Levene's Test for Equality of Variances

Check if two datasets have equal variances using Levene's Test. Enter your data below to calculate the F-statistic and P-value.

Enter Your Data

Enter comma-separated values for the first sample.

Enter comma-separated values for the second sample.

Results

F-Statistic:

P-Value:

Data Visualization

Visualize the sample data points and their means to understand the variance distribution.

Interpretation: A higher F-statistic and a low p-value (typically p < 0.05) suggest that the variances of the two groups are significantly different.

Understanding Levene's Test

Levene's test is used to check if two or more samples have equal variances. Equal variances, or homogeneity of variances, is an assumption for many statistical tests, such as t-tests and ANOVA. Levene's test is less sensitive than the Bartlett test to departures from normality. Thus, it is a more robust test if your data is not normally distributed.

Formula

The Levene's test statistic (F) is calculated using the formula:

$$ F = \frac{\sum_{i=1}^{k} n_i (\bar{Z}_{.i} - \bar{Z}_{..})^2 / (k-1)}{\sum_{i=1}^{k} \sum_{j=1}^{n_i} (Z_{ij} - \bar{Z}_{.i})^2 / (N-k)} $$
  • \( \(Z_{ij} = |Y_{ij} - \bar{Y}_{.i}| \) for mean deviation, or \( Z_{ij} = (Y_{ij} - \tilde{Y}_{.i})^2 \) for median deviation \)
  • \( \( \bar{Y}_{.i} \) is the mean of the \( i \)-th group, and \( \tilde{Y}_{.i} \) is the median of the \( i \)-th group.
  • \( \( \bar{Z}_{.i} \) is the mean of the \( Z_{ij} \) for the \( i \)-th group.
  • \( \( \bar{Z}_{..} \) is the overall mean of all \( Z_{ij} \).
  • \( \( k \) is the number of groups.
  • \( \( n_i \) is the number of samples in the \( i \)-th group.
  • \( \( N \) is the total number of samples.

Interpretation of P-value

A small p-value (typically ≤ 0.05) suggests that there is significant evidence to reject the null hypothesis that the variances of all groups are equal. In other words, the variances are significantly different. Conversely, a large p-value (> 0.05) indicates that there is not enough evidence to reject the null hypothesis, suggesting the variances are not significantly different.

Use Cases

  • Comparing the variability of product quality from different manufacturing processes.
  • Assessing whether different teaching methods lead to the same level of variation in student performance.
  • Verifying assumptions for t-tests and ANOVA in statistical research.

Source: Wikipedia - Levene's test