Φ ( u − q x ) ) k − 2 d u ] d x {displaystyle {egin{matrix}f(q;k,u )={ rac {{sqrt {2pi }}k(k-1)u ^{u /2}}{Gamma left({ rac {u }{2}}ight)2^{u /2-1}}}int _{0}^{infty }x^{u }varphi ({sqrt {u }}x) imes [0.5em]left[int _{-infty }^{infty }varphi (u)varphi (u-qx)(Phi (u)-Phi (u-qx))^{k-2},{ ext{d}}uight],{ ext{d}}xend{matrix}}} CDF F ( q ; k , ν ) = k ν ν / 2 Γ ( ν 2 ) 2 ν 2 − 1 ∫ 0 ∞ x ν − 1 e − ν x 2 / 2 × [ ∫ − ∞ ∞ φ ( u ) ( Φ ( u ) − Φ ( u − q x ) ) k − 1 d u ] d x {displaystyle {egin{matrix}F(q;k,u )={ rac {ku ^{u /2}}{Gamma left({ rac {u }{2}}ight)2^{{ rac {u }{2}}-1}}}int _{0}^{infty }x^{u -1}e^{-u x^{2}/2} imes [0.5em]left[int _{-infty }^{infty }varphi (u)(Phi (u)-Phi (u-qx))^{k-1},{ ext{d}}uight],{ ext{d}}xend{matrix}}} | ||
In probability and statistics, Studentized range distribution is a continuous probability distribution that arises when estimating the range of a normally distributed population in situations where the sample size is small and population standard deviation is unknown.
Contents
- Probability density function
- Cumulative distribution function
- Special cases
- How the studentized range distribution arises
- Uses
- References
Suppose that we take a sample of size n from each of k populations with the same normal distribution N(μ, σ) and suppose that
Probability density function
Differentiating the cumulative distribution function with respect to q gives the probability density function.
Cumulative distribution function
The cumulative distribution function is given by
Special cases
When the degrees of freedom approach infinity, the standard normal distribution can be used for the general equation above. If k is 2 or 3, the studentized range probability distribution function can be directly evaluated, where
When the degrees of freedom approaches infinity the studentized range cumulative distribution can be calculated at all k using the standard normal distribution.
How the studentized range distribution arises
For any probability distribution f, the range probability distribution is:
What this means, is that we are adding up the likelihood that, given k draws from a distribution, two of them differ by r, and the remaining k-2 draws all fall between the two extreme values. If we use u substitution where
In order to create the studentized range distribution, we first use the standard normal distribution for f and F, and change the variable r to q.
The chi distribution is:
If we apply a change of variables we see it can also be expressed as:
Multiplying the two and integrating over S gives:
Uses
Critical values of the studentized range distribution are used in Tukey's range test.