Skip to content

Bonferroni Adjustment

Bonferroni Adjustment:

Bonferroni adjustment is used in multiple comparison procedures to calculate an adjusted probability a of comparison-wise type I error from the desired probability aFW0 of family-wise type I error. The calculation guarantees that the use of the adjusted a in pairwise comparisons keeps the actual probability aFW of family-wise type I errors not higher than the desired level, as specified by the significance level of the test.

Bonferroni, an Italian mathematician, proved the following inequality:


1 (1 a)C £ C a; 0 £ a £ 1

for any value C. Since aFW and a are linked by the formula


aFW = 1 (1 a)C,

(see Family-wise Type I Error), where C is the total number of pairwise comparisons, the Bonferroni inequality gives the following approximate formula


a =  aFW0



Suppose 4 populations are to be compared, and the maximum allowed family-wise type I error is aFW0 = 0.05. Then, the total number of pairs C=4(4-1)/2=6, and Bonferroni adjustment gives a = 0.05/6 ? 0.0083. Now we may be sure that, if all the 4 populations have the same mean, and we test pairwise differences at significance level a = 0.0083, the probability aFW that you will erroneously conclude that the population means in at least one pair differ is not higher than 0.05.


If the symbols do not display properly, try
the graphic version of this page

Browse Other Glossary Entries


Test Yourself

Planning on taking an introductory statistics course, but not sure if you need to start at the beginning? Review the course description for each of our introductory statistics courses and estimate which best matches your level, then take the self test for that course. If you get all or almost all the questions correct, move on and take the next test.

Data Analytics

Considering becoming adata scientist, customer analyst or our data science certificate program?

Analytics Quiz

Advanced Statistics Quiz

Statistics Quiz


Looking at statistics for graduate programs or to enhance your foundational knowledge?

Statistics 1 Quiz

Regression Quiz

Regression Quiz


Entering the biostatistics field? Test your skill here.

Biostatistics Quiz

Advanced Statistics Quiz

Statistics 2 Quiz

Stay Informed

Our Blog

Read up on our latest blogs


Learn about our certificate programs


Find the right course for you

Contact Us

We'd love to answer your questions

Our mentors and academic advisors are standing by to help guide you towards the courses or program that makes the most sense for you and your goals.

300 W Main St STE 301, Charlottesville, VA 22903

(434) 973-7673

By submitting your information, you agree to receive email communications from All information submitted is subject to our privacy policy. You may opt out of receiving communications at any time.