Bonferroni Correction Explained: Managing Multiple Testing in Statistics

Imagine you’re running 20 different A/B tests on various elements of your product at the same time.

For each test, you use the standard 0.05 level for statistical significance. That means there’s a 5% chance of a false positive result on any single test, just by random chance.

Here’s where it gets more challenging—those 5% chances add up quickly when you have multiple comparisons. Across these 20 tests, the probability of getting at least one false positive is 1 - (0.95)^20 = 0.64, or 64%. That’s usually too high for most teams to trust their instincts.

The Bonferroni method adjusts the statistical significance level based on the number of tests performed. Instead of 0.05, the required significance level gets divided by the number of comparisons. For those 20 tests, the Bonferroni-adjusted significance level would be 0.05 / 20 = 0.0025 or 0.25%.

Making the threshold stricter reduces the chances of false positives slipping through, meaning you can trust the “wins” that get flagged.

Simply put, applying the Bonferroni method makes you less likely to get thrown off course during testing by random noise in the data.

False positives are hard to avoid when you have several A/B tests going on simultaneously—this is known as the multiple comparisons problem.

The problem can lead your team astray if you accidentally read too much into a “good” variation that simply caught a lucky break. For instance, you could end up rolling out an update that doesn’t improve your product or even worsens it.

Unfortunately, this is the situation many data-led businesses find themselves in. Every element of their website or app can and should be tested and optimized, meaning they often run a number of comparisons.

Using the Bonferroni test helps you reap the benefits of your multivariate data while avoiding false positives. It’s a relatively straightforward statistical technique that many teams use to overcome the multiple comparisons problem.

Like any statistical technique, the Bonferroni method has its pros and cons. You need to consider these when deciding when and how to implement it.

Let’s go through the main ones.

Pros

• Simple and intuitive: It’s one of the most straightforward ways to control the familywise error rate (FWER)—this is the probability of getting at least one false positive when multiple comparisons are tested. The formula is easy to understand and apply.

• Reduces false positives: By lowering the threshold for significance, Bonferroni gives you higher confidence that statistically significant results aren’t just due to chance.

• No extra testing required: Unlike other approaches, Bonferroni doesn’t need additional data collection or testing rounds. It works with your existing data.

Cons

• Potential power loss: Bonferroni corrections make it harder to get statistically significant results since the thresholds are stricter. This gives your test a lower power, meaning you’re more likely to miss real effects.

• Equal adjustments: The correction applies the same significance level adjustments across all comparisons, even if some are more important than others.

Over-correction issues: In cases with many comparisons or correlations between tests, Bonferroni can be too conservative, missing real effects to an unnecessary degree.

The Bonferroni method works best when running many tests or comparisons simultaneously on the same data sets.

Take experimentation platforms—these have dozens of A/B tests that evaluate different variations of pages, funnels, calls to action, and more across your website or app.

In these scenarios, the risk of false positives across all those tests rises if you simply use the standard 0.05 significance threshold everywhere.

The Bonferroni correction helps rein in those sky-high error rates by changing the significance level based on the number of comparisons made.

It’s also a good technique if your team isn’t as experienced with statistics. The adjustment formula is easy to grasp and takes the guesswork out of controlling familywise error rates. More complex corrections can be confusing to learn.

On the other hand, if you’re only running a handful of comparisons on entirely separate data sets, the costs of Bonferroni’s overcorrection may outweigh the benefits. You don’t want to make the significance thresholds so strict that you miss out on real, actionable insights.

It comes down to balancing your risk tolerances—how much are you willing to bet on a false positive taking you off course versus potentially overlooking a genuine effect?

For most businesses, Bonferroni is a solid middle-ground solution. It gives them more confidence in their A/B testing results without overhauling data workflows.

Let’s say you’re running an experiment to optimize the checkout flow for your SaaS product.

You test eight different variations, including:

1. Original checkout
2. Add progress tracker
3. Remove unnecessary fields
4. Bigger “Purchase” button
5. Add testimonial quotes
6. Summarize pricing details
7. Combined #3 and #4
8. Combined #2, #5, and #6

You’ve got the conversion rate data for each variation. Using the standard p < 0.05 significance level, variations #3, #4, and #7 were flagged as statistically significant wins compared to the original flow.

