Run the Test
Use raw observations or enter summary sample sizes and variances.
Example Data Table
| Observation | Group A | Group B | Group C |
|---|---|---|---|
| 1 | 10 | 8 | 14 |
| 2 | 12 | 9 | 15 |
| 3 | 9 | 12 | 13 |
| 4 | 11 | 10 | 16 |
| 5 | 13 | 11 | 15 |
| Group | n | Mean | Variance | Standard Deviation |
|---|---|---|---|---|
| Group A | 5 | 11.0 | 2.5 | 1.5811 |
| Group B | 5 | 10.0 | 2.5 | 1.5811 |
| Group C | 5 | 14.6 | 1.3 | 1.1402 |
Formula Used
For k groups with sample sizes n₁, n₂, ..., nₖ and sample variances s₁², s₂², ..., sₖ²:
1) Pooled variance
s_p² = [ Σ (nᵢ - 1)sᵢ² ] / (N - k)
2) Bartlett test statistic before correction
A = (N - k) ln(s_p²) - Σ (nᵢ - 1) ln(sᵢ²)
3) Correction factor
C = 1 + { 1 / [3(k - 1)] } × [ Σ 1/(nᵢ - 1) - 1/(N - k) ]
4) Corrected Bartlett chi-square statistic
χ² = A / C
5) Degrees of freedom
df = k - 1
6) P-value
p = P(Chi-square(df) ≥ observed χ²)
What the formula does: Bartlett’s test compares each group variance against a pooled variance estimate. If the variances differ enough, the test statistic becomes large and the p-value becomes small.
Important note: The method is powerful under normality, but it is sensitive to departures from normal distributions. For skewed data or outliers, a more robust alternative can be preferable.
How to Use This Calculator
- Choose Raw sample data if you have actual observations for every group.
- Choose Summary statistics if you only know each group’s sample size and sample variance.
- Enter a significance level such as 0.05.
- Add or remove groups as needed.
- Click Calculate Bartlett Test.
- Read the chi-square statistic, p-value, critical value, and final decision.
- Use the variance plot to compare group spread visually.
- Download the output as CSV or PDF for reporting.
FAQs
1. What does Bartlett’s test check?
It checks whether multiple groups have statistically equal variances. This is often used before ANOVA, pooled variance methods, and other procedures that assume homogeneity of variance.
2. When should I use raw data mode?
Use raw mode when you have the original observations for each group. The calculator will compute sample size, mean, variance, and standard deviation automatically before running the test.
3. When should I use summary mode?
Use summary mode when you only know each group’s sample size and sample variance. It is useful for reports, published studies, or condensed lab summaries.
4. What does a small p-value mean?
A small p-value suggests the variances are unlikely to be equal under the null hypothesis. At your chosen alpha level, that usually means rejecting variance homogeneity.
5. Why can Bartlett’s test be misleading sometimes?
It is sensitive to non-normal data. Strong skewness, heavy tails, or outliers can trigger significant results even when variance differences are not the main issue.
6. What is the correction factor doing?
The correction factor adjusts the test statistic so the chi-square approximation works better, especially when sample sizes are not very large or differ across groups.
7. Can I use this before ANOVA?
Yes. Bartlett’s test is commonly used to evaluate the equal variance assumption before classical ANOVA, provided the group distributions are reasonably close to normal.
8. What should I do if variances are unequal?
Consider methods that do not assume equal variances, such as Welch’s ANOVA, robust procedures, data transformation, or a nonparametric alternative when justified by the study design.