# Studentized Range Distribution (1 of 3)

The studentized range distribution is used for testing
all differences among pairs of means. It is similar to the

t distribution, differing only in
that it takes into account the number of means under
consideration. The more means under consideration, the
larger the

critical value of
t

_{s} (studentized t). This makes sense since the
more means there are, the greater the likelihood that at
least some differences between pairs of means will be large
due to chance alone. First, consider the case in which there
are only two means. The formula for t used to compare two
sample means is:

where is M

_{d} = M

_{1} - M

_{2},
µ

_{1}- µ

_{2} is the value
specified by the null hypothesis (almost always zero),
and

where

MSE is the means square
error and n

_{h} is the

harmonic mean of the two sample
sizes. If the null hypothesis is that µ

_{1} -
µ

_{2} = 0, then the formula for t can be
written as shown on the next page: