# Studentized range table - reported paper value does not match table

I'm reading a paper on multiple comparisons (citation below). The paper reports (page 324, step 5) that we need to:

"Find the percentage point of the studentized range (SR_n) corresponding to a and n (the number of samples) and for infinite degrees of freedom."

a = alpha = 0.05

n = number of samples = 5 (in this example in the paper)

The paper then reports that this value is 2.73.

When I look up the table (here, alpha=0.05, df=inf, k=5), I find 3.858

Naturally, these values are quite different and I don't think it's due to my using a table developed in the 21st century vs. the author's 1954 source.

Question: Why is my value (3.858) different from the reported value in the paper (2.73)? How did the author find this value?

Ryan, Thomas H. "Significance tests for multiple comparison of proportions, variances, and other statistics." Psychological bulletin 57.4 (1960): 318-328.

• Not sure if this is any help but the qtukey() function in R seems to agree with the table you link to. – mdewey Jan 9 at 14:55
• Thanks @mdewey, that's a great place to start. – StatsSorceress Jan 9 at 15:11

## 1 Answer

It appears that you and they disagree by a factor of $$\sqrt{2}$$. If you look at the table rows with $$k=2$$, they have the familiar 1.96.

R gives

> round(qtukey(.95,5:2,Inf),2)
 3.86 3.63 3.31 2.77
> round(qtukey(.95,5:2,Inf)/sqrt(2),2)
 2.73 2.57 2.34 1.96


where the 3.858 matches what you found in the reference table, and the second row matches the studentized ranges in the table in the paper

I think footnote 5 may be explaining the factor of $$\sqrt{2}$$?