Tukey's Honest Significant Difference (HSD) Test

Calculator

1. Load Your Data

Need to transform your data?

Select column for groups:

Select column for values:

Significance Level (α):

Exclude Outliers

Learn More

Tukey's HSD Test

Definition

Tukey's HSD (Honestly Significant Difference) Test is a post-hoc test used after ANOVA to determine which specific groups differ from each other. It controls the family-wise error rate while conducting multiple pairwise comparisons.

Formula

Test Statistic:

HSD = q_{\alpha,k,N-k}\sqrt{\frac{MSE}{n}}

Where:

$q_{\alpha,k,N-k}$ = studentized range statistic
$MSE$ = Mean Square Error from ANOVA, which is equivalent to Mean Square Within ( $MS_{Within}$ )
$n$ = sample size per group
$k$ = number of groups
$N$ = total sample size

Key Assumptions

Independence: Observations must be independent

Normality: Data within each group should be normally distributed

Homogeneity of Variance: Groups should have equal variances

Equal Sample Sizes: Preferably, groups should have equal sample sizes

Practical Example

Step 1: State the Data

Comparing three fertilizer treatments on plant growth (cm):

Treatment A	Treatment B	Treatment C
25	42	35
30	38	33
28	40	31
32	45	34
29	41	32

Step 2: Calculate Group Statistics

Treatment A: $\bar{x} = 28.8$ , $n = 5$
Treatment B: $\bar{x} = 41.2$ , $n = 5$
Treatment C: $\bar{x} = 33.0$ , $n = 5$
MSE (from ANOVA) = $5.3$

Step 3: Calculate HSD

For $\alpha = 0.05$ , $k = 3$ groups, $df = 12$ :

q_{0.05,3,12} = 3.77

HSD = 3.77\sqrt{\frac{5.3}{5}} = 3.884

Step 4: Compare Group Differences

$|B - A| = |41.2 - 28.8| = 12.4 > 3.884$ (significant)

$|C - A| = |33.0 - 28.8| = 4.2 > 3.884$ (significant)

$|B - C| = |41.2 - 33.0| = 8.2 > 3.884$ (significant)

Step 5: Draw Conclusions

All pairwise comparisons show significant differences at $\alpha = 0.05$ . Treatment B produced significantly higher growth than both A and C, and Treatment C produced significantly higher growth than Treatment A.

Code Examples

1library(emmeans)
2library(tidyverse)
3
4# Example data
5data <- data.frame(
6  group = rep(c("A", "B", "C"), each = 5),
7  values = c(25, 30, 28, 32, 29,  # Group A
8             42, 38, 40, 45, 41,  # Group B
9             35, 33, 31, 34, 32)  # Group C
10)
11
12data |>
13  group_by(group) |>
14  summarize(mean = mean(values), sd = sd(values), n = n())
15
16# Fit ANOVA model
17model <- aov(values ~ group, data = data)
18summary(model)
19
20q <- qtukey(0.95, nmeans = 3, df = 12)
21mse <- summary(model)[[1]]["Residuals", "Mean Sq"]
22hsd <- q * sqrt(mse / 5)
23print(hsd)
24
25# Get emmeans and perform Tukey's HSD
26emmeans_result <- emmeans(model, "group")
27pairs(emmeans_result, adjust = "tukey")

Python

1import pandas as pd
2from statsmodels.stats.multicomp import pairwise_tukeyhsd
3
4# Example data
5group1 = [25, 30, 28, 32, 29]
6group2 = [42, 38, 40, 45, 41]
7group3 = [35, 33, 31, 34, 32]
8
9# Create DataFrame
10data = pd.DataFrame({
11    'values': group1 + group2 + group3,
12    'group': ['A']*5 + ['B']*5 + ['C']*5
13})
14
15# Perform Tukey's HSD
16tukey = pairwise_tukeyhsd(
17    endog=data['values'],
18    groups=data['group'],
19    alpha=0.05
20)
21
22print(tukey)

Alternative Tests

Consider these alternatives:

Bonferroni Test: More conservative, controls family-wise error rate
Scheffé's Test: More flexible for complex comparisons
Games-Howell Test: When variances are unequal

Related Calculators

Help us improve

Found an error or have a suggestion? Let us know!