T-tests: Small sample statistical power

If you've ever tried to run an A/B test with just a handful of users, you know the sinking feeling - will your results actually mean anything? Small sample sizes are the reality for many teams, whether you're testing a feature with enterprise customers or working with limited traffic.

The good news is that t-tests, those statistical workhorses we all learned about in Stats 101, can still work with small samples. But there's a catch - you need to know when they'll fail you and what to do about it.

The basics of t-tests and small sample sizes

Let's start with a quick refresher. T-tests come in three flavors: one-sample (comparing your data to a known value), independent two-sample (comparing two separate groups), and paired (comparing the same group before and after). Pick the wrong one and you're already off to a bad start.

Here's the thing about small samples - when you have fewer than 30 data points, t-tests get finicky. They need your data to be roughly normal (that bell curve shape) and have similar variances between groups. Miss these assumptions and you might as well flip a coin.

But don't write them off just yet. T-tests can absolutely work for small samples if your data plays nice. The real challenge is statistical power - your ability to detect real differences when they exist. Think of it like trying to hear someone whisper in a noisy room. With small samples, you need a pretty loud whisper (big effect) to hear anything at all.

The folks at Statsig have built calculators that help you figure out exactly how loud that whisper needs to be. It's worth running the numbers before you start - nothing worse than finishing an experiment only to realize you never had a chance of detecting anything meaningful.

Challenges of using t-tests with small samples

Small samples create a perfect storm of statistical headaches. The biggest problem? You'll miss real effects that are actually there (what statisticians call Type II errors). It's like having a metal detector that only beeps for massive gold nuggets while ignoring all the coins.

P-values become especially tricky with small samples. That magical 0.05 threshold everyone obsesses over? It means a lot less when you're working with 15 data points instead of 1,500. Your confidence intervals will be so wide you could drive a truck through them.

The real danger comes when people misuse t-tests on obviously non-normal data. I've seen teams try to analyze conversion rates that are clearly skewed or have outliers that dominate the results. Just because you can run a t-test doesn't mean you should.

David Robinson's work on interpreting p-value histograms shows how messy this can get. When you run multiple tests on small samples, weird patterns emerge that signal something's wrong. It's a diagnostic tool more teams should use.

Strategies to improve statistical power in small-sample t-tests

When you're stuck with small samples, you need to squeeze every bit of signal from your data. Variance reduction is your best friend here - anything that makes your data less noisy helps you detect smaller effects.

One approach that's gained traction is CUPED (Controlled Experiments Using Pre-Existing Data). Instead of just comparing post-treatment values, CUPED uses historical data to adjust for pre-existing differences. It's like handicapping in golf - you account for each player's skill level before comparing scores.

Here's what actually moves the needle:

• Pick better metrics: Ditch those noisy, indirect measures. If you're testing a checkout flow, measure completion rate, not time on page

• Intensify your treatment: Make your changes bigger and bolder. Subtle tweaks won't show up in small samples

• Use within-subject designs: Test the same users before and after rather than comparing different groups

• Target responsive segments: Focus on users most likely to be affected by your change

The team at Statsig has documented several variance reduction techniques that can double or triple your effective sample size. It's like getting more data without actually collecting more data.

Alternative methods and advanced techniques for small sample analysis

Sometimes you need to abandon t-tests altogether. The t_α-test is specifically built for tiny samples - it's more conservative about claiming significance, which reduces false positives when you're data-poor.

Bayesian methods offer another escape route. Unlike traditional statistics that pretend you know nothing, Bayesian approaches let you incorporate what you already know. If you're testing a feature similar to something you've tried before, why throw away that knowledge? The examples in David Robinson's Bayesian A/B testing guide show how this works in practice.

Non-parametric tests like the Mann-Whitney U can handle weird distributions that would make t-tests choke. But here's the catch - they're less powerful, so you're trading robustness for sensitivity. It's like wearing boxing gloves instead of brass knuckles; safer but less effective.

The most underrated strategy? Better experimental design from the start. Too many teams treat power analysis as an afterthought. Run the numbers first:

1. Estimate your expected effect size (be realistic)

2. Calculate required sample size for 80% power

3. If it's too high, either find ways to reduce variance or accept you can't test this

Advanced techniques like stratification and blocking can also help, but they require planning. You can't retrofit good design after collecting bad data.

Closing thoughts

Working with small samples isn't ideal, but it's often reality. T-tests can still be valuable tools if you understand their limitations and plan accordingly. The key is being honest about what your data can and can't tell you.

Remember - statistical significance isn't everything. A well-designed small study that finds suggestive evidence is better than a poorly designed large study that finds "significant" noise.

For those diving deeper, check out Statsig's guides on avoiding false positives and negatives and experiment design. And if you're consistently struggling with small samples, it might be time to rethink your testing strategy entirely. Sometimes the best statistical technique is patience - waiting until you have enough data to test properly.

Hope you find this useful!