# On the Correct Interpretation of Confidence Intervals

“What is the correct interpretation of a 95% confidence interval (CI)?”

This was a question posed to students by a professor in an upper-level graduate statistics course. As expected (or perhaps surprisingly), some of the answers that students gave were not quite the ones he was looking for.

For example, let’s say we want to know the true average number of dates that Master’s students at Columbia University have been on in the past year. (Of course, we’d have to define what we mean by “date” but I’m sure there are tons of blog posts on the subject of “Does this count as a date?”)

Since it would be impractical to grab every graduate student at Columbia and ask them to (truthfully) report the number of dates they’ve been on, we can instead take a random sample of students and try to infer from this sample what the true mean is.

So, suppose we got a random sample of 100 Columbia graduate students and found that the average number of dates they went on in the past year was 7.6. The calculated 95% confidence interval was (4.4, 12.0). What is the correct interpretation of this confidence interval?

Let’s start with the incorrect interpretation that one might be tempted to give, which is that there is a 95% chance that the true mean is within this interval. Again, this is not the right interpretation. The true mean is either within or outside the confidence interval. The chances are 0% or 100%.

What the 95% CI does mean is that, if we repeatedly sampled 100 students from Columbia, and found the means and calculated the corresponding confidence intervals, we would expect the true mean to be within these CIs 95% of the time.

I’m aware that there are many sources out there explaining the answer to this question (“what is the correct interpretation of confidence intervals?”), but I thought it was worth repeating as it seems to come up in every statistics class. But the answers aren’t always right every time.

If you’re thinking, “My statistics professor might be satisfied with the ‘correct’ interpretation but how do I explain confidence intervals to people without any background in statistics?”

I’ll be addressing that in a future post, so check back soon!