Studentized Range - Description

Description

The value of the studentized range is most often represented by the variable q.

The studentized range computed from a list x1, ..., xn of numbers is given by the formulas

q _{n,\nu}= \frac{\max\{\,x_1,\ \dots \ x_n\,\} - \min\{\,x_1,\ \dots\ x_n\}}{s} = \max_{i,j=1, \dots, n}\left\{\frac{x_i - x_j}{s}\right\}

where

is the sample variance, the square of the sample standard deviation s, and

is the sample mean.

The critical value of q based on three factors:

  1. α (the probability of rejecting a true null hypothesis)
  2. n (the number of observations or groups)
  3. v (degrees of freedom in the second sample)

If X1, ..., Xn are independent identically distributed random variables that are normally distributed, the probability distribution of their studentized range is what is usually called the studentized range distribution. This probability distribution is the same regardless of the expected value and standard deviation of the normal distribution from which the sample is drawn: tables are available. This probability distribution has applications to hypothesis testing and multiple comparisons. For example, Tukey's range test and Duncan's new multiple range test (MRT), which uses q statistics, can be used as post-hoc analysis to test between which two groups there is a significant difference after rejecting null hypothesis by analysis of variance.

When only two groups need to be compared, the studentized range distribution is similar to the Student's 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 is. 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.

Read more about this topic:  Studentized Range

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