Original Alpha (α):
Number of Comparisons:
Bonferroni Corrected Alpha (α'):
Note: After applying the Bonferroni correction, all statistical tests must have p-values below to be considered significant.
Enter comma-separated p-values below to check if they meet the corrected significance threshold:
The Bonferroni correction is a statistical adjustment method used when multiple hypothesis tests are being performed at the same time. In traditional statistical testing, we often use a significance level (commonly denoted as alpha or α) of 0.05, meaning there is a 5% chance of incorrectly rejecting a true null hypothesis (Type I error). However, when several tests are conducted simultaneously, the probability of encountering at least one false positive increases substantially.
To address this, the Bonferroni correction adjusts the alpha level by dividing it by the number of tests or comparisons being made. The result is a more stringent threshold for statistical significance in each individual test. For example, if your original alpha is 0.05 and you're conducting 10 tests, the corrected alpha for each test would be 0.005 (0.05 ÷ 10). This means only p-values less than 0.005 would be considered statistically significant, thereby reducing the chance of false positives across all tests.
Although this method is very effective in reducing Type I errors, it can also increase the likelihood of Type II errors (failing to detect a true effect). As a result, while the Bonferroni correction offers a conservative approach to multiple comparisons, it may reduce statistical power in some situations. Despite this, it remains a widely accepted method due to its simplicity and clear interpretation.
Performing Bonferroni corrections manually can be time-consuming and error-prone, especially when dealing with large numbers of comparisons. This calculator provides a fast, accurate, and user-friendly way to apply the correction. It is designed for students, researchers, analysts, and anyone performing multiple hypothesis tests who needs to control the family-wise error rate.
By simply entering the original alpha level (e.g., 0.05) and the number of comparisons or tests, this calculator will:
Using this tool ensures your statistical conclusions are more reliable by accounting for the increased risk of false positives in multiple testing scenarios. It simplifies the process while maintaining transparency, making it ideal for educational, clinical, psychological, and data-driven applications where accuracy and rigor are critical.
Once you've entered your inputs and clicked "Calculate," the Bonferroni Correction Calculator will display a detailed summary of your results. Here's how to interpret each part:
The original alpha level (commonly denoted as α) is the significance threshold you set before performing your tests. This value reflects the probability of committing a Type I error (false positive) in a single test. The most common choice is 0.05, which implies a 5% chance of incorrectly rejecting a true null hypothesis. However, when multiple tests are conducted, this risk accumulates across all tests, which is why a correction is necessary.
This refers to the total number of hypothesis tests or comparisons you are making. For example, if you are comparing the effects of five different treatments, you might be performing 10 pairwise comparisons. The greater the number of comparisons, the higher the chance of obtaining at least one false-positive result. The Bonferroni method uses this number to determine how much to adjust your alpha level to control the overall error rate.
The corrected alpha level (α′) is the result of dividing your original alpha by the number of comparisons. This new threshold is used to judge the statistical significance of each individual test. For instance, if your original alpha is 0.05 and you perform 5 comparisons, your corrected alpha will be 0.01. This means each individual test must have a p-value less than or equal to 0.01 to be considered statistically significant.
Formula: α′ = α / m
Where α = original alpha and m = number of comparisons.
To make the results even easier to understand, the calculator displays a table that shows the corrected significance threshold for each test. Every test listed in the table will have the same adjusted alpha value, as the Bonferroni correction applies a uniform threshold across all tests.
This table is especially helpful when analyzing a set of p-values. You can compare each of your actual p-values against the corrected threshold to determine which results are statistically significant after applying the correction.
Understanding the logic behind the Bonferroni correction helps you apply it more effectively and interpret the results with greater confidence. This section breaks down the core formula and explains the key statistical concepts involved.
The Bonferroni correction adjusts the significance level (α) to account for multiple hypothesis tests. The formula is simple but powerful:
Corrected Alpha (α′) = α / m
Where:
This formula ensures that the overall chance of making a Type I error across all tests remains controlled at your chosen significance level.
One of the primary goals of the Bonferroni correction is to reduce Type I errors, which occur when a test incorrectly rejects a true null hypothesis. When multiple tests are conducted, the likelihood of encountering at least one Type I error increases. This is called the family-wise error rate (FWER).
By lowering the threshold for each individual test, the Bonferroni correction helps maintain the overall error rate at the desired level. For example, instead of allowing a 5% chance of error per test, it spreads that 5% across all tests, effectively lowering the risk of false positives.
While this correction is very effective at improving statistical reliability, it can also make it harder to detect real effects (increasing the chance of Type II errors). Therefore, it is most suitable when controlling false positives is a top priority.
The Bonferroni correction is a valuable tool in statistical analysis, especially when you're performing multiple comparisons. However, like any method, it has both strengths and weaknesses. Understanding these will help you decide when it's the right choice for your data.
You should consider using the Bonferroni correction in situations where:
In these scenarios, the Bonferroni correction provides a simple, transparent, and effective way to maintain confidence in your results.
Despite its strengths, the Bonferroni method also has some important limitations, particularly when the number of tests increases:
The Bonferroni correction is used to adjust the significance level (alpha) when performing multiple hypothesis tests. Its main purpose is to reduce the likelihood of Type I errors (false positives), which become more likely as the number of comparisons increases.
Simply divide your original alpha level by the number of comparisons (tests) you're conducting. The formula is:
Corrected Alpha = α / number of tests
You should use it when performing multiple statistical tests on the same dataset, especially if your goal is to control the family-wise error rate and avoid any false-positive results.
Yes, it can be. While it effectively reduces Type I errors, it can increase the risk of Type II errors (false negatives). This means you might miss detecting a real effect because the significance threshold becomes very strict, especially with a high number of comparisons.
Other correction methods include the Holm-Bonferroni method, the Benjamini-Hochberg procedure, and the Sidak correction. These methods can be less conservative and more powerful in detecting true effects while still controlling error rates.
Yes. The calculator is general-purpose and can be used with any set of hypothesis tests where you want to apply the Bonferroni correction. Just enter the total number of tests and your original alpha value.
The table shows the corrected alpha value for each individual test. Since the Bonferroni method uses the same threshold for all tests, each row will display the same value. You can compare your p-values to this threshold to determine significance.
If you have a very high number of comparisons (e.g., hundreds or thousands), the Bonferroni correction might be too strict. In that case, consider using a method that controls the false discovery rate (FDR), such as the Benjamini-Hochberg procedure, which is more appropriate for large-scale testing.