Calculate Tukey HSD and Tukey-Kramer critical differences to compare group means and test pairwise significance with q, MSW, and n.

HSD Calculator

Equal n (Tukey HSD)
Unequal n (Tukeyโ€“Kramer)
Pairwise Analyzer
Tip: q is the Studentized range critical value for your alpha, number of groups, and error degrees of freedom.

Related Calculators

Hsd (Honestly Significant Difference) Formula

The following formulas are used in Tukeyโ€™s multiple-comparisons procedure to calculate the critical difference between group means:

Equal n: HSD = qโˆš((MSW) / (n)) \ Unequal n (Tukeyโ€“Kramer): CDแตขโฑผ = qโˆš((MSW) / (2)((1) / (nแตข) + (1) / (nโฑผ)))

Variables:

  • HSD is the Honestly Significant Difference (the critical difference threshold in Tukeyโ€™s HSD procedure for equal group sizes)
  • q is the critical value from the Studentized range distribution (depends on ฮฑ, the number of groups a, and the within/error degrees of freedom)
  • MSW is the Mean Square Within (also called MSE, the pooled within-group variance estimate from ANOVA)
  • n is the sample size per group (for the equal-n case)
  • a is the number of groups being compared (used to select q)

Once you obtain q from the Studentized range distribution, compute the critical difference (HSD for equal n, or CDij for unequal n). A pairwise difference between means is typically declared statistically significant when |meani โˆ’ meanj| exceeds the critical difference.

What is an Hsd (Honestly Significant Difference)?

The Honestly Significant Difference (HSD) is the name commonly given to the critical difference threshold used in Tukeyโ€™s HSD post-hoc multiple-comparisons method. It is often used after an ANOVA (Analysis of Variance) indicates that at least one group mean differs, and you want to identify which pairs of group means differ while controlling the family-wise Type I error rate across many pairwise comparisons. Under Tukeyโ€™s method, a pairwise comparison is typically called significant when the absolute difference between two group means is greater than the HSD (or the Tukeyโ€“Kramer critical difference in the unequal-n case).

How to Calculate Hsd (Honestly Significant Difference)?

The following steps outline how to calculate the Honestly Significant Difference (HSD).

  1. Run (or review) the ANOVA and record MSW (MSE) and the within/error degrees of freedom.
  2. Choose a family-wise significance level (ฮฑ), such as 0.05.
  3. Determine the number of groups being compared (a) and the sample size per group (n) (or the group sample sizes ni for unequal sample sizes).
  4. Look up the Studentized range critical value q for your ฮฑ, number of groups a, and within/error degrees of freedom.
  5. Compute the critical difference: for equal n, HSD = q ร— โˆš(MSW / n); for unequal n, use the Tukeyโ€“Kramer form CDij = q ร— โˆš(MSW/2 ร— (1/ni + 1/nj)).
  6. For each pair of groups, compute |meani โˆ’ meanj| and conclude the pair differs significantly if |meani โˆ’ meanj| > (HSD or CDij). You can verify your result with the calculator above.

Example Problem:

Use the following variables as an example problem to test your knowledge.

Mean difference (between two group means) = 5

Pooled within-group standard deviation = 2 (so MSW = MSE = 22 = 4)

Sample size per group (n) = 20

Desired level of significance (alpha) = 0.05 (suppose the Studentized range critical value from the table is q = 4.00 for your number of groups and within/error degrees of freedom)