How to interpret a 95% confidence interval: A guide for analysts

Ever scratched your head over what a 95% confidence interval really means? You're not alone. Confidence intervals can seem like statistical magic, offering a range where the true value supposedly lies, but they're actually grounded in solid math. Understanding them can be a game-changer when interpreting experimental results or making data-driven decisions.

In this blog, we'll break down what confidence intervals are all about, how to construct them, and how to interpret them correctly. We'll also dive into how they apply in data analysis and A/B testing—a key area where tools like Statsig can help you make sense of your experiments. Ready to demystify confidence intervals? Let's get started!

Understanding confidence intervals

Confidence intervals are more than just fancy ranges in statistics—they're your best guess (with some wiggle room) about where the true population parameter lies. Instead of pinning all your hopes on a single point estimate, confidence intervals give you a range that likely contains the true value, giving you a clearer picture of precision and uncertainty.

So, how do you construct a 95% confidence interval? It's not as daunting as it sounds. You calculate the interval using the standard error of the mean and a critical value from the standard normal or t-distribution. The size of your interval depends on your sample size and variability. Bigger samples usually mean narrower intervals—more data gives you a better estimate—while more variability (think inconsistent data) leads to wider intervals.

But what does a 95% confidence interval really tell us? It's crucial to get this right. It doesn't mean there's a 95% chance the true parameter is in that interval for your specific sample. Instead, it means that if you repeated your experiment over and over, 95% of those confidence intervals would contain the true parameter. It's a subtle but important distinction that helps avoid common misconceptions.

In practice, confidence intervals are super handy for decision-making because they indicate how reliable your observed effects are. For example, in A/B testing (which you might be doing with Statsig!), if the interval for the difference in conversion rates between two variants doesn't include zero and is entirely above zero, you can be pretty confident there's a statistically significant improvement.

Constructing a 95% confidence interval

Calculating a 95% confidence interval for a mean isn't as intimidating as it might seem. Here's the step-by-step:

1. Find your sample mean and standard deviation.

2. Compute the standard error, which is the standard deviation divided by the square root of your sample size.

3. Determine the appropriate z-score or t-score based on your confidence level and sample size.

The standard error gives you an idea of how much your sample mean is expected to vary, and the z-scores and t-scores (critical values) adjust for how confident you want to be. Typically, you'll use z-scores for large samples (n > 30) or when you know the population standard deviation. If your sample is smaller or the population standard deviation is unknown (more on that here), t-scores are your go-to.

Remember, the confidence level you choose affects the width of your interval. Want to be more confident? Say, 99% confident instead of 95%? Your interval will get wider. That's because you're casting a wider net to be more certain. On the flip side, a lower confidence level like 90% gives you a narrower interval. It's all about balancing certainty and precision when you're interpreting confidence intervals.

Correctly interpreting a 95% confidence interval

It's easy to misunderstand what a 95% confidence interval actually means. Many people think it represents the probability that the true parameter lies within that interval—but that's not quite right, at least in the frequentist sense. In frequentist statistics, the true parameter is seen as fixed, and it's the interval that varies from sample to sample.

So what's the right way to interpret it? A 95% confidence interval means that if you were to repeat your sampling process over and over, about 95% of those intervals would contain the true parameter. This is called the long-run frequency interpretation. It's all about the performance of the method over repeated experiments, not the probability for a single interval.

When you're interpreting 95% confidence intervals, keep these points in mind:

• The interval gives you a range of plausible values for the parameter—it's not a definitive answer.

• Narrow intervals mean your estimate is more precise, while wide intervals indicate more uncertainty.

• If the interval includes the null value (like zero for a difference), your results might be inconclusive.

So, remember: confidence intervals are powerful tools for quantifying uncertainty, but they don't tell the whole story on their own. They should be used alongside other statistical measures to get a full understanding of your data. By grasping the correct interpretation of 95% confidence intervals, you can make smarter decisions based on your experimental results.

Applying confidence intervals in data analysis

Confidence intervals are super important when you're doing A/B testing or running experiments. They help you quantify the uncertainty around your estimates, which in turn helps you make informed decisions. For example, if your experiment shows a 5% increase in conversion rates with a 95% confidence interval of [2%, 8%], you can feel confident that the true effect is positive.

But confidence intervals are just one piece of the puzzle. When you combine them with p-values and effect sizes, you get a more complete picture of your results. P-values tell you about statistical significance, while effect sizes measure practical importance. Sometimes, a statistically significant result with a tiny effect size might not be worth acting on. Other times, a non-significant result with a large confidence interval might suggest you need more data.

It's also helpful to visualize confidence intervals to communicate your results effectively. Plotting your estimates with their intervals lets stakeholders quickly understand the uncertainty involved—check out this example. Tools like Statsig can make this process smoother, helping you interpret and share your experimental results.

Don't forget that explaining these statistical concepts to others can improve your own understanding. Writing about your analyses, maybe even starting a blog, is a great way to practice. You can get feedback from the community and deepen your grasp of topics like confidence intervals (here's how you can start blogging).

By applying confidence intervals in your data analysis, you're making your decisions more robust and data-driven. They offer a clear picture of the uncertainty around your estimates, helping you avoid over- or under-interpreting your results. Combining confidence intervals with other statistical measures and communicating effectively will take your experiments to the next level.

Closing thoughts

Confidence intervals are a powerful tool in your statistical toolbox—they help you understand the uncertainty around your estimates and make better decisions based on your data. By correctly interpreting them and applying them thoughtfully in your analyses, you can gain deeper insights from your experiments and A/B tests.

If you're looking to dive deeper, there are plenty of resources available to expand your understanding. And remember, tools like Statsig can assist you in making sense of confidence intervals and other statistical measures in your experiments.

Hope you found this helpful!