The first line is the control. Critical value 2.483036 against an unadjusted 2.048407 and a Bonferroni 2.546465. The 10mg and 20mg doses clear it; 40mg (t = 1.4410) does not.
3 treatments vs control, 28 error df
Critical |t| = 2.483036
The unadjusted two-sided t would be 2.048407 and Bonferroni across 3 comparisons would be 2.546465. Dunnett sits between them because the 3 comparisons all share the same control, which makes them correlated — Bonferroni assumes they are not, and pays for that assumption with 2.49% more required separation. 2 of 3 differ from control.
Dunnett critical |t|
2.483036
by integration, not a table
Unadjusted t
2.048407
wrong for more than one comparison
Bonferroni t
2.546465
2.55% stricter than needed
Control mean
53.5000
n = 8, MSE = 7.7054
Every treatment against the control, with its simultaneous confidence interval
Treatment
Mean
n
vs control
t
95% simultaneous CI
Verdict
Treatment 1
58.6250
8
+5.1250
3.6926
[1.679, 8.571]
differs from control
Treatment 2
62.6250
8
+9.1250
6.5746
[5.679, 12.571]
differs from control
Treatment 3
55.5000
8
+2.0000
1.4410
[-1.446, 5.446]
not distinguishable
The intervals are SIMULTANEOUS: all 3 of them hold together with 95% confidence, which is a stronger claim than 3 separate 95% intervals. That is what the wider critical value buys, and it is why reading these as ordinary confidence intervals understates them.
k comparisons, not k(k+1)/2 Uses the pooled error from every group Choose the control before you see the data
What this tool shows
The critical value here is computed, not looked up — a two-dimensional integration of the equicorrelated multivariate t — and the check that it is right needs no published table at all. With one treatment there is nothing to adjust for, so the answer must be the ordinary two-sided t critical value. It is, to 3.55×10⁻¹⁵ on the shipped example and to better than 2.5×10⁻⁹ across degrees of freedom from 5 to 240.
Dunnett's two-sided critical value, computed by numerical integration
The unadjusted t and the Bonferroni value beside it, so the gain is visible
Simultaneous 95% confidence intervals for every treatment against control
The pooled error mean square, from all groups including the control
k comparisons rather than the k(k+1)/2 an all-pairs procedure would run
A one-treatment case that reduces exactly to a two-sample t-test
Many vs one control Critical value computed Simultaneous intervals Beats Bonferroni
Updated 12 September 2026 · Works in any browser, no installation
Dunnett’s test compares several treatments against a single control while holding the family-wise error rate at the level you chose. It runs k comparisons rather than the k(k+1)/2 that Tukey HSD runs, and it accounts for the fact that those k comparisons all share the same control — which makes them correlated, and makes a Bonferroni correction stricter than necessary.
At a glance
Formula shown
Each comparison is t = (treatment mean − control mean) / √(MS_error·(1/n_t + 1/n_c)), and the critical value is the point where P(max|tᵢ| ≤ c) = 1 − α for k equicorrelated t statistics with ρ = ½ and the pooled error df. That probability is a double integral — over the chi scale factor and over the shared control deviation — and this tool evaluates it by Simpson quadrature and inverts it by the Illinois method rather than reading a table.
Scenario support
Dose-finding studies where every dose is compared to placebo, quality control against a reference batch, agricultural trials against an untreated plot, A/B/n tests where several variants are compared to the incumbent, and any design where one condition is privileged and the others are not being compared to each other.
Educational estimate
Planning support from the values you enter — not professional advice.
Why not Tukey, and why not Bonferroni
Both are correct procedures. Both answer a bigger question than the one a control-comparison study is asking, and both cost power for the privilege.
Tukey HSD controls the error rate across ALL pairwise comparisons. With four treatments and a control that is ten comparisons; Dunnett runs four. Paying for six comparisons you did not make is a real loss of power.
Bonferroni runs the right four comparisons but assumes they are independent. They are not: every one of them uses the same control mean, so they are positively correlated, and a bound built for independence is conservative.
On the tool’s nine-treatment preset, Dunnett needs |t| > 2.756883 and Bonferroni needs 2.861513 — 3.66% more separation, from an assumption that is simply false for this design.
The gap widens with k. At 20 error degrees of freedom it is 1.83% for two treatments, 2.76% for three, 3.89% for five and 5.17% for nine, because the more comparisons share the control, the more Bonferroni’s independence assumption gives away.
And the unadjusted t is simply wrong here. Running k separate t-tests at 0.05 gives a family-wise rate far above 0.05, which is the entire reason this family of procedures exists.
ρ = ½ is not an approximation
The correlation between two treatment-vs-control comparisons is exactly one half at equal group sizes, and that exactness is what makes the critical value computable rather than simulable.
Both comparisons contain the same control mean with the same coefficient. Cov(T₁ − C, T₂ − C) = Var(C), and each comparison has variance 2·Var(C) when n is equal, so the correlation is exactly 1/2.
That makes the joint distribution equicorrelated, which collapses a k-dimensional integral to a two-dimensional one — over the shared control deviation and over the chi-distributed scale factor of the pooled standard error.
With unequal group sizes the correlations differ between pairs and the exact critical value would require a genuine multivariate integration. This tool uses the equal-n value, which is the standard approach and is slightly conservative when the control is larger than the treatment groups.
Which is an argument for allocating more subjects to the control. A control group of about √k times the treatment size is the classic optimum, because the control mean enters every comparison and is where precision is worth most.
How a computed critical value is checked without a table
Anything numerically integrated needs verifying against something. The usual answer is a published table; there is a better one available here.
At k = 1 there is nothing to adjust for, so the Dunnett critical value MUST equal the ordinary two-sided t critical value. That is a hard identity, not a rough correspondence.
The integration reproduces it to 3.55×10⁻¹⁵ on the tool’s one-treatment preset, and to better than 2.5×10⁻⁹ at every degree of freedom from 5 to 240 — 3.9×10⁻¹⁴ or finer from 8 df upward. The worst case anywhere tested is 1.4×10⁻⁶ at 3 df.
A second check needs no external reference either: the value must lie strictly between the unadjusted t and the Bonferroni t at every k and df, since Dunnett corrects for multiplicity but does so knowing the comparisons are correlated. It does, everywhere tested.
And the published table is reproduced as a bonus. At 20 error degrees of freedom the tool gives 2.086, 2.379, 2.540, 2.651, 2.735 and 2.946 for k = 1 to 5 and 9 — the standard two-sided Dunnett table to every printed digit.
The intervals are simultaneous, which is a stronger claim
The confidence intervals in the table are not k separate 95% intervals. They are one 95% statement about all k comparisons at once, and the difference matters when a reader quotes one of them.
Individually, k separate 95% intervals have a joint coverage well below 95%. With four comparisons at 95% each, the chance that all four cover their true values is far lower, which is the same multiplicity problem restated in interval form.
A simultaneous set covers ALL of them with 95% probability, which is why each one is wider than it would be alone — here by the ratio of the Dunnett critical value to the unadjusted t.
Quoting one interval on its own understates it. Reporting “the 20mg dose raised the outcome by 9.13 (95% CI 5.68 to 12.57)” is conservative if that interval came from a simultaneous set, and it is worth saying which it is.
The intervals and the tests always agree. A comparison is significant exactly when its interval excludes zero, because both use the same critical value — which is not automatic when a p-value and an interval come from different approximations.
It inherits the ANOVA assumptions, and one more
The pooled error mean square comes from every group including the control, which is what makes the procedure powerful and what makes one assumption load-bearing.
Equal variances across all groups. The pooled MS_error is only a valid error term if every group has the same spread; a Levene test is the standard check. Where variances differ, the Games-Howell test gives each pair its own error term instead.
Independent observations within and between groups, which is a design question rather than a diagnostic one and cannot be recovered from the data.
Approximate normality within groups, which the test tolerates reasonably well at moderate group sizes.
And the one that is specific to Dunnett: the control must be chosen in advance. Picking the lowest-scoring group as the “control” after seeing the data turns a k-comparison procedure into an all-pairs one with a hidden selection step, and the error rate the critical value promises no longer applies.
Reporting a Dunnett test
Four things, and the second is what distinguishes this from a string of t-tests.
