The following tables provide the critical value for *q*(*k, df, α*) for *α* = .10, .05 and .01 and values of *k* up to 20. Following these tables, we provide additional tables for *α* =.05, .025, .01, .005 and .001 and values of *k* up to 40, but with less granularity for higher values of *df*. See Unplanned Comparisons for ANOVA for more details.

**First group of tables**

**Alpha = 0.10**

**Alpha = 0.01**

**Alpha = 0.05**

**Alpha = 0.005**

Duncan’s Multiple Range Test DMRT still valid. If yes, its formilar

Tok,

You can use the DMRT, but some have criticized the test as being too liberal. Tukey’s HSD is commonly used instead and so I have focused on this test instead of DMRT.

Charles

Hi Charles,

Hi Charles,

I’m sorry to insist so much on Tukey’s HSD, but I really want to understand.

Does what you wrote mean that Tukey’s HSD is not set up to compare say the group with the second smallest mean with say the group with the third largest mean?

The point is that after having made an analysis of variance, and found that there is, at least one significant difference among the means, it is essential to find out which of the various differences among the means may be statistically significant. One thing you know for sure (at a given level of confidence) and that is that there may be significance between the smallest and the largest mean…

So what to do if you want to calculate the smallest difference that should exist between two significantly different means since Fishers LSD’s error rate becomes too high for more than three (I read) means?

The answer to this question needs to show a fine balance between reliability and precision, I feel.

Thank you once more for your precious time,

Erik

Hi Erik,

Tukey’s HSD will identify a significant difference between any two groups, not just the smallest mean with the largest. The way it reduces experimentwise error is based on the difference between the largest and smallest means, but this is perhaps a detail that you may not find that important.

Charles

Hi Charles,

1. I found in my wine tastings Tukey’s HSD test being so severe that it often makes conclusions disappointing at 95% of confidence level. As I wrote to you I believe a one sided test is applicable since all differences between means are positive, would you aggree with that? I really am interested to know weather your tables refere to a one tailed test or not. If they were two sided one could use de 90% tables and have a confidence level of 95%

2. Why do you not support Fisher’s LSD test? Is it because it uses the Residual Variance and a t-value while HSD uses the Within Wines Variance and a q-value? For my wine tastings it gave mostly plausible, say “believable” values. But is gut feeling of course.

Thank you sincerely for answer,

Erik

Hi Erik,

1. As I just wrote to you, I believe that Tukey’s HSD is a two tailed test.

2. I don’t support Fisher’s LSD since it doesn’t handle experimentwise error rate very well. Tukey’s HSD is much better at this.

Charles

hello. may i ask?. How can be the q critical value be computed without depending on the Duncan’s distribution table? Is there an existing formula that can be used? what is it? thanks for your response 🙂

Carlo,

You can compute the q critical value by using the QINV worksheet formula supplied by the Real Statistics Resource Pack, as described on the following webpage>

http://www.real-statistics.com/students-t-distribution/studentized-range-distribution/

The resource pack can be downloaded for free from the website.

Charles

I downloaded the package…but the QINV or QCRIT is not available…

Carlo,

What version of the software are you using? The Mac version?

Charles

Hi Charles,

I have been trying to come to a better understanding of Tukey’s HSD and wonder if you can aggree with what I wrote hereunder:

Given a panel of n judges scoring each of k wines means X1, …, Xi, … Xk were found.

After sorting those means from largest to smallest, the test hypotheses for the k(k-1)/2 differences are:

Ho: mu-i minus mu-j = 0

H1: mu-i minus mu-j larger than 0, and not “different from zero” like I found on some sites.

So reject Ho if Xi – Xj >= q-alpha*S-w/square root n, seems to me being a one-sided test. Is this correct ? Is your table 3 made up in that sense ?

Thank you very much, this is a brillant site,

Erik

Erik,

I am very please that you like the site and I appreciate your calling it “brilliant”.

I believe that Tukey’s HSD is a two-tailed test and not a one-tailed test.

The reason for the sorting that you refer to is that the test is essentially set up to compare the group with the smallest mean with the group with the largest mean.

Charles

good day. i have this little question. How can be the q critical value be computed without depending on the tukey’s distribution table? Is there an existing formula that can be used? what is it? thanks for your response 🙂

The Real Statistics Resource Pack provides the QINV function which calculates the critical values. See the webpage

Studentized Range Distribution

Charles

How can i get the critical value if my df=42 and n=3, with alpha=0.05? I can see 40 and 60 but I don’t know how I can get the exact value for that? What is my critical value then? Thanks!

Iris,

I can offer you two choices:

(1) interpolate the values in the table between the values for 40 and 60. Since 42 is a lot closer to 40 than 60, the appropriate value should be closer to the value in the table for 40.

(2) Install the Real Statistics Resource Pack. You can then use either the QCRIT or QINV function. QCRIT will do the interpolation for you and QINV will directly calculate an estimated value for df = 42.

Charles

Are these tables also applicable to games-howell post hoc tests?

Currently my df values are below 4 and a similar table I found only went to 5 as a minimum, which proved hard finding the correct Q value.

If so, when choosing the the Q value for example if the df was 4.65, would 4 or 5 on the table be most accurate?

I understand with games howell, you must first calculate the df, between each pair, followed by the qcrit value for each pair, before using in the final equation (see link): http://www.unt.edu/rss/class/Jon/ISSS_SC/Module009/isss_m91_onewayanova/img138.png to get the minimum significant difference.

