The Studentized Range (q) Distribution
The Tukey method uses the studentized range distribution. Suppose we have r independent observations y1, ..., yr from a normal distribution with mean μ and variance σ2. Let w be the range for this set; i.e., the maximum minus the minimum. Now suppose that we have an estimate s2 of the variance σ2 which is based on ν degrees of freedom and is independent of the yi (i = 1,...,r). The studentized range is defined as
Tukey's test is based on the comparison of two samples from the same population. From the first sample, the range (calculated by subtracting the smallest observation from the largest, or, where Yi represents all of the observations) is calculated, and from the second sample, the standard deviation is calculated. The studentized range ratio is then calculated:
where q = studentized range, and s = standard deviation of the second sample.
This value of q is the basis of the critical value of q, based on three factors:
- α (the Type I error rate, or the probability of rejecting a true null hypothesis)
- n (the number of degrees of freedom in the first sample (the one from which range was calculated))
- v (the number of degrees of freedom in the second sample (the one from which s was calculated))
The distribution of q has been tabulated and appears in many textbooks on statistics. In addition, R offers a cumulative distribution function (ptukey) and a quantile function (qtukey) for q.
Read more about this topic: Tukey's Range Test
Famous quotes containing the words range and/or distribution:
“In the range of things toddlers have to learn and endlessly review—why you can’t put bottles with certain labels in your mouth, why you have to sit on the potty, why you can’t take whatever you want in the store, why you don’t hit your friends—by the time we got to why you can’t drop your peas, well, I was dropping a few myself.”
—Mary Kay Blakely (20th century)
“My topic for Army reunions ... this summer: How to prepare for war in time of peace. Not by fortifications, by navies, or by standing armies. But by policies which will add to the happiness and the comfort of all our people and which will tend to the distribution of intelligence [and] wealth equally among all. Our strength is a contented and intelligent community.”
—Rutherford Birchard Hayes (1822–1893)