Name the control and say it was pre-specified. The whole error-rate guarantee rests on that, and it is one clause.
Give the critical value or the adjusted p-values. Raw t statistics with no adjustment stated read as k independent tests, which is exactly the interpretation the procedure exists to prevent.
Report the simultaneous intervals, and say they are simultaneous. They carry the size of each effect, and their joint interpretation is what makes them wider than ordinary intervals.
Report every comparison, including the ones that failed. A dose-response study that prints only the doses that worked has made the multiplicity correction meaningless.
Sources and methodology
References for Dunnett's procedure and its critical values.
Method. The critical value is obtained by evaluating P(max |Tᵢ| ≤ c) for k equicorrelated t statistics at ρ = ½ — a double integral over the shared control deviation and the chi-distributed scale factor — and inverting it by the Illinois variant of regula falsi rather than by bisection, which reaches the quadrature’s own accuracy floor in about a dozen probes instead of forty. The quadrature grid was chosen by measurement against a 400×400 reference: the panel count is derived from the integration range so the spacing stays constant whatever the degrees of freedom are, since the scale factor concentrates around 1 with a standard deviation of about 1/√(2ν). Correctness is asserted against an identity rather than a table — at k = 1 the value must equal the two-sided t critical value, and it does to 3.55×10⁻¹⁵ — and against the bracketing property that Dunnett lies strictly between the unadjusted and Bonferroni values at every k and ν. A missing control, a single-value group and a design with no within-group variation all return no result. That engine is verified on every change against 96 assertions. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
Tukey HSDEvery pairwise comparison after an ANOVA with simultaneous confidence intervals — and the studentized range integrated rather than interpolated from a table, so any group count, df and level works. At two groups the critical q is exactly √2 times the critical t.
Games-Howell TestEvery pair with its own standard error and Welch degrees of freedom, plus the measured cost of pooling: with all means equal Tukey fires 29.92% of the time in one variance pattern and 3.00% in another.
Bonferroni CorrectionAdjusts p-values by Bonferroni, Holm, Šidák, Holm-Šidák and Benjamini-Hochberg at once — and shows that Holm controls exactly what Bonferroni controls while never rejecting fewer, which makes plain Bonferroni dominated.
One-Way ANOVAThe full F table with eta and omega squared, plus every pairwise gap — because a significant F says something differs and never says which, and ten groups tested pairwise carry a 90% false-positive rate.
Levene's TestComputes both centre conventions side by side and shows the median is what makes it robust: on skewed data the mean-centred original rejects 15.00% of true nulls against the median version's 4.47%.
t-testOne-sample, two-sample and paired t-tests defaulting to Welch, with Student's pooled version printed beside it — and a warning when the two disagree on the verdict.
An educational tool. The critical value assumes equal group sizes (where the correlation between comparisons is exactly one half) and a common variance across all groups including the control, and the entire error-rate guarantee depends on the control having been pre-specified rather than chosen after seeing the data.
Published a many-against-one-control procedure whose critical value is computed rather than looked up: a two-dimensional numerical integration of the equicorrelated multivariate t at rho = 1/2, which is exactly the correlation between two treatment-vs-control comparisons at equal n.
Verified the integration against an identity rather than a table. At one treatment there is nothing to adjust for, so the answer must be the ordinary two-sided t critical value — and it is, to 3.55e-15 on the shipped example and to better than 2.5e-9 across degrees of freedom from 5 to 240.
Reproduced the published two-sided Dunnett table as a by-product: 2.086, 2.379, 2.540, 2.651, 2.735 and 2.946 at 20 error degrees of freedom for k = 1 to 5 and 9.
Printed the unadjusted t and the Bonferroni value beside it, which quantifies what Bonferroni's independence assumption costs here: 1.83% more required separation at two treatments, rising to 5.17% at nine, because every comparison shares the same control and they are therefore correlated.
Made the whole thing fast enough to be a page rather than a batch job — the quadrature weights are built once per degrees of freedom and the root is found by the Illinois method rather than bisection, which took one critical value from 3.4 seconds to under 200 milliseconds.
Add this calculator to your site
Responsive embed — and private: nothing your visitors type leaves their browser.