Can someone please confirm if this equation is correct?

I followed the guidelines from this website: http://www.unt.edu/rss/class/Jon/ISSS_SC/Module009/isss_m91_onewayanova/node7.html

However in the final calculation of minimum significant difference for games howell, (down the bottom of the webpage) I noticed in place of their qcrit, they used the calculated df value instead, despite in the formula saying q crit. I wonder if this is a mistake, or if this is what one is meant to do.

Any guidance would be greatly appreciated and if anyone has a better calculations example for games-howell please share.

Warm regards, Jess

Jess,

Yes, these tables are applicable to Games-Howell. If your df is between 4 and 5 you can interpolate between the values in the table.

Sorry but I haven’t had time to look at the references you provided regarding your other questions.

Charles

Dear Charles,

Thank you kindly for that!

I have since downloaded the program onto my mac and have attempted to use QCRIT formula, however every time I type the function with the values, separated by commas, it shows up as #NAME and says ‘compile error in module: look up’.

Any guidance please?

Apologies the message is #VALUE!, not name that shows up.

Jess,

I have just checked on my Windows.based computer and the QCRIT function works fine. I can’t think of any reason why it wouldn’t work on the Mac, but

unfortunately, since I don’t own a Mac I can’t test the QCRIT on the Mac until the next time I borrow a Mac from a friend. In any case, you can try to use the

QINVfunction which estimates the value of the inverse Studentized Range value without doing a table lookup.Charles

Thank you for your service! I am interested to known about ANOVA post-hoc Q test. Is it possible to run this test for more than 20 treatments (groups)? Where do I get a Q table that provides values for a>20? Can I calculate table Q value? How? Is there an “infinite a” Q value? Will you please help me?

Gurumani,

The Real Statistics Resource Pack provides the function QINV(p, k, df, tails) which calculates the table value for any value of k (= # of groups), even for k > 20. The values of Q for infinite df are given in the table on the referenced webpage. You can download the Real Statistics Resource Pack for free from the website.

Charles

what mean of tukey HSD and tukey-b

I don’t understand your question. Please explain further.

Charles

thanks a lot prof. i can finish my task about this :).

Can someone please tell me what the Q statistic is for alpha=0.05, df=156 and k=4? I can’t find it in the reference table.

Sarah,

You would need to interpolate between the table values for df = 120 and df = 240. Q-crit for df = 120 is 3.685 and the Q-crit for df = 240 id 3.659. A linear interpolation would give the value 3.6772, which can be calculated using the Real Statistics formula =QCRIT(4,156,0.05,2).

The Real Statistics formula =QINV(0.05,4,156,2), which does not use the table, will usually give a more accurate answer, which in this case is 3.6726.

Charles

Second group of q tables 0.01 k=40

thanks

This comes from reference [Ha] on the Bibliography webpage of the Real Statistics website.

Charles

how found Second group of tables 0.05 k=40

thanks

This comes from reference [Ha] on the Bibliography webpage of the Real Statistics website.

Charles

how can i get df 45, i can only see 40 and then 60. thanks

You have to interpolate. E.g. if the table value for 40 is .2 and the table value for 60 is .4 then the value for 45 would be .25.

Alternatively you can use the QCRIT or QINV functions provided in the Real Statistics Resource Pack which carry out this work for you. See the webpage http://www.real-statistics.com/students-t-distribution/studentized-range-distribution/.

Charles

Can anyone please tell me what the Q statistic is for alpha 0.05, df 75 and k 23? I have not found it in any reference table.

These are outside the range of values found in the table. You can use the Real Statistics Resource Pack’s

QINVfunction to find the approximate value. The formula =QINV(0.05,23,75) gives the value 5.306308907. More details can be found at http://www.real-statistics.com/students-t-distribution/studentized-range-distribution/.Charles

hello

Can you please tell me what the Q statistic is for two way anova (25×2), alpha 0.05, and df 200 ? Thx 😀

I don’t know of a q statistic for two-way Anova. The q statistic is used in various Anova follow-up tests (e.g. Tukey HSD). These are described elsewhere on the website, but they apply to one-way as well as two-way Anova, although perhaps you are referring to some test that I am not familiar with.. For more information see http://www.real-statistics.com/one-way-analysis-of-variance-anova/unplanned-comparisons/ and http://www.real-statistics.com/two-way-anova/contrasts-two-factor-anova/.

Charles

This question is regarding to one of the components of Tukey’s HSD procedure which is studentised range statistic i.e.q(k, df, α). Here I have k=9, df=441 and α=0.05.So in order to find q, I need to refer the critical values table and look for k=9 and df=441. But all the tables I get have values of df till 120 or 240 and then infinity. Since i am looking for df=441 which row of df should I use: 240 or Infinity? Hope my conveyance of the question is clear to you. Please help. Thanks

Sarthak,

After 240 you should use infinity. In the next release of the software you will be able to enter df = 441, but there will only be a small difference between that value and the value for infinity.

Charles

Can you please tell me what the Q statistic is for alpha 0.05, df 21 and k 3? The numbers are so small I’m not sure I am reading them correctly.

Actually, is there a place where I can download these with greater resolution so that I can read all the numbers?

Sonja, if you download the Excel workbook with all the examples in this website, you will also find the table of Q statistics. Just go to http://www.real-statistics.com/free-download/real-statistics-examples-workbook/ for the free download. Charles

Sonja, it is 3.565. Charles