However, with eight total comparisons made, the probability of at least one false positive is around 34% if you use that 0.05 threshold. This is too high to assume those three variations genuinely improved conversions.

This is where you’d apply the Bonferroni correction. With eight tests, the significance level is adjusted to 0.05 / 8 = 0.00625. Using this much stricter 0.625% threshold, only variation #7 remains statistically significant.

By using the Bonferroni method, you’ve increased confidence that the “Remove Fields” and “Bigger Button” variation is a combination worth implementing while avoiding potential false positives on the others.

The tradeoff is that variation #3 or #4 alone may have also improved the conversion rate but weren’t statistically significant. However, it’s much better to be cautious than to mislead your entire product roadmap.

The Bonferroni test is one of the most common methods for correcting for multiple comparisons, but it’s not the only one available.

Here’s a quick look at how it stacks up against other popular techniques.

Bonferroni vs. Tukey

Tukey’s range is more specialized for comparing every possible pair of means while controlling FWER.

It’s a tad more complex than Bonferroni, but it can make sense if you deal with many pairwise comparisons between groups.

Bonferroni vs. Šidák

The Šidák correction is quite similar to Bonferroni. Still, it tends to be slightly more powerful (and more willing to reject the null hypothesis) when you have many comparisons.

The adjustment formula is also marginally different. However, for smaller test sets, the differences are negligible.

Bonferroni vs. Scheffé

Scheffé’s method controls the FWER across all possible contrasts or comparisons, not just the ones you’re explicitly testing.

This makes it even more cautious than Bonferroni, making it harder to get statistical significance. The trade-off is higher confidence in your test results but potentially overlooked true effects.

Which technique should you choose?

All these methods try to solve the same core multiple comparison issues, only with slightly different mathematical approaches and assumptions.

For most businesses running A/B tests and experimentation programs, the classic Bonferroni corrections offer a balance between simplicity and statistical power.

Unless you have extensive expertise and familiarity with the other methods, Bonferroni is a good default choice. It is straightforward to calculate and explain, making it an easy “no false positive” backstop that keeps your optimization roadmap clear and accurate.

Implementing the Bonferroni method isn’t too technically challenging, especially if you’re familiar with running A/B tests and analyzing data.

As already outlined, the core idea is to adjust the threshold for statistical significance based on the number of comparison tests made. Instead of using the 0.05 p value threshold, divide that by the total number of comparisons.

So, if you’re running 20 A/B tests simultaneously, the Bonferroni-adjusted significance level would be 0.05 / 20 = 0.0025.

Variations that don’t meet that stricter 0.25% threshold wouldn’t be considered statistically significant wins after correcting for multiple comparisons.

Most A/B testing tools allow you to specify the total number of comparisons or desired FWER. The software can then automatically apply the Bonferroni adjustment when determining statistical significance.

If you’re using your own analysis, you must manually calculate the adjusted alpha threshold and then use that when running statistical tests instead of 0.05. Remember to document that you’ve applied the Bonferroni correction.

The adjustment is straightforward enough that you could even implement it after your tests finish, like when reporting results—simply note which variations pass the stricter significance threshold.

Getting used to the Bonferroni mindset of “raising the bar for significance” can take some adjustment. However, it’s well worth it to avoid troublesome false positives that can send you down optimization rabbit holes.

Managing multiple comparisons is one of the statistical realities you must deal with as a data-driven company. The more you experiment and optimize your product, the higher the risk of false positives taking you down the wrong path.

That’s why having a solid process for correction methods like the Bonferroni adjustment is essential. It gives you confidence that when a test does get highlighted as statistically significant, it’s not random. You can release your product updates knowing you’re making genuine improvements.

Thankfully, you don’t have to build those capabilities from scratch.

Amplitude’s product experimentation and A/B testing platform has multiple comparisons corrected for right out of the box, including Bonferroni.

Amplitude Experiment enables you to run reliable A/B, multivariate, and multi-page funnel tests while automatically applying sound statistical methodology, like false discovery rate control. All you have to do is design your experiment and let Amplitude handle the analytics rigorously.

Spend less time worrying about statistics and more time innovating and creating exceptional product experiences.